**Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement**

Georgios A. Tsengenes and Georgios A. Adamidis

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48394

## **1. Introduction**

518 Fuzzy Controllers – Recent Advances in Theory and Applications

965-970

(4), pp. 563–577

Wesley Longman

Chown, G.A. and Hartman, R.C. (1998). *Design and experiment with a fuzzy controller for automatic generation control (AGC)*, *IEEE Trans. Power Systems,* (Aug. 1998), Vol. 13, pp.

Fosha, C.E. and Elgerd, O.I. (1970). T*he megawatt frequency control problem: a new approach via optimal control theory*, IEEE *Trans. Power Apparatus and Systems*, (April 1970), Vol. PAS-89

Passino, K.M. and Yourkovich, S. (1998), *Fuzzy Control,* Menlo Park, California: Addison

In recent years, the increasing use of power electronics in the commercial and industry processes results in harmonics injection and lower power factor to the electric power system [Emanuel A E. (2004)]. Conventionally, in order to overcome these problems, passive R-L-C filters have been used. The use of this kind of filters has several disadvantages. Recently, due to the evolution in modern power electronics, new device called "shunt active power filter (SAPF)" was investigated and recognized as a viable alternative to the passive filters. The principle operation of the SAPF is the generation of the appropriate current harmonics required by the non-linear load.

For the reference currents generation, one of the best known and effective technique is the 'instantaneous reactive power theory' or '*p-q* theory' [Czarnecki L S. (2006)]. In the literature, various modifications of *p-q* theory for the reference currents generation have been proposed [Salmeron P, Herrera R S, Vazquez J R. (2007)], [Kilic T, Milun S, et al. (2007)]. It is a common phenomenon in an electric power system, the grid voltages to be non-ideal [Segui-Chilet S S, Gimeno-Sales F J, et al. (2007)]. In such condition, the *p-q* theory is ineffective. To improve the efficiency of the *p-q* theory various reference currents generation techniques [Kale M, Ozdemir E. (2005)], [Tsengenes G, Adamidis G. (2011)] have been proposed. Except from the reference currents generation method, the current control method plays an important role to the overall system's performance. Plenty of methods have been used in the current control loop [Buso S, Malesani L, et al. (1998)] (e.g. ramp comparison, space vector modulation), one of which is the hysteresis current control. Hysteresis current controller compared to other current control methods has a lot of advantages such as robustness and simplicity [Tsengenes G, Adamidis G. (2010)].

© 2012 Tsengenes and Adamidis, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The conventional reference currents generation techniques use a PI controller in order to regulate the dc bus voltage. The tuning of the PI controller requires precise linear mathematical model of the plant, which is very difficult to obtained, and it fails to perform satisfactorily under parameters variations, non-linearities, etc. To overcome these disadvantages, in recent years controllers which use artificial intelligent techniques have been implemented, like fuzzy logic controllers (FLCs) and artificial neural network (ANN) [Saad A, Zellouma L. (2009)], [El-Kholy E E, El-Sabbe, et al. (2006)], [Han ., Khan M M, Yao et al. (2010)], [Skretas S B, Papadopoulos D P (2009)]. The FLC surpasses the conventional PI controller due to its ability to handle non-imparities, its superior perform with a nonaccurate mathematical model of the systems, and its robustness. In the literature some papers which implement FLCs in SAPF in order to improve the efficiency of the reference currents generation technique [Han Y, Khan M M, Yao et al. (2010)], [Jain S K, Agrawal P, et al. (2002)] and the current control loop have been reported [Mekri F, Machmoum M. (2010)], [Lin B R, Hoft R G. (1996)].

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 521

per phase as shown in figure 2. These virtual voltages will have the same amplitude as the fundamental harmonic (50 Hz) of the grid voltage, and will be synchronized with zero

**Figure 1.** Synoptic diagram of the proposed electric power system and the control system

Mathematical equations for the virtual grid voltages *a-b-c* reference frame are given by

2 120 *\* o*

The modified *p-q* theory, for the reference currents generation, will use the virtual grid

*sc u* and not the actual grid voltage. The load currents and the virtual grid

Where *Vs* is the root-mean-square (rms) value of the grid voltage ( <sup>222</sup>

and *θ* is the angular frequency of the grid voltages (θ=2∙*π*∙*fgrid*=2∙*π*∙50).

*sa s u V sin( θ)* (1)

<sup>2</sup> <sup>120</sup> *\* <sup>ο</sup> sb s u V sin( <sup>θ</sup> )* (2)

*sc s u V sin( θ )* (3)

*s sa sb sc V uuu* ),

2 *\**

phase shifting compared with the corresponding grid voltages.

**Figure 2.** Generation of three virtual grid voltages

equations (1), (2) and (3).

voltages *\**

*sa <sup>u</sup>* , *\* sb <sup>u</sup>* , *\**

In this chapter a fuzzy logic controlled SAPF for current harmonics elimination is presented. The control scheme is based on two FLCs, the first one controls the dc bus voltage and the second one controls the output current of the inverter. Furthermore for the reference currents generation a modified version of the *p-q* theory is proposed, in order to improve the performance of the SAPF under non-ideal grid voltages. The performance of the proposed control scheme is evaluated through computer simulations using the software Matlab/Simulink under steady state and transient response. The superiority of the proposed fuzzy logic control scheme over the conventional control scheme is established both in steady state and transient response for current harmonics elimination and dc bus voltage of the SAPF respectively.

At the end a proposal for future investigation is presented. A combination between the fuzzy and the PI control is proposed. The new controller is called "fuzzy-tuned PI controller". The theoretical analysis and some simulation results are illustrated in order to verify the efficiency of the fuzzy-tuned PI controller.

## **2. Description of the proposed fuzzy control scheme**

The main function of the SAPF is the current harmonics elimination and the reactive power compensation of the load. The general block diagram of a grid connected SAPF, as well as the detailed model of the control scheme is illustrated in Fig.1. The Reference currents generation method includes the dc bus voltage control which is the outer control loop. The current control method is the internal control loop which generates the appropriate switching pattern.

## **2.1. Reference currents generation method**

For the reference currents generation a modified version of the *p-q* theory is used. One of the disadvantages of the *p-q* theory is the very poor efficiency of the method under non-ideal grid voltages. In this chapter the generation of three virtual grid voltages is proposed, one per phase as shown in figure 2. These virtual voltages will have the same amplitude as the fundamental harmonic (50 Hz) of the grid voltage, and will be synchronized with zero phase shifting compared with the corresponding grid voltages.

**Figure 1.** Synoptic diagram of the proposed electric power system and the control system

**Figure 2.** Generation of three virtual grid voltages

520 Fuzzy Controllers – Recent Advances in Theory and Applications

[Lin B R, Hoft R G. (1996)].

the SAPF respectively.

verify the efficiency of the fuzzy-tuned PI controller.

**2.1. Reference currents generation method** 

**2. Description of the proposed fuzzy control scheme** 

The conventional reference currents generation techniques use a PI controller in order to regulate the dc bus voltage. The tuning of the PI controller requires precise linear mathematical model of the plant, which is very difficult to obtained, and it fails to perform satisfactorily under parameters variations, non-linearities, etc. To overcome these disadvantages, in recent years controllers which use artificial intelligent techniques have been implemented, like fuzzy logic controllers (FLCs) and artificial neural network (ANN) [Saad A, Zellouma L. (2009)], [El-Kholy E E, El-Sabbe, et al. (2006)], [Han ., Khan M M, Yao et al. (2010)], [Skretas S B, Papadopoulos D P (2009)]. The FLC surpasses the conventional PI controller due to its ability to handle non-imparities, its superior perform with a nonaccurate mathematical model of the systems, and its robustness. In the literature some papers which implement FLCs in SAPF in order to improve the efficiency of the reference currents generation technique [Han Y, Khan M M, Yao et al. (2010)], [Jain S K, Agrawal P, et al. (2002)] and the current control loop have been reported [Mekri F, Machmoum M. (2010)],

In this chapter a fuzzy logic controlled SAPF for current harmonics elimination is presented. The control scheme is based on two FLCs, the first one controls the dc bus voltage and the second one controls the output current of the inverter. Furthermore for the reference currents generation a modified version of the *p-q* theory is proposed, in order to improve the performance of the SAPF under non-ideal grid voltages. The performance of the proposed control scheme is evaluated through computer simulations using the software Matlab/Simulink under steady state and transient response. The superiority of the proposed fuzzy logic control scheme over the conventional control scheme is established both in steady state and transient response for current harmonics elimination and dc bus voltage of

At the end a proposal for future investigation is presented. A combination between the fuzzy and the PI control is proposed. The new controller is called "fuzzy-tuned PI controller". The theoretical analysis and some simulation results are illustrated in order to

The main function of the SAPF is the current harmonics elimination and the reactive power compensation of the load. The general block diagram of a grid connected SAPF, as well as the detailed model of the control scheme is illustrated in Fig.1. The Reference currents generation method includes the dc bus voltage control which is the outer control loop. The current control

For the reference currents generation a modified version of the *p-q* theory is used. One of the disadvantages of the *p-q* theory is the very poor efficiency of the method under non-ideal grid voltages. In this chapter the generation of three virtual grid voltages is proposed, one

method is the internal control loop which generates the appropriate switching pattern.

Mathematical equations for the virtual grid voltages *a-b-c* reference frame are given by equations (1), (2) and (3).

$$
\mu\_{sa}^\* = \sqrt{2} \cdot V\_s \cdot \sin(\Theta) \tag{1}
$$

$$
\mu\_{sb}^\* = \sqrt{2} \cdot V\_s \cdot \sin(\theta - 120^\circ) \tag{2}
$$

$$
\mu\_{\rm sc}^\* = \sqrt{2} \cdot V\_{\rm s} \cdot \sin(\Theta + 120^o) \tag{3}
$$

Where *Vs* is the root-mean-square (rms) value of the grid voltage ( <sup>222</sup> *s sa sb sc V uuu* ), and *θ* is the angular frequency of the grid voltages (θ=2∙*π*∙*fgrid*=2∙*π*∙50).

The modified *p-q* theory, for the reference currents generation, will use the virtual grid voltages *\* sa <sup>u</sup>* , *\* sb <sup>u</sup>* , *\* sc u* and not the actual grid voltage. The load currents and the virtual grid

voltages are transformed in *α-β* reference frame according to the transformer matrix of equation (4). The virtual grid voltages and the load current in *α-β* reference frame are given by equations (5), (6).

$$
\begin{bmatrix} \mathbb{C} \\ \mathbb{C} \end{bmatrix}\_{abc \to \alpha \beta} = \sqrt{\frac{2}{3}} \cdot \begin{vmatrix} 1 & -\frac{1}{2} & \frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{vmatrix} \tag{4}
$$

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 523

1 0 21 3 3 22

*ca,ref <sup>α</sup>,ref cb,ref <sup>β</sup>,ref*

*<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup>*

*cc,ref*

**Figure 3.** General structure of the fuzzy logic controller for dc bus voltage control

the trapezoidal membership function is determined by equation (11.b).

(Positive Large). The fuzzy logic controller characteristics used in this section are:

*i*

**2.2. Fuzzy logic dc bus voltage controller** 

the input and output control signals.

trapezoidal membership functions.

Fuzzification using continuous universe of discourse.

 Implications using Mamdani's 'min' operator. Defuzzification using the 'centroid' method.

1 3 2 2

(10)

For the dc bus voltage control a FLC is implemented. Figure 3 shows the synoptic block diagram of the proposed FLC. As inputs to FLC the error between the sensed and the reference dc bus voltage ( *dc,ref dc eV V* ) and the error variation ( *e e(k) e(k )* 1 ) at *kth* sampling instant are used. The output of the fuzzy logic controller is considered as the active power losses of the inverter ( *loss p* ). The coefficients G1, G2 and G3 are used to adjust

The FLC converts the crisp variables into linguistic variables. To implement this process it uses the following seven fuzzy sets, which are: NL (Negative Large), NM (Negative Medium), NS (Negative Small), Z (Zero), PS (Positive Small), PM (Positive Medium), PL

Seven fuzzy sets for each input (e, Δe) and output (Δploss) with triangular and

Figure 4 shows the normalized triangular and trapezoidal membership functions for the input and output variables. The degree of fuzziness/membership (*μδ,tri*(*x*)) of the triangular membership function is determined by equation (11.a). The degree of fuzziness (*μδ,tra*(*x*)) of

SAPF and reduce the total harmonic distortion (THDi) of the current a FLC for the dc bus voltage control and a FLC for the current control are implemented. The current controller handles the reference and the actual currents in *a-b-c* reference frame. As a result, the inverse *α-β* transformation of equation (10) is used in order to transform the reference currents in *a-*

*b-c* reference frame.

$$
\begin{bmatrix} \boldsymbol{\mu}\_{\boldsymbol{s}\alpha}^{\bullet} \\ \boldsymbol{\mu}\_{\boldsymbol{s}\beta}^{\bullet} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\mathcal{C}} \end{bmatrix}\_{abc\to\alpha\beta} \cdot \begin{bmatrix} \boldsymbol{\mu}\_{\boldsymbol{s}\boldsymbol{b}}^{\bullet} \\ \boldsymbol{\mu}\_{\boldsymbol{s}\boldsymbol{b}}^{\bullet} \\ \boldsymbol{\mu}\_{\boldsymbol{s}\boldsymbol{c}}^{\bullet} \end{bmatrix} \tag{5}
$$

$$
\begin{bmatrix} i\_{l\alpha} \\ i\_{l\beta} \end{bmatrix} = \begin{bmatrix} \mathbf{C} \end{bmatrix}\_{abc\rightarrow \alpha\beta} \cdot \begin{bmatrix} i\_{la} \\ i\_{lb} \\ i\_{lc} \end{bmatrix} \tag{6}
$$

The instantaneous active and reactive powers of the electric power system are calculated via the following equation:

$$
\begin{bmatrix} p \\ q \end{bmatrix} = \begin{bmatrix} \boldsymbol{u}\_{s\alpha}^{\*} & \boldsymbol{u}\_{s\beta}^{\*} \\ -\boldsymbol{u}\_{s\beta}^{\*} & \boldsymbol{u}\_{s\alpha}^{\*} \end{bmatrix} \cdot \begin{bmatrix} \boldsymbol{i}\_{l\alpha} \\ \boldsymbol{i}\_{l\beta} \end{bmatrix} \tag{7}
$$

The instantaneous powers p and q are composed from a dc part ( ) and an ac part ( ) corresponding to fundamental and harmonic current respectively. Equation (8) gives the instantaneous active and reactive power respectively.

$$\begin{array}{c} p = \widetilde{p} + \overline{p} \quad \text{(a)}\\ q = \widetilde{q} + \overline{q} \quad \text{(b)} \end{array} \tag{8}$$

The ac component of the active power is extracted using a low pass filter. Using the *p-q* theory current harmonics are eliminated and the reactive power of the load is compensated. Therefore the reference currents of the SAPF in *α-β* reference frame are:

$$
\begin{bmatrix}
\dot{i}\_{c\alpha,ref} \\
\dot{i}\_{c\beta,ref}
\end{bmatrix} = \frac{1}{(\boldsymbol{u}\_{s\alpha}^{\bullet})^2 + (\boldsymbol{u}\_{s\beta}^{\bullet})^2} \cdot \begin{bmatrix}
\boldsymbol{u}\_{s\alpha}^{\bullet} & -\boldsymbol{u}\_{s\beta}^{\bullet} \\
\boldsymbol{u}\_{s\beta}^{\bullet} & \boldsymbol{u}\_{s\alpha}^{\bullet}
\end{bmatrix} \cdot \begin{bmatrix}
\stackrel{\frown}{p-p\_{loss}} \\
q
\end{bmatrix} \tag{9}
$$

Where *loss p* are related to the inverter operating losses. The grid should cover the *loss p* in order to keep the capacitor voltage constant. Conventionally *loss p* are calculated using a dc bus voltage sensor and a PI controller. In order to improve the dynamic performance of SAPF and reduce the total harmonic distortion (THDi) of the current a FLC for the dc bus voltage control and a FLC for the current control are implemented. The current controller handles the reference and the actual currents in *a-b-c* reference frame. As a result, the inverse *α-β* transformation of equation (10) is used in order to transform the reference currents in *ab-c* reference frame.

$$
\begin{bmatrix} i\_{ca,ref} \\ i\_{cb,ref} \\ i\_{cc,ref} \end{bmatrix} = \sqrt{\frac{2}{3}} \cdot \begin{bmatrix} 1 & 0 \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \cdot \begin{bmatrix} i\_{\alpha,ref} \\ i\_{\beta,ref} \end{bmatrix} \tag{10}
$$

### **2.2. Fuzzy logic dc bus voltage controller**

522 Fuzzy Controllers – Recent Advances in Theory and Applications

by equations (5), (6).

the following equation:

voltages are transformed in *α-β* reference frame according to the transformer matrix of equation (4). The virtual grid voltages and the load current in *α-β* reference frame are given

*<sup>s</sup><sup>α</sup> sb abc αβ <sup>s</sup><sup>β</sup> sc*

*la <sup>l</sup><sup>α</sup> lb abc αβ <sup>l</sup><sup>β</sup> lc <sup>i</sup> <sup>i</sup> C i <sup>i</sup>*

The instantaneous active and reactive powers of the electric power system are calculated via

*p u u i q i u u*

 

The instantaneous powers p and q are composed from a dc part ( ) and an ac part ( ) corresponding to fundamental and harmonic current respectively. Equation (8) gives the

> 

*ppp qqq* 

The ac component of the active power is extracted using a low pass filter. Using the *p-q* theory current harmonics are eliminated and the reactive power of the load is compensated.

*<sup>c</sup>α,ref* <sup>1</sup> *<sup>s</sup><sup>α</sup> <sup>s</sup><sup>β</sup> loss*

Where *loss p* are related to the inverter operating losses. The grid should cover the *loss p* in order to keep the capacitor voltage constant. Conventionally *loss p* are calculated using a dc bus voltage sensor and a PI controller. In order to improve the dynamic performance of

*i (u ) (u ) u u q*

 

Therefore the reference currents of the SAPF in *α-β* reference frame are:

2 2

*cβ,ref sα sβ sβ sα*

*i u u*

 

> *sα sβ lα <sup>l</sup><sup>β</sup> <sup>s</sup><sup>β</sup> <sup>s</sup><sup>α</sup>*

> > (a) (b)

 

*C u*

*abc αβ <sup>C</sup>* 

*u*

*u*

instantaneous active and reactive power respectively.

1 1 <sup>1</sup> <sup>2</sup> 2 2

*sb*

(5)

(6)

(7)

(8)

*p p*

(9)

*u*

*u*

*i*

(4)

<sup>3</sup> 3 3 <sup>0</sup> 2 2

> For the dc bus voltage control a FLC is implemented. Figure 3 shows the synoptic block diagram of the proposed FLC. As inputs to FLC the error between the sensed and the reference dc bus voltage ( *dc,ref dc eV V* ) and the error variation ( *e e(k) e(k )* 1 ) at *kth* sampling instant are used. The output of the fuzzy logic controller is considered as the active power losses of the inverter ( *loss p* ). The coefficients G1, G2 and G3 are used to adjust the input and output control signals.

**Figure 3.** General structure of the fuzzy logic controller for dc bus voltage control

The FLC converts the crisp variables into linguistic variables. To implement this process it uses the following seven fuzzy sets, which are: NL (Negative Large), NM (Negative Medium), NS (Negative Small), Z (Zero), PS (Positive Small), PM (Positive Medium), PL (Positive Large). The fuzzy logic controller characteristics used in this section are:


Figure 4 shows the normalized triangular and trapezoidal membership functions for the input and output variables. The degree of fuzziness/membership (*μδ,tri*(*x*)) of the triangular membership function is determined by equation (11.a). The degree of fuzziness (*μδ,tra*(*x*)) of the trapezoidal membership function is determined by equation (11.b).

$$\mu\_{\delta, trri}(\alpha) = \begin{cases} 0 & \text{x < a, x > c} \\ \frac{\chi - a}{b - a} & a \le \chi < b \\ \frac{c - \chi}{c - b} & b \le \chi \le c \end{cases} \quad \text{(a)}$$

$$\mu\_{\delta, trai}(\alpha) = \begin{cases} 0 & \text{x < a, x > d} \\ \frac{\chi - a}{b - a} & a \le \chi \le b \\ 1 & b < \chi < c \end{cases} \quad \text{(b)}$$

$$\begin{cases} \frac{d - \chi}{d - c} & c \le \chi \le d \\ \end{cases} \quad \text{c)} \le \chi \le d$$

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 525

*μCA C (x) φ μ (x),μ (x) μ ( )* (13)

(14)

If *Cμ (x)* denotes the degree of membership of the membership functions *Cμ ( )* of the

In the defuzzification procedure the centroid method with a discretized universe of discurse

*i out i*

*x μ (x )*

*out i*

*μ (x )*

1

*i*

*i out n*

*x*

understanding of the SAPF behavior [Jain S K, Agrawal P, et al. (2002)].

*n*

1

Where *xout* is crisp output value *xi* is the output crisp variable and *out i μ (x )* is the degree of membership of the output fuzzy value, and *i* is the number of output discrete elements in

In the design of the fuzzy control algorithm, the knowledge of the systems behavior is very important. This knowledge is put in the form of rules of inference. The rule table which is shown in Table 1 contains 49 rules. The elements of the rule table are obtained from an

Δe NB NM NS Z PS PM PB NB NB NB NB NB NM NS Z NM NB NB NB NM NS Z PS NS NB NB NM NS Z PS PM Z NB NM NS Z PS PM PB PS NM NS Z PS PM PB PB PM NS Z PS PM PB PB PB PB Z PS PM PB PB PB PB

One of the best known and most effective current control methods is the hysteresis band control technique. Some of its advantages are the simplicity of the construction combined with the excellent dynamic response. Apart from the significant advantages, this method

For the reduction of the THDi index, the implementation of a fuzzy logic hysteresis current controller is proposed. The synoptic diagram of fuzzy logic hysteresis controller for the

output fuzzy sets C, where *x X* , equation 13 is used.

can be expressed as:

the universe of discourse.

**Table 1.** Fuzzy control rule table.

**2.3. Fuzzy logic Hysteresis current controller** 

has some drawbacks such as the high THDi index.

e

Where *x*, *a*, *b*, *c*, and d belong to the universe of discourse (*X*).

**Figure 4.** Membership functions for a) input variable e (pu), b) input variable Δe (pu), and c) output variable Δploss (pu)

Let *Aμ (x)* and *<sup>B</sup> μ (x)* denote the degree of membership of the membership functions *Aμ ( )* and *<sup>B</sup> μ ( )* of the input fuzzy sets A and B, where *x X* . Mamdani's logic operator is described as:

$$\left[\phi\left[\mu\_A(\mathbf{x}), \mu\_\mathbf{B}(\mathbf{x})\right]\right] = \min\left[\mu\_A(\mathbf{x}), \mu\_\mathbf{B}(\mathbf{x})\right] = \mu\_A(\mathbf{x}) \land \mu\_\mathbf{B}(\mathbf{x})\tag{12}$$

If *Cμ (x)* denotes the degree of membership of the membership functions *Cμ ( )* of the output fuzzy sets C, where *x X* , equation 13 is used.

$$
\mu\_{\mathbf{C}}(\mathbf{x}) = \phi \big| \left[ \mu\_{A}(\mathbf{x}), \mu\_{\mathbf{B}}(\mathbf{x}) \right] \cdot \mu\_{\mathbf{C}}(\cdot) \tag{13}
$$

In the defuzzification procedure the centroid method with a discretized universe of discurse can be expressed as:

$$\alpha\_{out} = \frac{\sum\_{i=1}^{n} \boldsymbol{\alpha}\_i \cdot \mu\_{out}(\boldsymbol{\alpha}\_i)}{\sum\_{i=1}^{n} \mu\_{out}(\boldsymbol{\alpha}\_i)} \tag{14}$$

Where *xout* is crisp output value *xi* is the output crisp variable and *out i μ (x )* is the degree of membership of the output fuzzy value, and *i* is the number of output discrete elements in the universe of discourse.

In the design of the fuzzy control algorithm, the knowledge of the systems behavior is very important. This knowledge is put in the form of rules of inference. The rule table which is shown in Table 1 contains 49 rules. The elements of the rule table are obtained from an understanding of the SAPF behavior [Jain S K, Agrawal P, et al. (2002)].


**Table 1.** Fuzzy control rule table.

524 Fuzzy Controllers – Recent Advances in Theory and Applications

*δ,tri*

*δ,tra*

Where *x*, *a*, *b*, *c*, and d belong to the universe of discourse (*X*).

0

 

*x a <sup>μ</sup> (x) a x b b a*

(a)

*x a,x c*

*x a,x d*

*c x bxc c b*

*d x cxd d c*

(a) (b)

(b)

(11)

0

1

**Figure 4.** Membership functions for a) input variable e (pu), b) input variable Δe (pu), and c) output

(c)

Let *Aμ (x)* and *<sup>B</sup> μ (x)* denote the degree of membership of the membership functions *Aμ ( )* and *<sup>B</sup> μ ( )* of the input fuzzy sets A and B, where *x X* . Mamdani's logic operator is

*φ μA AA (x),μ (x) min μ (x),μ (x) μ (x) μ (x)*

(12)

variable Δploss (pu)

described as:

*x a axb b a <sup>μ</sup> (x) bxc*

#### **2.3. Fuzzy logic Hysteresis current controller**

One of the best known and most effective current control methods is the hysteresis band control technique. Some of its advantages are the simplicity of the construction combined with the excellent dynamic response. Apart from the significant advantages, this method has some drawbacks such as the high THDi index.

For the reduction of the THDi index, the implementation of a fuzzy logic hysteresis current controller is proposed. The synoptic diagram of fuzzy logic hysteresis controller for the

phase-a is shown in figure 5. The same controller is applied to the other two-phases (b and c). As inputs to FLC the error between the reference current and the sensed current ( *ca,ref ca ei i* ) and the error variation ( *e e(k) e(k )* 1 ) at *kth* sampling instant are used. The output of the FLC is considered as the amplitude of the current error. The coefficients F1 and F2 are used to adjust the input control signals. The saturations blocks are used for limiting the initial error.

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 527

It is a common phenomenon in electric power system the grid voltages to be non-ideal. This problem is particularly important in Greek electric power distribution system, mainly due to the large increase of power electronic devices. In this section the mathematical model of the

2 120

*<sup>o</sup> sc*

When the three-phase grid voltages are unbalanced ( *su u* ), the grid voltages can be

*sua sua sua sub sub sub suc suc suc*

Where *usua* , *sub u* , and *usuc* are positive sequence components and *usua* , *sub u* , and

It is a very common phenomenon in electric power distribution systems, voltages having non-ideal waveforms, and different levels of harmonics. When the three-phase grid voltages are distorted (*usd*), the grid voltages have harmonics components. In this scenario the

> *sda sda, f sda,h sdb sdb, f sdb,h sdc sdc, f sdc,h*

Where, *sdb, f u* , and *sdc, f u* are positive sequence components and , *sdb,h u* , and *sdc,h u* are

When the three-phase grid voltages are distorted and unbalanced (*udu*), the grid voltages contain harmonic components and unbalances. For this case, the distorted and unbalanced

> *sdua sda, f sda,h sua sua sda,h sua sdub sdb, f sdb,h sub sub sdb,h sub sduc sdc,h suc suc sdc,h suc sdc, f*

*u u u u uu u uu u uuu u u u u u uu u*

*u u u uu u u u u* 

*uu u uu u uu u*

expressed as positive and negative sequence components as shown in equation (12).

*u sin(ωt) u V sin(ωt ) u sin(ωt )* 

120

 

*o*

(16)

(17)

 

(18)

(15)

grid voltages under several non-ideal cases will briefly be presented.

*sa*

The ideal grid voltages have sinusoidal waveform and can be represented as:

*sb s*

The ideal grid voltages have only the fundamental frequency component.

*usuc* are negative sequence components.

distorted grid voltage can be represented as:

harmonics components of the grid voltages.

three-phase grid voltages are expressed as:

**3. Non-ideal grid voltages** 

The fuzzy logic controller characteristics used in this section are the same as in the previous section (2.2). Figure 6 shows the triangular and trapezoidal membership functions for the input and output variables. The rule table for the hysteresis fuzzy logic controller is the same with Table 1.

**Figure 5.** General structure of the hysteresis FLC for the current control loop

**Figure 6.** Membership functions for a) input variable e, b) input variable Δe, and b) output variable Ierror

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 527

## **3. Non-ideal grid voltages**

526 Fuzzy Controllers – Recent Advances in Theory and Applications

limiting the initial error.

same with Table 1.

phase-a is shown in figure 5. The same controller is applied to the other two-phases (b and c). As inputs to FLC the error between the reference current and the sensed current ( *ca,ref ca ei i* ) and the error variation ( *e e(k) e(k )* 1 ) at *kth* sampling instant are used. The output of the FLC is considered as the amplitude of the current error. The coefficients F1 and F2 are used to adjust the input control signals. The saturations blocks are used for

The fuzzy logic controller characteristics used in this section are the same as in the previous section (2.2). Figure 6 shows the triangular and trapezoidal membership functions for the input and output variables. The rule table for the hysteresis fuzzy logic controller is the

**Figure 6.** Membership functions for a) input variable e, b) input variable Δe, and b) output variable Ierror

(c)

(a) (b)

**Figure 5.** General structure of the hysteresis FLC for the current control loop

It is a common phenomenon in electric power system the grid voltages to be non-ideal. This problem is particularly important in Greek electric power distribution system, mainly due to the large increase of power electronic devices. In this section the mathematical model of the grid voltages under several non-ideal cases will briefly be presented.

The ideal grid voltages have sinusoidal waveform and can be represented as:

$$
\begin{bmatrix} u\_{sa} \\ u\_{sb} \\ u\_{sc} \end{bmatrix} = \sqrt{2} \cdot V\_s \cdot \begin{bmatrix} \sin(\omega t) \\ \sin(\omega t - 120^\circ) \\ \sin(\omega t + -120^\circ) \end{bmatrix} \tag{15}
$$

The ideal grid voltages have only the fundamental frequency component.

When the three-phase grid voltages are unbalanced ( *su u* ), the grid voltages can be expressed as positive and negative sequence components as shown in equation (12).

$$
\begin{bmatrix} u\_{sua} \\ u\_{sub} \\ u\_{sub} \end{bmatrix} = \begin{bmatrix} u\_{sua+} \\ u\_{sub+} \\ u\_{sub\cdot +} \end{bmatrix} + \begin{bmatrix} u\_{sua-} \\ u\_{sub\cdot -} \\ u\_{sub\cdot -} \end{bmatrix} \tag{16}
$$

Where *usua* , *sub u* , and *usuc* are positive sequence components and *usua* , *sub u* , and *usuc* are negative sequence components.

It is a very common phenomenon in electric power distribution systems, voltages having non-ideal waveforms, and different levels of harmonics. When the three-phase grid voltages are distorted (*usd*), the grid voltages have harmonics components. In this scenario the distorted grid voltage can be represented as:

$$
\begin{bmatrix} u\_{sda} \\ u\_{sdb} \\ u\_{sdc} \end{bmatrix} = \begin{bmatrix} u\_{sda,f} \\ u\_{sdb,f} \\ u\_{sdc,f} \end{bmatrix} + \begin{bmatrix} u\_{sda,h} \\ u\_{sdb,h} \\ u\_{sdc,h} \end{bmatrix} \tag{17}
$$

Where, *sdb, f u* , and *sdc, f u* are positive sequence components and , *sdb,h u* , and *sdc,h u* are harmonics components of the grid voltages.

When the three-phase grid voltages are distorted and unbalanced (*udu*), the grid voltages contain harmonic components and unbalances. For this case, the distorted and unbalanced three-phase grid voltages are expressed as:

$$
\begin{bmatrix} u\_{\text{sda},f} \\ u\_{\text{sda},f} \\ u\_{\text{sda},f} \end{bmatrix} = \begin{bmatrix} u\_{\text{sda},f} \\ u\_{\text{sdb},f} \\ u\_{\text{sdb},f} \end{bmatrix} + \begin{bmatrix} u\_{\text{sda},h} \\ u\_{\text{sdb},h} \\ u\_{\text{sdc},h} \end{bmatrix} + \begin{bmatrix} u\_{\text{sda}-} \\ u\_{\text{sdb}-} \\ u\_{\text{sdc}-} \end{bmatrix} = \begin{bmatrix} u\_{\text{sda}+} \\ u\_{\text{sdb}+} \\ u\_{\text{sdc}+} \end{bmatrix} + \begin{bmatrix} u\_{\text{sda},h} \\ u\_{\text{sdb},h} \\ u\_{\text{sdc},h} \end{bmatrix} + \begin{bmatrix} u\_{\text{sda}-} \\ u\_{\text{sdb}-} \\ u\_{\text{sdc}-} \end{bmatrix} \tag{18}
$$

## **4. Simulation results**

In this section the electric power system of figure 1 will be simulated. The simulation will be carried out via Matlab/Simulink. The characteristics of the electric power system are shown in Table 2. Four practical scenarios were examined in which the grid voltages are ideal, unbalanced, distorted and distorted-unbalanced. For the worst case, where the grid voltages are distorted-unbalanced the performance of the electric power system will be analyzed using the conventional and the fuzzy logic control system. The behavior of the PI controller and the FLC will be compared based on the dc bus voltage control.

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 529

Phases *a b c*  THDi (Load\_1) 18.84 26.62 21.92 THDi (Load\_1 + Load\_2) 5.61 10.02 5.61

**Figure 7.** Distorted-unbalanced grid voltages

**Figure 8.** Grid currents without the application of the SAPF

**Figure 9.** Reactive power of the grid without the SAPF

**Table 3.** Grid current THDi index.

Besides, the behavior of the conventional hysteresis controller and the hysteresis FLC will be compared based on the inverter output current control. For the comparison of the performance between the conventional control methods and the control methods with fuzzy logic theory the THDi index in steady state, and the oscillation of the dc bus voltage during the transient response will be considered. Thereafter, the non-linear load will be called "Load\_1" and the linear load will be called "Load\_2". In all cases the transient response occurs at the same time (t=0.4 sec). It was considered that time t=0.4 sec in the electric power system, additionally to the initial non-linear load (Load\_1) a linear load (Load\_2) is connected.


**Table 2.** Parameters of the electric power system.

#### **4.1. Distorted-Unbalanced grid voltages**

In this case the grid voltages are considered to be distorted-unbalanced, and they are expressed as:

$$
\begin{bmatrix} \boldsymbol{\mu}\_{\text{sda}} \\ \boldsymbol{\mu}\_{\text{sdb}} \\ \boldsymbol{\mu}\_{\text{sdc}} \end{bmatrix} = \sqrt{2} \cdot \boldsymbol{V}\_{\text{s}} \cdot \begin{bmatrix} \sin(\omega t) \\ \sin(\omega t - 120^{\circ}) \\ \sin(\omega t + -120^{\circ}) \end{bmatrix} + 1 \cdot \begin{bmatrix} \sin(\omega t) \\ \sin(\omega t - 120^{\circ}) \\ \sin(5\omega t + 120^{\circ}) \end{bmatrix} + 2 \cdot \begin{bmatrix} \sin(5\omega t) \\ \sin(7\omega t - 120^{\circ}) \\ \sin(7\omega t + 120^{\circ}) \end{bmatrix} \tag{19}
$$

Figure 7 shows the distorted-unbalanced grid voltages. Figures 8 and 9 show the grid currents (*isa*, *isb*, *isc*) and the reactive power of the grid respectively, without the application of the SAPF. In Table 3 the THDi index of the grid currents for the loads 'Load\_1' and 'Load\_1+Load\_2' is denoted.

**Figure 7.** Distorted-unbalanced grid voltages

(rms) Vs=230V Non-linear

**Table 2.** Parameters of the electric power system.

**4.1. Distorted-Unbalanced grid voltages** 

and the FLC will be compared based on the dc bus voltage control.

In this section the electric power system of figure 1 will be simulated. The simulation will be carried out via Matlab/Simulink. The characteristics of the electric power system are shown in Table 2. Four practical scenarios were examined in which the grid voltages are ideal, unbalanced, distorted and distorted-unbalanced. For the worst case, where the grid voltages are distorted-unbalanced the performance of the electric power system will be analyzed using the conventional and the fuzzy logic control system. The behavior of the PI controller

Besides, the behavior of the conventional hysteresis controller and the hysteresis FLC will be compared based on the inverter output current control. For the comparison of the performance between the conventional control methods and the control methods with fuzzy logic theory the THDi index in steady state, and the oscillation of the dc bus voltage during the transient response will be considered. Thereafter, the non-linear load will be called "Load\_1" and the linear load will be called "Load\_2". In all cases the transient response occurs at the same time (t=0.4 sec). It was considered that time t=0.4 sec in the electric power system, additionally to the initial non-linear load (Load\_1) a linear load (Load\_2) is

inductance Ls=0.1mH Linear Load L2=1mH dc side capacitor Cdc=3mF

Firing angle α=10o Linear Load R2=2Ω dc bus voltage Vdc=1 kV

load L1=1mH Non-linear load side impedance LL=1mH

In this case the grid voltages are considered to be distorted-unbalanced, and they are

2 120 13 120 23 5 120 8 7 120

<sup>120</sup>*<sup>o</sup> )*

*oo o*

Figure 7 shows the distorted-unbalanced grid voltages. Figures 8 and 9 show the grid currents (*isa*, *isb*, *isc*) and the reactive power of the grid respectively, without the application of the SAPF. In Table 3 the THDi index of the grid currents for the loads 'Load\_1' and

120 120 5 120 7

*sin(ωt) sin(ωt) sin( ωt) sin( ωt) V sin(ωt ) sin(ωt ) sin( ωt ) sin( ωt ) sin(ωt ) sin(ωt ) sin( ωt ) sin( ωt*

*o o oo*

load R1=4Ω SAPF inductance Lc=1mH

5 7

(19)

**4. Simulation results** 

connected.

Grid

Grid voltage

Non-linear

expressed as:

*sda sdb sdc*

*u u u*

*s*

'Load\_1+Load\_2' is denoted.

**Figure 8.** Grid currents without the application of the SAPF

**Figure 9.** Reactive power of the grid without the SAPF


**Table 3.** Grid current THDi index.

## *4.1.1. Conventional controller and fuzzy controller for dc bus voltage control*

In this section the control scheme consists of the PI dc bus voltage controller and the hysteresis current controller (called "PI-HYS") will be compared with the control scheme consists of the fuzzy dc bus voltage controller (from section 2.2) and the hysteresis current controller (called "FUZ-HYS"). Figure 10 shows the dc bus voltage transient response at time t=0.4 sec for both control schemes.

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 531

**Figure 11.** Grid currents with the application of the SAPF, for both control schemes

**Figure 12.** Grid reactive power after the compensation, for both control schemes

successfully eliminates the high order harmonics from the grid currents.

grid currents in the case of dc bus voltage FLC is observed.

From figure 11 it is obvious that the faster response of the dc bus voltage using the fuzzy logic controller has a positive effect on the grid current, as the grid currents return to steady state operation faster (figure 11, time t1 for FLC, and time t2 for PI controller). It should be noted that, from figure 11 and Table 4 no significant change in the harmonic distortion of the

From the simulation results it is observed that the performance of the SAPF is satisfactory in the case where the grid voltages are distorted-unbalanced. This fact is a consequence of the modified version of the *p-q* theory, which was proposed in this chapter. The SAPF

It is also observed that the SAPF compensates the reactive power of the load. As shown in figure 9 the reactive power of the grid without compensation for the 'Load\_1' is Q=27 kVAr, then adding the 'Load\_2' is increased to Q=58 kVAr. By using the active power filter, reactive power compensation is achieved for both initial and final load (the compensated

**Figure 10.** Dc bus voltage response for both control schemes

From figure 10 it is obvious that the dc bus voltage fuzzy logic controller outperforms the conventional PI controller. In particular, it is noted that the oscillation of the dc bus voltage with the application if the fuzzy logic controller is smaller compared to the PI one. Likewise the recovery time until the dc bus voltage returns to steady state is fairly smaller when the FLC is applied. This result has an effect on the time needed by the grid currents to return to steady state operation.

Figures 11 and 12 show the grid currents and the reactive power of the grid when the SAPF connected. Figures 11and 12 illustrate the results for both control schemes. Table 4 shows the THDi index of the grid currents considering the loads 'Load\_1' and 'Load\_1+Load\_2'.


**Table 4.** Grid current THDi index.

**Figure 11.** Grid currents with the application of the SAPF, for both control schemes

time t=0.4 sec for both control schemes.

**Figure 10.** Dc bus voltage response for both control schemes

steady state operation.

'Load\_1+Load\_2'.

**Table 4.** Grid current THDi index.

*4.1.1. Conventional controller and fuzzy controller for dc bus voltage control* 

In this section the control scheme consists of the PI dc bus voltage controller and the hysteresis current controller (called "PI-HYS") will be compared with the control scheme consists of the fuzzy dc bus voltage controller (from section 2.2) and the hysteresis current controller (called "FUZ-HYS"). Figure 10 shows the dc bus voltage transient response at

From figure 10 it is obvious that the dc bus voltage fuzzy logic controller outperforms the conventional PI controller. In particular, it is noted that the oscillation of the dc bus voltage with the application if the fuzzy logic controller is smaller compared to the PI one. Likewise the recovery time until the dc bus voltage returns to steady state is fairly smaller when the FLC is applied. This result has an effect on the time needed by the grid currents to return to

Figures 11 and 12 show the grid currents and the reactive power of the grid when the SAPF connected. Figures 11and 12 illustrate the results for both control schemes. Table 4 shows the THDi index of the grid currents considering the loads 'Load\_1' and

Phases *a b c a b c* 

THDi (Load\_1) 4.05 3.81 3.07 4.35 4.12 3.32 THDi (Load\_1 + Load\_2) 1.83 1.96 1.60 1.86 1.96 1.61

PI-HYS FYZ-HYS

**Figure 12.** Grid reactive power after the compensation, for both control schemes

From figure 11 it is obvious that the faster response of the dc bus voltage using the fuzzy logic controller has a positive effect on the grid current, as the grid currents return to steady state operation faster (figure 11, time t1 for FLC, and time t2 for PI controller). It should be noted that, from figure 11 and Table 4 no significant change in the harmonic distortion of the grid currents in the case of dc bus voltage FLC is observed.

From the simulation results it is observed that the performance of the SAPF is satisfactory in the case where the grid voltages are distorted-unbalanced. This fact is a consequence of the modified version of the *p-q* theory, which was proposed in this chapter. The SAPF successfully eliminates the high order harmonics from the grid currents.

It is also observed that the SAPF compensates the reactive power of the load. As shown in figure 9 the reactive power of the grid without compensation for the 'Load\_1' is Q=27 kVAr, then adding the 'Load\_2' is increased to Q=58 kVAr. By using the active power filter, reactive power compensation is achieved for both initial and final load (the compensated

reactive power of the grid is Q=28 VAr) as shown in figure 12. Comparing the two control schemes, PI-HYS and FYZ-HYS, similar behavior for the reactive power compensation is detected. For both control schemes there is a short transient period during the change of load.

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 533

Phases *a b c a b c a b c*  THDi (1) 4.05 3.81 3.07 4.35 4.12 3.32 3.37 3.53 3.04 THDi (2) 1.83 1.96 1.60 1.86 1.96 1.61 1.62 1.77 1.57

From figure 14 it is evident that the faster response of the dc bus voltage with the application of the FUZ-FUZ HYS control scheme has a positive effect on the grid current, as they return to steady state operation in smaller interval time (figure 14, time instant t1 for FUZ-FUZ HYS control scheme, and time instant t2 for PI-HYS control scheme). It should be noted that, from figure 14 smaller harmonic distortion of the grid currents is observed using the FUZ-FUZ HYS control scheme. In figure 14 some of the points where improvement is

**Figure 14.** Grid currents with the application of the SAPF, for both control schemes

**Figure 15.** Grid reactive power after the compensation, for both control schemes

**Table 5.** Grid current THDi index.

observed are highlighted using circles.

PI‐HY FYZ-HYS FUZ-FYZ HYS

## *4.1.2. Conventional dc bus voltage and ac current controller compared with fuzzy controller*

For the reduction of the THDi index of the grid currents, authors propose a control scheme consists of a fuzzy logic dc bus voltage controller together with fuzzy logic hysteresis current controllers (as in section 2.3) (called "FUZ-FYZ HYS").

In this section the control scheme consists of the PI controller for the dc bus voltage control and the hysteresis controller for current control (PI-HYS) will be compared with the control scheme consists of the fuzzy controller for the dc bus voltage control and the fuzzy logic hysteresis controller for current control (FUZ-FUZ HYS). Figure 13 shows the dc bus voltage response at time t=0.4 sec for both control schemes.

From figure 13 it is obvious that the use of fuzzy logic for and ac output current control outperforms the control scheme of PI dc bus voltage control and hysteresis current control. In particular we observe that the oscillation of the dc bus voltage is smaller when the fuzzy logic dc bus voltage control and the fuzzy hysteresis current control is used. Likewise the interval time until the dc bus voltage returns to steady state operation is fairly smaller when the fuzzy logic scheme is applied. Comparing the results of figure 10 with those of figure 10, it is observed that the control schemes of FUZ- HYS and FUZ-FUZ HYS have no significant difference in the control of the dc bus voltage.

**Figure 13.** Dc bus voltage response, for both control schemes

Figures 14 and 15 show the grid currents and the reactive power of the grid with the application of the SAPF for both control schemes (PI-HYS and FUZ-FUZ HYS). Table 5 shows the grid currents THDi index for the loads 'Load\_1' and 'Load\_1+Load\_2'.


**Table 5.** Grid current THDi index.

532 Fuzzy Controllers – Recent Advances in Theory and Applications

load.

*controller* 

reactive power of the grid is Q=28 VAr) as shown in figure 12. Comparing the two control schemes, PI-HYS and FYZ-HYS, similar behavior for the reactive power compensation is detected. For both control schemes there is a short transient period during the change of

For the reduction of the THDi index of the grid currents, authors propose a control scheme consists of a fuzzy logic dc bus voltage controller together with fuzzy logic hysteresis

In this section the control scheme consists of the PI controller for the dc bus voltage control and the hysteresis controller for current control (PI-HYS) will be compared with the control scheme consists of the fuzzy controller for the dc bus voltage control and the fuzzy logic hysteresis controller for current control (FUZ-FUZ HYS). Figure 13 shows the dc bus voltage

From figure 13 it is obvious that the use of fuzzy logic for and ac output current control outperforms the control scheme of PI dc bus voltage control and hysteresis current control. In particular we observe that the oscillation of the dc bus voltage is smaller when the fuzzy logic dc bus voltage control and the fuzzy hysteresis current control is used. Likewise the interval time until the dc bus voltage returns to steady state operation is fairly smaller when the fuzzy logic scheme is applied. Comparing the results of figure 10 with those of figure 10, it is observed that the control schemes of FUZ- HYS and FUZ-FUZ HYS have no significant

Figures 14 and 15 show the grid currents and the reactive power of the grid with the application of the SAPF for both control schemes (PI-HYS and FUZ-FUZ HYS). Table 5

shows the grid currents THDi index for the loads 'Load\_1' and 'Load\_1+Load\_2'.

*4.1.2. Conventional dc bus voltage and ac current controller compared with fuzzy* 

current controllers (as in section 2.3) (called "FUZ-FYZ HYS").

response at time t=0.4 sec for both control schemes.

difference in the control of the dc bus voltage.

**Figure 13.** Dc bus voltage response, for both control schemes

From figure 14 it is evident that the faster response of the dc bus voltage with the application of the FUZ-FUZ HYS control scheme has a positive effect on the grid current, as they return to steady state operation in smaller interval time (figure 14, time instant t1 for FUZ-FUZ HYS control scheme, and time instant t2 for PI-HYS control scheme). It should be noted that, from figure 14 smaller harmonic distortion of the grid currents is observed using the FUZ-FUZ HYS control scheme. In figure 14 some of the points where improvement is observed are highlighted using circles.

**Figure 14.** Grid currents with the application of the SAPF, for both control schemes

**Figure 15.** Grid reactive power after the compensation, for both control schemes

From the analysis of the simulation results, the improvement in the THDi index of the grid current using the FUZ-FUZ HYS control scheme is observed, as shown in Table 5. For phase-a, the improvement in the THDi index with the FUZ-FUZ HYS control scheme is about 22.5%. For phases-b and -c, the improvement in the THDi index with the FUZ-FUZ HYS control scheme is about 14.3% and 8.5% respectively. Investigating the electric power system, the unbalances in the THDi index is the result of the unbalances in the grid voltages.

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 535

The reference voltage in *α-β* reference frame will be used by the Space Vector Modulation

The operation of the fuzzy-tuned PI controller is based on the use of a FLC for on-line tuning of the gains *Kp* and *Ki* of the PI controller, as shown in equation (20). Then the PI controller uses the adjusted gains *Kp*, *Ki* and the current error (e) to create the reference

> *p pp i ii*

tuned PI controller the triangular function was used as input and output fuzzy sets.

**Figure 17.** Membership functions for a) input variables e, Δe, and b) output variables Δ*Ki* (pu) and

(a) (b)

*K K K (k ) K K K (k )* 

As inputs to the fuzzy-tuned PI controller the error *c,ref c ei i* and the error variation Δe=e(k)-e(k-1) are determined. As outputs from the fuzzy part, the gains Δ*Ki* (pu) and Δ*Kp*  (pu) of the PI controller, are determined. Using the gains *Ki* and *Ki*, the PI controller outputs the reference output voltage of the inverter ( *c,ref u* ). The scaling factors G1, G2, G3 and G4 are used to normalize the input and output signals. In figure 17.a seven membership functions are used for each input (NL-Negative Large, NM-Negative Medium, NS-Negative Small, ZE-Zero, PS-Positive Small, PM-Positive Medium, and PL-Positive Large). In figure 17.b two membership functions are used for each output (B for Big and S for Small). For the fuzzy-

1 1

(20)

algorithm for the switching pattern generation.

**Figure 16.** General structure of the fuzzy-tuned PI controller

output control signals (referencevoltage).

Δ*Kp* (pu)

From the comparison of the PI-HYS and FYZ-FUZ HYS control schemes, similar behavior for the reactive power compensation is observed. For both control schemes there is a short transition period during the load change. As shown in figure 9 the reactive power of the grid considering only the 'Load\_1', without compensation for the is Q=27 kVAr, then adding the 'Load\_2' the reactive power is increased to Q=58 kVAr. Using the SAPF, reactive power compensation is achieved for both initial and final load (in this case, the reactive power of the grid is approximately Q=29 VAr) as shown in figure 15.

From the simulation results it is observed that considering the above mentioned case the performance of the SAPF is excellent, as well as the performance of the SAPF is not affected by the distorted-unbalanced grid voltages. This fact is a consequence of the modified version of the *p-q* theory.

## **5. Future research**

The fuzzy logic controller outperforms the conventional PI controller due to robustness and the superior transient response. However FLC have some significant disadvantages. The main drawback of the FLC is the requirement of an expert for the design of the membership functions and the fuzzy rules. To overcome this disadvantage, a novel artificial intelligent controller called "fuzzy-tuned PI controller" has been proposed in the literature of automation control [De Carli A, Linguori P, et al. (1994)], [Zhao Z-Y, Tomizuka M, et al. (1993)]. The fuzzy-tuned PI controller in figure 16 is a combination of the fuzzy controller and the PI controller. Using the fuzzy part we can estimate the gains *Κp* and *Kp* of the PI controller. Then the PI controller based on these gains outputs the reference signal. The fuzzy-tuned PI controller was initially applied for the speed control of the induction motor drives [Chen Y, Fu b, et al. (2008)] and the dc bus voltage control of the grid connected inverters [Suryanarayana H, Mishra MK (2008)].

No significant work, comparing the performance of the PI and fuzzy-tuned PI controller for the current control of a grid connected inverter, has been reported. In this section the PI and the fuzzy-tuned PI controller are applied to the inner current control loop. The criterion for the comparison of the two controllers are based on the transient response.

## **5.1. Fuzzy-tuned PI controller analysis for current control**

The synoptic block diagram of the proposed fuzzy-tuned PI controller is illustrated in figure 16. As inputs to the fuzzy-tuned PI current controller are the actual and the reference currents. The current controller outputs the appropriate reference signal (reference voltage). The reference voltage in *α-β* reference frame will be used by the Space Vector Modulation algorithm for the switching pattern generation.

**Figure 16.** General structure of the fuzzy-tuned PI controller

534 Fuzzy Controllers – Recent Advances in Theory and Applications

the grid is approximately Q=29 VAr) as shown in figure 15.

inverters [Suryanarayana H, Mishra MK (2008)].

version of the *p-q* theory.

**5. Future research** 

From the analysis of the simulation results, the improvement in the THDi index of the grid current using the FUZ-FUZ HYS control scheme is observed, as shown in Table 5. For phase-a, the improvement in the THDi index with the FUZ-FUZ HYS control scheme is about 22.5%. For phases-b and -c, the improvement in the THDi index with the FUZ-FUZ HYS control scheme is about 14.3% and 8.5% respectively. Investigating the electric power system, the unbalances in the THDi index is the result of the unbalances in the grid voltages. From the comparison of the PI-HYS and FYZ-FUZ HYS control schemes, similar behavior for the reactive power compensation is observed. For both control schemes there is a short transition period during the load change. As shown in figure 9 the reactive power of the grid considering only the 'Load\_1', without compensation for the is Q=27 kVAr, then adding the 'Load\_2' the reactive power is increased to Q=58 kVAr. Using the SAPF, reactive power compensation is achieved for both initial and final load (in this case, the reactive power of

From the simulation results it is observed that considering the above mentioned case the performance of the SAPF is excellent, as well as the performance of the SAPF is not affected by the distorted-unbalanced grid voltages. This fact is a consequence of the modified

The fuzzy logic controller outperforms the conventional PI controller due to robustness and the superior transient response. However FLC have some significant disadvantages. The main drawback of the FLC is the requirement of an expert for the design of the membership functions and the fuzzy rules. To overcome this disadvantage, a novel artificial intelligent controller called "fuzzy-tuned PI controller" has been proposed in the literature of automation control [De Carli A, Linguori P, et al. (1994)], [Zhao Z-Y, Tomizuka M, et al. (1993)]. The fuzzy-tuned PI controller in figure 16 is a combination of the fuzzy controller and the PI controller. Using the fuzzy part we can estimate the gains *Κp* and *Kp* of the PI controller. Then the PI controller based on these gains outputs the reference signal. The fuzzy-tuned PI controller was initially applied for the speed control of the induction motor drives [Chen Y, Fu b, et al. (2008)] and the dc bus voltage control of the grid connected

No significant work, comparing the performance of the PI and fuzzy-tuned PI controller for the current control of a grid connected inverter, has been reported. In this section the PI and the fuzzy-tuned PI controller are applied to the inner current control loop. The criterion for

The synoptic block diagram of the proposed fuzzy-tuned PI controller is illustrated in figure 16. As inputs to the fuzzy-tuned PI current controller are the actual and the reference currents. The current controller outputs the appropriate reference signal (reference voltage).

the comparison of the two controllers are based on the transient response.

**5.1. Fuzzy-tuned PI controller analysis for current control** 

The operation of the fuzzy-tuned PI controller is based on the use of a FLC for on-line tuning of the gains *Kp* and *Ki* of the PI controller, as shown in equation (20). Then the PI controller uses the adjusted gains *Kp*, *Ki* and the current error (e) to create the reference output control signals (referencevoltage).

$$\begin{aligned} K\_p &= \Delta K\_p + K\_p(k-1) \\ K\_i &= \Delta K\_i + K\_i(k-1) \end{aligned} \tag{20}$$

As inputs to the fuzzy-tuned PI controller the error *c,ref c ei i* and the error variation Δe=e(k)-e(k-1) are determined. As outputs from the fuzzy part, the gains Δ*Ki* (pu) and Δ*Kp*  (pu) of the PI controller, are determined. Using the gains *Ki* and *Ki*, the PI controller outputs the reference output voltage of the inverter ( *c,ref u* ). The scaling factors G1, G2, G3 and G4 are used to normalize the input and output signals. In figure 17.a seven membership functions are used for each input (NL-Negative Large, NM-Negative Medium, NS-Negative Small, ZE-Zero, PS-Positive Small, PM-Positive Medium, and PL-Positive Large). In figure 17.b two membership functions are used for each output (B for Big and S for Small). For the fuzzytuned PI controller the triangular function was used as input and output fuzzy sets.

**Figure 17.** Membership functions for a) input variables e, Δe, and b) output variables Δ*Ki* (pu) and Δ*Kp* (pu)

The fuzzy rules which are shown in Table 6 for the ΔKp and in Table 7 for the ΔΚi, are determined using the standard form of fuzzy rules: *IF e is Ai and Δe is Bj, THEN ΔΚP,i is Cij and ΔΚI,i is Dij.* 

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 537

From figure 18 we can observe that the current error in *a-b-c* reference frame is very big when the PI controller is used (the power of the inverter increases). When the fuzzy-tuned PI controller is applied the error becomes almost zero while the recovery time is smaller compared to the PI. This fact has a direct impact to the output currents, which in the case of fuzzy-tuned PI controller have smoother behavior, while in the case of PI controller have

From the dynamic response of the electric power system is concluded that the fuzzy-tuned

(a)

**Figure 18.** Current in ac side of the inverter during the dynamic response using the a) PI, and b) fuzzy-

(b)

PI controller is best suited for the inner current control loop.

rougher behavior.

tuned PI current controller


**Table 6.** Fuzzy control rules table for ΔKP.


**Table 7.** Fuzzy control rules table for ΔKI.

### **5.2. Comparison between fuzzy-tuned PI controller and the PI controller**

In this section the behavior of the two current controllers will be compared based on the dynamic response. At the time instant t=0.4 sec a sudden variation of the output power of the inverter occurs. Figures 18.a and 18.b show the output currents of the inverter in *a-b-c* reference frame for the PI and the fuzzy-tuned PI current controller, respectively.

From figure 18 we can observe that the current error in *a-b-c* reference frame is very big when the PI controller is used (the power of the inverter increases). When the fuzzy-tuned PI controller is applied the error becomes almost zero while the recovery time is smaller compared to the PI. This fact has a direct impact to the output currents, which in the case of fuzzy-tuned PI controller have smoother behavior, while in the case of PI controller have rougher behavior.

536 Fuzzy Controllers – Recent Advances in Theory and Applications

e

Δe

**Table 6.** Fuzzy control rules table for ΔKP.

**Table 7.** Fuzzy control rules table for ΔKI.

e

Δe

*ΔΚI,i is Dij.* 

The fuzzy rules which are shown in Table 6 for the ΔKp and in Table 7 for the ΔΚi, are determined using the standard form of fuzzy rules: *IF e is Ai and Δe is Bj, THEN ΔΚP,i is Cij and*

> NL B B B B B B B NM S B B B B B S

> NS S S B B B S S ZE S S S B S S S

> PS S S B B B S S PM S B B B B B S PL B B B B B B B

> > NL NM NS ZE PS PM PL

NL B B B B B B B

NM B S S S S S B NS B B S S S B B

ZE B B B S B B B PS B B S S S B B

PM B S S S S S B PL B B B B B B B

In this section the behavior of the two current controllers will be compared based on the dynamic response. At the time instant t=0.4 sec a sudden variation of the output power of the inverter occurs. Figures 18.a and 18.b show the output currents of the inverter in *a-b-c*

**5.2. Comparison between fuzzy-tuned PI controller and the PI controller** 

reference frame for the PI and the fuzzy-tuned PI current controller, respectively.

NL NM NS ZE PS PM PL

From the dynamic response of the electric power system is concluded that the fuzzy-tuned PI controller is best suited for the inner current control loop.

**Figure 18.** Current in ac side of the inverter during the dynamic response using the a) PI, and b) fuzzytuned PI current controller

## **6. Conclusion**

In this chapter a modified version of the *p-q* theory was proposed, in order to improve the performance of the SAPF in the case of non-ideal grid voltages. For the performance improvement of the control scheme, the fuzzy logic theory was applied. A fuzzy logic controller for the dc bus voltage control was used. From the computer simulations and the analysis of the results, smaller amplitude and duration of the dc bus voltage oscillations during the transient response has been demonstrated. A further investigation of the system was carried out applying fuzzy logic hysteresis controller to control the output current of the inverter. From the investigation of the control system using fuzzy logic controller both for dc bus voltage and inverter output current control, the dc bus voltage during the transient response and the THDi index of the grid currents are obviously improved.

Performance Evaluation of PI and Fuzzy Controlled Power Electronic Inverters for Power Quality Improvement 539

Jain S K, Agrawal P, Gupta H O. (2002). Fuzzy logic controlled shunt active power filter for

Kale M, Ozdemir E. (2005). Harmonic and reactive power compensation with shunt active power filter under non-ideal mains voltage. *Electr Power Syst Reseach*, Vol.74, No.3

Kilic T, Milun S, Petrovic G. (2007). Design and implementation of predictive filtering system for current reference generation of active power filter. Int *Journ of Electr Power* 

Lin B R, Hoft R G. (1996). Analysis of power converter control using neural network and rule-based methods. *Electr Power Comp and Systems*, Vol.24, No.7, pp.695-

Mekri F, Machmoum M. (2010). A comparative study of voltage controllers for series active

Saad A, Zellouma L. (2009). Fuzzy logic controller for three-level shunt active filter compensating harmonics and reactive power. *Electr Power Syst Reseach*, Vol. 79, No.10,

Salmeron P, Herrera R S, Vazquez J R. (2007). A new approach for three-phase compensation based on the instantaneous reactive power theory. *Electr Power Syst* 

Segui-Chilet S S, Gimeno-Sales F J, Orts S, Garcera G, Figueres E, Alcaniz M, Masot R. (2007). Approach to unbalance power active compensation under linear load unbalances and fundamental voltage asymmetries. *Int J Electr Power Energy Syst*, Vol.29,

Singh G K, Singh A K, Mitra R. (2007). A simple fuzzy logic based robust active power filter for harmonics minimization under random load variation. *Electr Power Syst Reseach*,

Skretas S B, Dimitrios D. P. (2009). Efficient design and simulation of an expandable hybrid (wind–photovoltaic) power system with MPPT and inverter input voltage regulation features in compliance with electric grid requirements. *Electr Power Syst Reseach*, Vol.79,

Suryanarayana H, Mishra M K (2008). Fuzzy logic based supervision of dc link pi control in

Tsengenes G, Adamidis G. (2010). An Improved Current Control Technique for the Investigation of a Power System with a Shunt Active Filter. *In: International Symposium* 

Tsengenes G, Adamidis G. (2011). Investigation of the behavior of a three phase gridconnected photovoltaic system to control active and reactive power. *Elec Power Syst* 

Zhao Z-Y, Tomizuka M, Isaka S (1993). Fuzzy gain scheduling of pid controllers. *IEEE Trans* 

*on Power Electronics, Electrical Drives, Automation and Motion*, Pisa, Italy.

power filter. *Electr Power Syst Reseach*, Vol.80, No. , pp.615-626.

power quality improving. *IEE Electr Power Appl*, Vol.149, No.5, pp.317-328.

pp.363-370.

720.

pp.1337-1341.

No.7, pp.526-539.

No.9 , pp.1271-1285.

Vol.77, No.8 , pp.1101-1111.

a dstatcom. *In: India conference*, Kanpur

*Reser*, Vol. 81, No. 1, pp.177-184, 2011.

*Systems Man and Cybern*, Vol.23, No.4, pp. 1392-1398.

*Energy Syst*, Vol.29, No.2, pp.106-112.

*Reseach*, Vol.78, No.4, pp.605-617.

Furthermore, authors use a combination of the PI and the fuzzy control known as "fuzzytuned PI" control. The performance of the inverter in case of the fuzzy-tuned PI control is used on the current control loop was investigated.

## **Author details**

Georgios A. Tsengenes and Georgios A. Adamidis *Department of Electrical and Computer Engineering, Democritus University of Thrace, Greece* 

## **7. References**


Jain S K, Agrawal P, Gupta H O. (2002). Fuzzy logic controlled shunt active power filter for power quality improving. *IEE Electr Power Appl*, Vol.149, No.5, pp.317-328.

538 Fuzzy Controllers – Recent Advances in Theory and Applications

used on the current control loop was investigated.

Georgios A. Tsengenes and Georgios A. Adamidis

In this chapter a modified version of the *p-q* theory was proposed, in order to improve the performance of the SAPF in the case of non-ideal grid voltages. For the performance improvement of the control scheme, the fuzzy logic theory was applied. A fuzzy logic controller for the dc bus voltage control was used. From the computer simulations and the analysis of the results, smaller amplitude and duration of the dc bus voltage oscillations during the transient response has been demonstrated. A further investigation of the system was carried out applying fuzzy logic hysteresis controller to control the output current of the inverter. From the investigation of the control system using fuzzy logic controller both for dc bus voltage and inverter output current control, the dc bus voltage during the transient response and the THDi index of the grid currents are obviously

Furthermore, authors use a combination of the PI and the fuzzy control known as "fuzzytuned PI" control. The performance of the inverter in case of the fuzzy-tuned PI control is

*Department of Electrical and Computer Engineering, Democritus University of Thrace, Greece* 

Active Filter Applications. *IEEE Trans Ind Elect*, Vol.45, No.5, pp.722-729.

Three-Phase System*. IEEE Trans Power Del*, Vol.21, No.1, pp.362-367.

Conditions. *IEEE Trans Ind Appl*, Vol.40, No.3, pp.869-876.

control. *Simulat Model Pract Theor*, Vol.16, No.9, pp.1215-1238.

Buso S, Malesani L, Mattavelli P. (1998). Comparison of Current Control Techniques for

Chen Y, Fu b, Li Q (2008). Fuzzy logic based auto-modulation of parameters pi control for active power filter. In: World congress on intelligent control and automation,

Czarnecki L S. (2006). Instantaneous Reactive Power p-q Theory and Power Properties of

De Carli A, Linguori P, Marroni A (1994). A fuzzy-pi control strategy. *Control Eng Practice*,

El-Kholy E E, El-Sabbe, El-Hefnawy, Mharous M H. (2006). Three-phase active power filter based on current controlled voltage source inverter. Int J Electr Power Energy Syst,

Emanuel A E. (2004). Summary of IEEE Standard 1459: Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced or Unbalanced

Han Y, Khan M M, Yao G, Zhou L-D, Chen C. (2010). A novel harmonic-free power factor corrector based on T-type APF with adaptive linear neural network (ADALINE)

**6. Conclusion** 

improved.

**Author details** 

**7. References** 

Chongqing, 2008.

Vol.2, No.1, pp.147-153.

Vol.28, No.8 , pp.537-547.

	- Zhou K, Wang D. (2002). Relationship between space-vector modulation and three-phase carrier-based PWM: A comprehensive analysis. *IEEE Trans Ind Electron*, Vol.49, No.1, pp.186-196.

**Chapter 23** 

© 2012 Ibrahim et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Discrete-Time Cycle-to-Cycle Fuzzy Logic** 

**Control of FES-Induced Swinging Motion** 

B. S. K. K. Ibrahim, M. O. Tokhi, M. S. Huq and S. C. Gharooni

Functional electrical stimulation (FES) can be used to restore motor function to individuals with spinal cord injuries (SCI). FES involves artificially inducing a current in specific motor neurons to generate a skeletal muscle contraction. FES induced movement control is a significantly challenging area for researchers. The challenge mainly arises due to muscle response characteristics such as fatigue, time-varying properties and nonlinear dynamics of paralyzed muscles [1]. Another challenge is due to certain motor reflexes such as spasticity. Spasticity is a reflex or uncontrolled response to something that excites the nerve endings and produces muscle contractions. These reflexes are often unpredictable and may impede

Primarily due to the complexity of the system (nonlinearities, time-variation) practical FES systems are predominantly open-loop where the controller receives no information about the actual state of the system [3]. In its basic form, these systems require continuous user input. Practical success of this open-loop control strategy is still, however, seriously limited due to the fixed nature of the associated parameters. The problem arises especially due to the existing parameter variations (e.g., muscle fatigue), inherent time-variance, and strong nonlinearities present in the neuromuscular-skeletal system or the plant to be controlled. Besides, in such open-loop control approach, the actual movement is not assessed in real time and any mechanism of adapting the stimulation pattern in response to unforeseen circumstances such as external perturbations or muscle spasms is absent [4]. These prominent problems can be resolved by having a suitable closed-loop adaptive control mechanism. Such approach has several advantages over open-loop schemes, including better tracking performance and smaller sensitivity to the modeling errors, parameter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48488

**1. Introduction** 

joint movements [2].

variations, and external disturbances [5].

## **Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion**

B. S. K. K. Ibrahim, M. O. Tokhi, M. S. Huq and S. C. Gharooni

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48488

## **1. Introduction**

540 Fuzzy Controllers – Recent Advances in Theory and Applications

pp.186-196.

Zhou K, Wang D. (2002). Relationship between space-vector modulation and three-phase carrier-based PWM: A comprehensive analysis. *IEEE Trans Ind Electron*, Vol.49, No.1,

> Functional electrical stimulation (FES) can be used to restore motor function to individuals with spinal cord injuries (SCI). FES involves artificially inducing a current in specific motor neurons to generate a skeletal muscle contraction. FES induced movement control is a significantly challenging area for researchers. The challenge mainly arises due to muscle response characteristics such as fatigue, time-varying properties and nonlinear dynamics of paralyzed muscles [1]. Another challenge is due to certain motor reflexes such as spasticity. Spasticity is a reflex or uncontrolled response to something that excites the nerve endings and produces muscle contractions. These reflexes are often unpredictable and may impede joint movements [2].

> Primarily due to the complexity of the system (nonlinearities, time-variation) practical FES systems are predominantly open-loop where the controller receives no information about the actual state of the system [3]. In its basic form, these systems require continuous user input. Practical success of this open-loop control strategy is still, however, seriously limited due to the fixed nature of the associated parameters. The problem arises especially due to the existing parameter variations (e.g., muscle fatigue), inherent time-variance, and strong nonlinearities present in the neuromuscular-skeletal system or the plant to be controlled. Besides, in such open-loop control approach, the actual movement is not assessed in real time and any mechanism of adapting the stimulation pattern in response to unforeseen circumstances such as external perturbations or muscle spasms is absent [4]. These prominent problems can be resolved by having a suitable closed-loop adaptive control mechanism. Such approach has several advantages over open-loop schemes, including better tracking performance and smaller sensitivity to the modeling errors, parameter variations, and external disturbances [5].

© 2012 Ibrahim et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In controlling cyclical movement, one can try to follow pre-set joint angle trajectories. Although the trajectory-based closed-loop control has been developed but it has not been used yet in clinical FES gait because of difficulties in achieving accurate tracking performance [6]. Moreover, in the swing phase of gait, following exact trajectories is unimportant and inefficient, leading to fatigue due to large forces that must be exerted to precisely control the high inertia body segments [3]. For these reasons, cycle-to-cycle control method is expected to be an alternative to trajectory based closed-loop FES control. The cycle-to-cycle control delivers electrical stimulation in the form of open-loop control in each cycle without reference trajectory but it is still closed-loop control. In this control strategy, movement parameters at the end of each cycle are compared as in the desired set point, and the stimulation for the next cycle is adjusted on the basis of the error in the preceding cycle.

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 543

mapping of a number of inputs can be specified easily using fuzzy linguistic variables and fuzzy rules [12]. The control signal is computed by rule evaluation called fuzzy inference instead of by mathematical equations. In order to compensate the non-linearity of the musculo-skeletal system responses, the cycle-to-cycle control was implemented using fuzzy

This chapter presents the development of strategies for swinging motion control by controlling the amount of stimulation pulsewidth to the quadriceps muscle of the knee joints. The capability of the controller to control knee joint movements is first assessed in computer simulations using a musculo-skeletal knee joint model. The knee joint model developed in Matlab/Simulink, as described in [13], is used to develop an FLC-based cycleto-cycle control strategy for the knee joint movement. The FLC output is the controlled FES stimulation pulsewidth signal which stimulates the knee extensors providing torque to the knee joint. The swinging movement is performed by only controlling stimulation pulsewidth to the knee extensors to extent the knee and then the knee is left freely to flex in the flexion period. The controllers are then tested through experimental work on a paraplegic in terms of swinging performance and compensation of muscle fatigue and

The shank-quadriceps dynamics are modelled as the interconnection of passive and active properties of muscle model and the segmental dynamics. The total knee-joint moment is

*M*i gsd *MMMM <sup>a</sup>* (1)

where *Mi* refers to inertial moment, *Mg* is gravitational moment, *Ma* refers to an active knee joint moment produced by electrical stimulation, *Ms* is the knee joint elastic moment and *Md* is the viscous moment representing the passive behaviour of the knee joint. In this research the *Mi* and *Mg* are represented by the equations of motion for dynamic model of the lower limb while *Ma* and *Ms*+*Md* are represented by a fuzzy model as active properties of quadriceps muscle and passive viscoelasticity respectively. A schematic representation of the knee joint model consisting of active properties, passive viscoelasticity and equations of motion of the lower limb is shown in Figure 1. The active joint moment is added with the passive joint moment as an input (torque) to the lower limb model and this will produce the knee angle as the output. The subject participating in this work was a 48 year-old T2&T3 incomplete paraplegic male with 20 years post-injury with height = 173cm and weight =

A schematic diagram of the lower limb model is shown in Figure 5, where 2 *q* = shank length,

*r* = position of COM along the foot, 1

=knee angle

controller. Thus FLC with is the preferred option in the current work

spasticity.

given as [14]:

1

**2. Materials and method** 

80kg. Informed consent was obtained from the subject.

*r* = position of COM along the shank, 2

**2.1. Model of knee joint** 

In fact, FES induced movements have traditionally been achieved through application of stimulus bursts rather than continuous tracking control. The burst of stimulus signal would drive the joint to its desired orientation through ballistic movement and thus traversing a trajectory defined purely by the physics of the segment combination [5]. The cycle-to-cycle control approach retains this basic mechanism of movement generation through stimulus burst and comes into action when the movement is repetitive or cyclical, through automatic adjustment of the burst parameters to maintain the desired target orientation at each cycle [7]. While the trajectory based closed-loop control for knee joint angle of paraplegic has been criticized for having poor tracking and oscillatory responses and even its inability to reach full knee extension angle [8]. the ability of cycle-to-cycle control approach to realize the target joint orientation has been demonstrated in experimental tests of controlling maximum knee extension angle [9] or hip joint range [7].

Researches as in [7] and [9] investigated a discrete-time proportional-integral-derivative (PID) feedback controller for cycle-to-cycle adaptation of an experimentally initialized stimulation signal with a view to compensate for the fatigue-induced time variation of muscle output. For practical use of cycle-to-cycle control, realization in multi-joint control is crucial, in which problems seen in the PID controller such as drawback in determination of controller parameter values and a lack of capability in compensating muscle fatigue [7] have to be solved. It is difficult to establish the control parameters for these systems since they are not always the same under different circumstances [10]. Therefore, traditional control approaches, such as PID control might not perform satisfactorily if the system to be controlled is of highly nonlinear and uncertain nature [5].

On the other hand, fuzzy logic control (FLC) has long been known for its ability to handle a complex nonlinear system without developing a mathematical model of the system. FLC is the fastest growing soft computing tool in medicine and biomedical engineering [11]. It is being used successfully in an increasing number of application areas in the control community. FLCs are rule-based systems that use fuzzy linguistic variables to model human rule-of-thumb approaches to problem solving, and thus overcoming the limitations that classical expert systems may face because of their inflexible representation of human decision making. The major strength of fuzzy control also lies in the way a nonlinear output mapping of a number of inputs can be specified easily using fuzzy linguistic variables and fuzzy rules [12]. The control signal is computed by rule evaluation called fuzzy inference instead of by mathematical equations. In order to compensate the non-linearity of the musculo-skeletal system responses, the cycle-to-cycle control was implemented using fuzzy controller. Thus FLC with is the preferred option in the current work

This chapter presents the development of strategies for swinging motion control by controlling the amount of stimulation pulsewidth to the quadriceps muscle of the knee joints. The capability of the controller to control knee joint movements is first assessed in computer simulations using a musculo-skeletal knee joint model. The knee joint model developed in Matlab/Simulink, as described in [13], is used to develop an FLC-based cycleto-cycle control strategy for the knee joint movement. The FLC output is the controlled FES stimulation pulsewidth signal which stimulates the knee extensors providing torque to the knee joint. The swinging movement is performed by only controlling stimulation pulsewidth to the knee extensors to extent the knee and then the knee is left freely to flex in the flexion period. The controllers are then tested through experimental work on a paraplegic in terms of swinging performance and compensation of muscle fatigue and spasticity.

## **2. Materials and method**

#### **2.1. Model of knee joint**

542 Fuzzy Controllers – Recent Advances in Theory and Applications

knee extension angle [9] or hip joint range [7].

controlled is of highly nonlinear and uncertain nature [5].

In controlling cyclical movement, one can try to follow pre-set joint angle trajectories. Although the trajectory-based closed-loop control has been developed but it has not been used yet in clinical FES gait because of difficulties in achieving accurate tracking performance [6]. Moreover, in the swing phase of gait, following exact trajectories is unimportant and inefficient, leading to fatigue due to large forces that must be exerted to precisely control the high inertia body segments [3]. For these reasons, cycle-to-cycle control method is expected to be an alternative to trajectory based closed-loop FES control. The cycle-to-cycle control delivers electrical stimulation in the form of open-loop control in each cycle without reference trajectory but it is still closed-loop control. In this control strategy, movement parameters at the end of each cycle are compared as in the desired set point, and the stimulation for the next cycle is adjusted on the basis of the error in the preceding cycle. In fact, FES induced movements have traditionally been achieved through application of stimulus bursts rather than continuous tracking control. The burst of stimulus signal would drive the joint to its desired orientation through ballistic movement and thus traversing a trajectory defined purely by the physics of the segment combination [5]. The cycle-to-cycle control approach retains this basic mechanism of movement generation through stimulus burst and comes into action when the movement is repetitive or cyclical, through automatic adjustment of the burst parameters to maintain the desired target orientation at each cycle [7]. While the trajectory based closed-loop control for knee joint angle of paraplegic has been criticized for having poor tracking and oscillatory responses and even its inability to reach full knee extension angle [8]. the ability of cycle-to-cycle control approach to realize the target joint orientation has been demonstrated in experimental tests of controlling maximum

Researches as in [7] and [9] investigated a discrete-time proportional-integral-derivative (PID) feedback controller for cycle-to-cycle adaptation of an experimentally initialized stimulation signal with a view to compensate for the fatigue-induced time variation of muscle output. For practical use of cycle-to-cycle control, realization in multi-joint control is crucial, in which problems seen in the PID controller such as drawback in determination of controller parameter values and a lack of capability in compensating muscle fatigue [7] have to be solved. It is difficult to establish the control parameters for these systems since they are not always the same under different circumstances [10]. Therefore, traditional control approaches, such as PID control might not perform satisfactorily if the system to be

On the other hand, fuzzy logic control (FLC) has long been known for its ability to handle a complex nonlinear system without developing a mathematical model of the system. FLC is the fastest growing soft computing tool in medicine and biomedical engineering [11]. It is being used successfully in an increasing number of application areas in the control community. FLCs are rule-based systems that use fuzzy linguistic variables to model human rule-of-thumb approaches to problem solving, and thus overcoming the limitations that classical expert systems may face because of their inflexible representation of human decision making. The major strength of fuzzy control also lies in the way a nonlinear output The shank-quadriceps dynamics are modelled as the interconnection of passive and active properties of muscle model and the segmental dynamics. The total knee-joint moment is given as [14]:

$$M\_i = M\_a + M\_\mathbf{g} + M\_\mathbf{s} + M\_\mathbf{d} \tag{1}$$

where *Mi* refers to inertial moment, *Mg* is gravitational moment, *Ma* refers to an active knee joint moment produced by electrical stimulation, *Ms* is the knee joint elastic moment and *Md* is the viscous moment representing the passive behaviour of the knee joint. In this research the *Mi* and *Mg* are represented by the equations of motion for dynamic model of the lower limb while *Ma* and *Ms*+*Md* are represented by a fuzzy model as active properties of quadriceps muscle and passive viscoelasticity respectively. A schematic representation of the knee joint model consisting of active properties, passive viscoelasticity and equations of motion of the lower limb is shown in Figure 1. The active joint moment is added with the passive joint moment as an input (torque) to the lower limb model and this will produce the knee angle as the output. The subject participating in this work was a 48 year-old T2&T3 incomplete paraplegic male with 20 years post-injury with height = 173cm and weight = 80kg. Informed consent was obtained from the subject.

A schematic diagram of the lower limb model is shown in Figure 5, where 2 *q* = shank length, 1 *r* = position of COM along the shank, 2 *r* = position of COM along the foot, 1 =knee angle and 2 =ankle angle. Hence, the dynamics of motion can be represented in the simpler form based on Kane's equations as in [13]. The gravitational ( *M*<sup>g</sup> ) moment is represented by:-

> 1 11 2 1 2 cos cos *Mg mg r mg q*

(2)

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 545

Segment Length (m) Shank length 0.426 Foot length 0.068

Approximated position of COM of shank 0.213 Approximated position of COM of soot 0.034

Researcher as in [15] highlighted cycle-to-cycle control as a method for using feedback to improve product quality for processes that are inaccessible within a single processing cycle but can be changed between cycles. The same concept has been applied in this study, where only reaching a target joint orientation through ballistic movement is taken into consideration rather than rigorously following a trajectory. The muscle is stimulated by single burst of controlled stimulation pulsewidth for each cycle to induce joint movement reaching the target extension knee angle. Therefore, the method is different from the

An outline of the discrete-time fuzzy control based cycle-to-cycle control is shown in Figure3. The controlled maximum joint angle of the previous cycle is delivered as feedback signal. Error is defined as difference between the target and measured joint angle. The controller will regulate the duration of stimulation pulsewidth based on the error and

traditional closed-loop control such as tracking control of desired angle trajectory.

**Table 1.** Anthropometric data of subject

previous flexion angle.

**2.2. Cycle to cycle controller development** 

**Figure 3.** Discrete-time FLC based cycle to cycle control

**Figure 1.** Schematic representation of the knee joint model

The inertial ( *M*<sup>i</sup> ) moment of the lower limb is represented as follows:-

$$M\_i = -m\_2 q\_2 \dot{\theta}\_1^2 r\_2 - I\_1 \ddot{\theta}\_1 - m\_1 r\_1^2 \ddot{\theta}\_1 - m\_2 q\_2^2 \ddot{\theta}\_1 \tag{3}$$

where, *m*<sup>1</sup> = shank mass, *m*<sup>2</sup> =foot mass, 1*I* = moment of inertia about COM, 1 =knee velocity, 1 =knee acceleration, *g* =gravity=9.81 m/s2.

Anthropometric measurements of length of the lower limb were made and this is shown in Table 1.

**Figure 2.** Lower limb model [4]

The knee joint model input is the stimulation pulsewidth as would be delivered in practice by an electrical stimulator. The complete model of knee joint thus developed is utilized as platform for simulation of the system and development of control approaches.


**Table 1.** Anthropometric data of subject

544 Fuzzy Controllers – Recent Advances in Theory and Applications

*M <sup>a</sup>*

**Figure 1.** Schematic representation of the knee joint model

The inertial ( *M*<sup>i</sup> ) moment of the lower limb is represented as follows:-

=knee acceleration, *g* =gravity=9.81 m/s2.

=ankle angle. Hence, the dynamics of motion can be represented in the simpler form

*MM* **ds**

1

(3)

<sup>2</sup>0

2*r*

, *M***<sup>i</sup>** *M <sup>g</sup>*

(2)

=knee

based on Kane's equations as in [13]. The gravitational ( *M*<sup>g</sup> ) moment is represented by:-

1 11 2 1 2 cos cos *Mg mg r mg q* 

2 22 *Mi mq r I mr mq* 2 2 1 2 1 1 11 1 2 2 1

Anthropometric measurements of length of the lower limb were made and this is shown in

*q*2

The knee joint model input is the stimulation pulsewidth as would be delivered in practice by an electrical stimulator. The complete model of knee joint thus developed is utilized as

platform for simulation of the system and development of control approaches.

1 *r*

where, *m*<sup>1</sup> = shank mass, *m*<sup>2</sup> =foot mass, 1*I* = moment of inertia about COM, 1

and 2 

velocity, 1 

Table 1.

**Figure 2.** Lower limb model [4]

## **2.2. Cycle to cycle controller development**

Researcher as in [15] highlighted cycle-to-cycle control as a method for using feedback to improve product quality for processes that are inaccessible within a single processing cycle but can be changed between cycles. The same concept has been applied in this study, where only reaching a target joint orientation through ballistic movement is taken into consideration rather than rigorously following a trajectory. The muscle is stimulated by single burst of controlled stimulation pulsewidth for each cycle to induce joint movement reaching the target extension knee angle. Therefore, the method is different from the traditional closed-loop control such as tracking control of desired angle trajectory.

An outline of the discrete-time fuzzy control based cycle-to-cycle control is shown in Figure3. The controlled maximum joint angle of the previous cycle is delivered as feedback signal. Error is defined as difference between the target and measured joint angle. The controller will regulate the duration of stimulation pulsewidth based on the error and previous flexion angle.

**Figure 3.** Discrete-time FLC based cycle to cycle control

## *2.2.1. Maximum flexion and extension detector*

The maximum flexion time detector will detect the time the angle reaches the peak of knee flexion for each cycle. The maximum extension signal and time detector will detect the peak angle of knee extension and the time the knee angle reaches this point for each cycle. The extension and flexion stages of the knee angle are shown in Figure 4.

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 547

the total pulses passed through the comparators and amplifying the signal with 220µs. The higher the controlled signal the more gates can be passed through and the wider the duration of pulse. Therefore the output of this converter is a single burst of controlled

The FLC-based cycle-to-cycle control was designed to achieve the following objectives:

Measured output of the controlled musculoskeletal system of the previous cycle is delivered as feedback signal. Proper value of signal is determined and regulated automatically by a Sugeno-type fuzzy controller using control rules as shown in Table 2. Input membership function is expressed as triangle fuzzy sets. Output membership function is expressed as fuzzy singletons. Input of fuzzy controller is aggregated by fuzzy inference using fuzzy

The fuzzy rules base directs control action based on error and flexion angle. The error will be higher when the muscle fatigues, in which case the response to a stimulation burst will

ii. Able to reach target extension angle thus maintain a steady swinging motion.

stimulation pulse duration with constant amplitude (220µs) for each cycle.

**Figure 5.** Signal comparator and shifting technique

*2.2.4. Controller objectives* 

i. Able to reach full knee extension

iii. Compensate for muscle fatigue iv. Compensate for spasticity

*2.2.5. Fuzzy controller design* 

rules to produce control action.

**Figure 4.** Extension and flexion stages

## *2.2.2. Activation switch*

Activation switch consists of hold and multiplier block to be active only when the knee angle reaches maximum flexion. The activation switch will hold the error signal and produce the output whenever it receives a signal from the maximum flexion time detector.

## *2.2.3. Amplitude to time duration convertor*

Amplitude to time duration converter is linearly converting the controlled signal (amplitude) from controller to time duration using signal comparator and shifting technique as shown in Figure 6.3. In this technique, first the controlled signal is compared with specific constant values for low to high in the parallel structure. Each comparator compares the controlled signal with the specific constant, if the controlled signal is greater than or equal to the specific constant then a single pulse will pass through the comparator. The first comparator compares the control signal with zero, if there is any signal from controller then the output will be a pulse with 0.05s width. The second comparator compares the control signal with specific constant and shift 0.05s and the next comparator compares and shifts by a further 0.05s. Then the resultant pulse duration for each cycle is obtained by summing up the total pulses passed through the comparators and amplifying the signal with 220µs. The higher the controlled signal the more gates can be passed through and the wider the duration of pulse. Therefore the output of this converter is a single burst of controlled stimulation pulse duration with constant amplitude (220µs) for each cycle.

**Figure 5.** Signal comparator and shifting technique

## *2.2.4. Controller objectives*

546 Fuzzy Controllers – Recent Advances in Theory and Applications

*2.2.1. Maximum flexion and extension detector* 

**Figure 4.** Extension and flexion stages

*2.2.3. Amplitude to time duration convertor* 

*2.2.2. Activation switch* 

extension and flexion stages of the knee angle are shown in Figure 4.

The maximum flexion time detector will detect the time the angle reaches the peak of knee flexion for each cycle. The maximum extension signal and time detector will detect the peak angle of knee extension and the time the knee angle reaches this point for each cycle. The

Activation switch consists of hold and multiplier block to be active only when the knee angle reaches maximum flexion. The activation switch will hold the error signal and produce the output whenever it receives a signal from the maximum flexion time detector.

Amplitude to time duration converter is linearly converting the controlled signal (amplitude) from controller to time duration using signal comparator and shifting technique as shown in Figure 6.3. In this technique, first the controlled signal is compared with specific constant values for low to high in the parallel structure. Each comparator compares the controlled signal with the specific constant, if the controlled signal is greater than or equal to the specific constant then a single pulse will pass through the comparator. The first comparator compares the control signal with zero, if there is any signal from controller then the output will be a pulse with 0.05s width. The second comparator compares the control signal with specific constant and shift 0.05s and the next comparator compares and shifts by a further 0.05s. Then the resultant pulse duration for each cycle is obtained by summing up The FLC-based cycle-to-cycle control was designed to achieve the following objectives:


#### *2.2.5. Fuzzy controller design*

Measured output of the controlled musculoskeletal system of the previous cycle is delivered as feedback signal. Proper value of signal is determined and regulated automatically by a Sugeno-type fuzzy controller using control rules as shown in Table 2. Input membership function is expressed as triangle fuzzy sets. Output membership function is expressed as fuzzy singletons. Input of fuzzy controller is aggregated by fuzzy inference using fuzzy rules to produce control action.

The fuzzy rules base directs control action based on error and flexion angle. The error will be higher when the muscle fatigues, in which case the response to a stimulation burst will change. To compensate for this changing system response, the stimulation burst time has to be increased such that shank can reach the desired angle in every cycle. The flexion angle was taken into account to rule out the disturbance due to spasticity. Combination of the information about error and knowledge about flexion angle will be necessary for controller to give an appropriate stimulation pulsewidth in compensation of muscle fatigue and motor reflexes. pulsewidth in compensation of muscle fatigue and motor reflexes.

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 549

was controlled using fuzzy controller to reach full extension angle. The full extension angle that can be achieved by paraplegic was defined as 10°. The computer simulation test was performed with stimulation course of 50 cycles. The first 5 cycles of the controlled swinging leg test of the full extension knee angle are shown in Figure 6.4. As can be seen, using FLCbased cycle-to-cycle control approach the first objective was achieved; to reach the full knee extension. The knee reached full extension at 3rd cycle and was able to maintain the

swinging motion without any predefined trajectory.

**Figure 6.** Controlled swinging leg for desired angle at 10° (full extension)

**3.1.2.1. Without presence of musle fatique and voluntary activation** 

The desired extension knee angle was set to be at 65° as considered in [16]. The test was initiated with stimulation pulse of 220µs amplitude with 0.25s burst duration for the first

The test is to achieve the target extension angle and thus maintain a steady swing of the shank without presence of musle fatique and voluntary activation. In the each test computer simulation was performed with stimulation course of 50 cycles. The first 10 cycles of the controlled swinging leg test of the knee joint at 65° is shown in Figure 7. As can be seen the cycle-to-cycle control approach can achieve the target extension angle at 3rd and thus maintain a steady swing of the shank. It is noted that the performance of the controller was

*3.1.2. Target extension angle and maintain steady swinging* 

cycle before activating the controller.

quite good and acceptable.


**Table 2.** Rules of Sugeno-type FLC

## **3. Results and discussion**

This discrete-time fuzzy logic cycle to cycle control technique emphasizes the view to overcome some drawbacks of trajectory based closed-loop FES control such as poor tracking, oscillating response and inability to reach full knee extension angle (Hatwell, 1991). Then the capabilities in compensating for muscle fatigue and spastisity are investigated. The ability of this control approach to realize the target joint orientation is assessed in simulation and experimental test as follows:-

## **3.1. Controllers' performance in simulation environment**

A complete set of non-linear dynamic equations of the knee joint model comprising the passive properties and active properties have been used in the simulations for purposes of controller development. Computer simulations are performed to assess the performance of the designed discrete-time fuzzy logic cycle-to-cycle control approach in generating stimulation burst durations for the desired extension angle. The simulations were carried out within the Matlab/Simulink environment. The muscle model was controlled by changing the pulse width; however the amplitude and the frequency of the stimulation pulses were constant. Here only the knee extensors are controlled by applying regulated stimulation pulsewidth to the quadriceps muscle model.

## *3.1.1. Full knee extension angle*

The test was initiated with stimulation pulse of 240µs amplitude with 0.3s burst duration for the first cycle of swing in gait before activating the controller. FES induced swinging motion was controlled using fuzzy controller to reach full extension angle. The full extension angle that can be achieved by paraplegic was defined as 10°. The computer simulation test was performed with stimulation course of 50 cycles. The first 5 cycles of the controlled swinging leg test of the full extension knee angle are shown in Figure 6.4. As can be seen, using FLCbased cycle-to-cycle control approach the first objective was achieved; to reach the full knee extension. The knee reached full extension at 3rd cycle and was able to maintain the swinging motion without any predefined trajectory.

548 Fuzzy Controllers – Recent Advances in Theory and Applications

**Error** 

Positive

Positive

Positive

**Table 2.** Rules of Sugeno-type FLC

**3. Results and discussion** 

*3.1.1. Full knee extension angle* 

assessed in simulation and experimental test as follows:-

stimulation pulsewidth to the quadriceps muscle model.

**3.1. Controllers' performance in simulation environment** 

change. To compensate for this changing system response, the stimulation burst time has to be increased such that shank can reach the desired angle in every cycle. The flexion angle was taken into account to rule out the disturbance due to spasticity. Combination of the information about error and knowledge about flexion angle will be necessary for controller to give an appropriate stimulation pulsewidth in compensation of muscle fatigue and motor

Negative Low Low Low Low Zero Zero Zero Low Low Low Low Zero Zero

Small Low Normal Normal Very Low Very Low Zero

Medium Low Normal Normal Normal Very Low Zero

This discrete-time fuzzy logic cycle to cycle control technique emphasizes the view to overcome some drawbacks of trajectory based closed-loop FES control such as poor tracking, oscillating response and inability to reach full knee extension angle (Hatwell, 1991). Then the capabilities in compensating for muscle fatigue and spastisity are investigated. The ability of this control approach to realize the target joint orientation is

A complete set of non-linear dynamic equations of the knee joint model comprising the passive properties and active properties have been used in the simulations for purposes of controller development. Computer simulations are performed to assess the performance of the designed discrete-time fuzzy logic cycle-to-cycle control approach in generating stimulation burst durations for the desired extension angle. The simulations were carried out within the Matlab/Simulink environment. The muscle model was controlled by changing the pulse width; however the amplitude and the frequency of the stimulation pulses were constant. Here only the knee extensors are controlled by applying regulated

The test was initiated with stimulation pulse of 240µs amplitude with 0.3s burst duration for the first cycle of swing in gait before activating the controller. FES induced swinging motion

Big Normal Normal Normal Normal Normal Zero

**Flexion angle** Very High High Normal Low Very Low Extremely Low

reflexes. pulsewidth in compensation of muscle fatigue and motor reflexes.

**Figure 6.** Controlled swinging leg for desired angle at 10° (full extension)

## *3.1.2. Target extension angle and maintain steady swinging*

The desired extension knee angle was set to be at 65° as considered in [16]. The test was initiated with stimulation pulse of 220µs amplitude with 0.25s burst duration for the first cycle before activating the controller.

#### **3.1.2.1. Without presence of musle fatique and voluntary activation**

The test is to achieve the target extension angle and thus maintain a steady swing of the shank without presence of musle fatique and voluntary activation. In the each test computer simulation was performed with stimulation course of 50 cycles. The first 10 cycles of the controlled swinging leg test of the knee joint at 65° is shown in Figure 7. As can be seen the cycle-to-cycle control approach can achieve the target extension angle at 3rd and thus maintain a steady swing of the shank. It is noted that the performance of the controller was quite good and acceptable.

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 551

**Figure 8.** Controller with simulated fatigue

knee angle (degree)

**Figure 9.** Controlled swinging leg with presence of fatigue

Since in practice spasticity is unpredictable, the motor reflex was simulated by multiplying a random number (between 0.1 to 1) with flexion angle as the second controller's input as in Figure 10. Controller has to compensate for the presence of spasticity in order to maintain swinging motion. Figure 11 shows the knee joint response obtained using this controller with the influence of muscle fatigue and voluntary activation. The presence of spasticity can be noted at 3rd cycle and for almost 5sec, then shank returning back to rest angle. When

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>50</sup>

time (s)

**Figure 7.** Controlled swinging leg for desired angle at 65° (normal extension)

#### **3.1.2.2. With presence of musle fatique and without presence of voluntary activation**

Muscle fatigue is an inevitable pitfall in FES induced control of movements. Fatigue is defined as the inability of a muscle to continue to generate a required force. It limits the duration FES can be effective by drastically reducing the muscle force output. It thus should be considered as an important criterion for FES induced movement control. The fatigue resistance of the control approaches were analyzed based on relative drop in knee extension by simulating the fatigue as in Figure 8. The fatigue simulation was to reduce muscle torque output to the 80% of total torque output at the end of simulation [5]. Figure 6.7 shows the knee joint response obtained using this controller, but with FES torque dropped down to 80% of its normal value. The effect of fatigue can be noted at 3rd cycle, the shank was unable to reach the target angle. Then controller has taken action to overcome this by increasing the stimulation burst time. Then, after 6th cycle the shank reached the target angle thus maintaining the swinging motion. As can be seen the controller performed very well in terms of robustness of the FLC in the presence of muscle fatigue.

#### **3.1.2.3. With presences of musle fatique and spasticity**

SCI muscle often exhibits spasticity [17] and may vary with time during cyclical motion [18]. Spasticity is defined as motor disorder characterized by a velocity-dependent increase in tonic stretch reflexes with exaggerated tendon jerks [19]. With regard to FES-approaches for function restoration, spasticity certainly represents a disadvantage especially flexion spasticity [20]. Muscle fatigue (as in previous test) and motor reflex to represent spasticity were simulated to assess the ability of controller to tackle these influences.

**Figure 8.** Controller with simulated fatigue

**knee angle (degree)**

**Figure 7.** Controlled swinging leg for desired angle at 65° (normal extension)

terms of robustness of the FLC in the presence of muscle fatigue.

were simulated to assess the ability of controller to tackle these influences.

**3.1.2.3. With presences of musle fatique and spasticity** 

**3.1.2.2. With presence of musle fatique and without presence of voluntary activation** 

Muscle fatigue is an inevitable pitfall in FES induced control of movements. Fatigue is defined as the inability of a muscle to continue to generate a required force. It limits the duration FES can be effective by drastically reducing the muscle force output. It thus should be considered as an important criterion for FES induced movement control. The fatigue resistance of the control approaches were analyzed based on relative drop in knee extension by simulating the fatigue as in Figure 8. The fatigue simulation was to reduce muscle torque output to the 80% of total torque output at the end of simulation [5]. Figure 6.7 shows the knee joint response obtained using this controller, but with FES torque dropped down to 80% of its normal value. The effect of fatigue can be noted at 3rd cycle, the shank was unable to reach the target angle. Then controller has taken action to overcome this by increasing the stimulation burst time. Then, after 6th cycle the shank reached the target angle thus maintaining the swinging motion. As can be seen the controller performed very well in

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>50</sup>

**time (s)**

SCI muscle often exhibits spasticity [17] and may vary with time during cyclical motion [18]. Spasticity is defined as motor disorder characterized by a velocity-dependent increase in tonic stretch reflexes with exaggerated tendon jerks [19]. With regard to FES-approaches for function restoration, spasticity certainly represents a disadvantage especially flexion spasticity [20]. Muscle fatigue (as in previous test) and motor reflex to represent spasticity

**Figure 9.** Controlled swinging leg with presence of fatigue

Since in practice spasticity is unpredictable, the motor reflex was simulated by multiplying a random number (between 0.1 to 1) with flexion angle as the second controller's input as in Figure 10. Controller has to compensate for the presence of spasticity in order to maintain swinging motion. Figure 11 shows the knee joint response obtained using this controller with the influence of muscle fatigue and voluntary activation. The presence of spasticity can be noted at 3rd cycle and for almost 5sec, then shank returning back to rest angle. When

severe spasticity happens, the controller will stop the stimulation because it may seriously hinder the swinging activity as well as for reason of safety. Once knee angle reaches the rest angle, the controller starts the stimulation. However, as can be seen in Figure 11 muscle spasm is noted leading to muscle fatigue. The controller was able to compensate for the presence of fatigue after few cycles. The controller performed well in terms of robustness of the FLC overcoming the influence of these phenomena after few cycles.

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 553

The laboratory apparatus built to study the knee joint control by FES is shown in Figure 12. The subject sat on a chair, which allowed the lower leg to swing freely, while the ankle angle was fixed at 0°. The knee extensors (quadriceps muscle group) were stimulated by a pair of surface electrodes (2"x5"). The cathode was placed on the motor point of rectus femoris and the anode was placed distally at the quadriceps tendon. Knee angle was defined in Figure 2,

The computer-controlled stimulator system consisted of a personal computer, computercontrolled interface (including analog-to-digital converter), current controlled stimulator and electro-goniometer (see Figure 12). The stimulation pulsewidth is generated by FLC based on the error by comparing the actual extension angle and the desired ones. All these operations were performed in the Matlab/Simulink environment in the computer. The Hasomed stimulator device was connected to PC via USB interface port. The knee joint angle was measured via the Biometric flexible electroganiometer mounted at approximate center of rotation of the knee joint. Stimulation pulsewidth ranged from 0 to 230µs and stimulation current was fixed to 40mA with a biphasic type pulse. The stimulation frequency was set to 25Hz and the knee joint angle sampling time was 0.05s. The experimental validation tests of the discrete-time fuzzy logic cycle to cycle control (based on simulation study) was assessed

The test was initiated with stimulation pulse of 240µs amplitude with 0.3s burst duration for the first cycle of swing in gait before activating the controller as in the simulation study.

**3.2. Experimental validation of controller** 

**Figure 12.** The equipment setup of this study

*3.2.1. Full knee extension angle* 

with when the lower leg was at rest during knee flexion (i.e., 90).

the capability of the controller to control the swinging motion as desired.

**Figure 10.** Controller with simulated fatigue and motor reflex

**Figure 11.** Controlled swinging leg with influence of fatigue and voluntary activation

## **3.2. Experimental validation of controller**

552 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 10.** Controller with simulated fatigue and motor reflex

knee angle (degree)

**Figure 11.** Controlled swinging leg with influence of fatigue and voluntary activation

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>50</sup>

time (s)

severe spasticity happens, the controller will stop the stimulation because it may seriously hinder the swinging activity as well as for reason of safety. Once knee angle reaches the rest angle, the controller starts the stimulation. However, as can be seen in Figure 11 muscle spasm is noted leading to muscle fatigue. The controller was able to compensate for the presence of fatigue after few cycles. The controller performed well in terms of robustness of

the FLC overcoming the influence of these phenomena after few cycles.

The laboratory apparatus built to study the knee joint control by FES is shown in Figure 12. The subject sat on a chair, which allowed the lower leg to swing freely, while the ankle angle was fixed at 0°. The knee extensors (quadriceps muscle group) were stimulated by a pair of surface electrodes (2"x5"). The cathode was placed on the motor point of rectus femoris and the anode was placed distally at the quadriceps tendon. Knee angle was defined in Figure 2, with when the lower leg was at rest during knee flexion (i.e., 90).

The computer-controlled stimulator system consisted of a personal computer, computercontrolled interface (including analog-to-digital converter), current controlled stimulator and electro-goniometer (see Figure 12). The stimulation pulsewidth is generated by FLC based on the error by comparing the actual extension angle and the desired ones. All these operations were performed in the Matlab/Simulink environment in the computer. The Hasomed stimulator device was connected to PC via USB interface port. The knee joint angle was measured via the Biometric flexible electroganiometer mounted at approximate center of rotation of the knee joint. Stimulation pulsewidth ranged from 0 to 230µs and stimulation current was fixed to 40mA with a biphasic type pulse. The stimulation frequency was set to 25Hz and the knee joint angle sampling time was 0.05s. The experimental validation tests of the discrete-time fuzzy logic cycle to cycle control (based on simulation study) was assessed the capability of the controller to control the swinging motion as desired.

**Figure 12.** The equipment setup of this study

## *3.2.1. Full knee extension angle*

The test was initiated with stimulation pulse of 240µs amplitude with 0.3s burst duration for the first cycle of swing in gait before activating the controller as in the simulation study.

FES-induced swinging motion was controlled using fuzzy controller to reach full extension angle at 10°. The shank was able to reach full knee extension at 4th cycle as can be seen in Figure 13. The ability of the cycle-to-cycle control approach to reach full knee extension has been demonstrated similar to that in [9].

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 555

was fully relaxed. Figure 14 shows the response of knee angle in the first scenario. The controller was able to perform a steady swinging motion of shank after 5s with ability to extend the knee to the desired extension angle. Hence this controller achieved the main objective; to maintain a steady swinging of the lower limb as desired but the controller

**3.2.2.2. With presence of musle fatique and without presence of voluntary activation** 

In the second part of test, the controller was validated when the muscle fatigue happened due to high stimulation intensity. This scenario can be seen by monitoring the reduction in the extension of knee joint. Few cycles of the stimulation course before and after fatigue were considered in this test as shown in Figure 15. It can been seen that at the beginning the knee angle was unable to reach the desired value due to fatigue, and then with the controller action by increasing the stimulation burst time the shank was able move a bit higher. After few cycles the knee angle reached the desired extension angle and swinging motion was maintained. Therefore, this controller achieved the third objective; to maintain a steady swinging of the lower limb as desired in presence of muscle fatigue. Only a small amount of

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>60</sup>

time (s)

In the final validation test, the controller was validated with presence of spasticity and muscle fatigue. This scenario can be seen by monitoring the knee angle at which spasticity stops the natural swing. Furthermore, high EMG activity was apparent throughout the trace. These uncontrolled muscle movements were brought by spasms causing fatigue. Thus, the reduction in the extension of knee joint can be seen after spasm. 15 cycles of the stimulation course before and after spasm were considered in this test as shown in Figure

needed more time to achieve this.

65

70

75

80

knee angle (degree)

85

90

95

**Figure 14.** Controlled swinging leg (Experimental work)

EMG activity was recorded in this scenario.

**3.2.2.3. With presence of musle fatique and voluntary activation** 

**Figure 13.** Controlled swinging leg for desired angle at 10° (full extension)

#### *3.2.2. Target extension angle and maintaining steady swinging*

The same procedure as in the simulation work was applied with the test initiated with 220µs amplitude with 0.15s burst durations of stimulation pulse. The controller performed in high intense stimulation course of 100 cycles as shown in Figure 14 in order to get influences of muscle fatigue and spasticity. EMG signal via surface electrodes on quadriceps muscle was recorded in this test to monitor EMG activity. The controller was tested in three scenarios, as in the simulations. Few trials were conducted in order to make sure the presence of these three scenarios was on the same stimulation course. An intra-trial interval for 120s was used to reduce the effect of fatigue in the beginning of the stimulation. Finally the best stimulation course with presence of both phenomena; muscle fatigue and spasticity were recorded. The recorded stimulation course was for almost 100 cycles and this was divided into three scenarios as follows:

#### **3.2.2.1. Without presence of musle fatique and voluntary activation**

In the first part of test, the controller was validated without fatigue and spasticity by considering only the first 10 cycles of the stimulation course. Before beginning the test, the subject was asked to relax as much as possible. There was no EMG present when the subject was fully relaxed. Figure 14 shows the response of knee angle in the first scenario. The controller was able to perform a steady swinging motion of shank after 5s with ability to extend the knee to the desired extension angle. Hence this controller achieved the main objective; to maintain a steady swinging of the lower limb as desired but the controller needed more time to achieve this.

**Figure 14.** Controlled swinging leg (Experimental work)

554 Fuzzy Controllers – Recent Advances in Theory and Applications

**Figure 13.** Controlled swinging leg for desired angle at 10° (full extension)

*3.2.2. Target extension angle and maintaining steady swinging* 

**3.2.2.1. Without presence of musle fatique and voluntary activation** 

into three scenarios as follows:

0

10

20

30

40

50

knee angle (degree)

60

70

80

90

100

The same procedure as in the simulation work was applied with the test initiated with 220µs amplitude with 0.15s burst durations of stimulation pulse. The controller performed in high intense stimulation course of 100 cycles as shown in Figure 14 in order to get influences of muscle fatigue and spasticity. EMG signal via surface electrodes on quadriceps muscle was recorded in this test to monitor EMG activity. The controller was tested in three scenarios, as in the simulations. Few trials were conducted in order to make sure the presence of these three scenarios was on the same stimulation course. An intra-trial interval for 120s was used to reduce the effect of fatigue in the beginning of the stimulation. Finally the best stimulation course with presence of both phenomena; muscle fatigue and spasticity were recorded. The recorded stimulation course was for almost 100 cycles and this was divided

time (s)

0 2 4 6 8 10 12

In the first part of test, the controller was validated without fatigue and spasticity by considering only the first 10 cycles of the stimulation course. Before beginning the test, the subject was asked to relax as much as possible. There was no EMG present when the subject

been demonstrated similar to that in [9].

FES-induced swinging motion was controlled using fuzzy controller to reach full extension angle at 10°. The shank was able to reach full knee extension at 4th cycle as can be seen in Figure 13. The ability of the cycle-to-cycle control approach to reach full knee extension has

#### **3.2.2.2. With presence of musle fatique and without presence of voluntary activation**

In the second part of test, the controller was validated when the muscle fatigue happened due to high stimulation intensity. This scenario can be seen by monitoring the reduction in the extension of knee joint. Few cycles of the stimulation course before and after fatigue were considered in this test as shown in Figure 15. It can been seen that at the beginning the knee angle was unable to reach the desired value due to fatigue, and then with the controller action by increasing the stimulation burst time the shank was able move a bit higher. After few cycles the knee angle reached the desired extension angle and swinging motion was maintained. Therefore, this controller achieved the third objective; to maintain a steady swinging of the lower limb as desired in presence of muscle fatigue. Only a small amount of EMG activity was recorded in this scenario.

#### **3.2.2.3. With presence of musle fatique and voluntary activation**

In the final validation test, the controller was validated with presence of spasticity and muscle fatigue. This scenario can be seen by monitoring the knee angle at which spasticity stops the natural swing. Furthermore, high EMG activity was apparent throughout the trace. These uncontrolled muscle movements were brought by spasms causing fatigue. Thus, the reduction in the extension of knee joint can be seen after spasm. 15 cycles of the stimulation course before and after spasm were considered in this test as shown in Figure 16. It can be seen that spasticity caused uncontrolled movement. The controller automatically stopped the stimulation due to spasticity. Once there was no motion of the knee, the controller started the stimulation with widest stimulation burst time to compensate for muscle fatigue. After 5s the knee angle reached the desired extension angle and swinging motion was maintained. Therefore, this controller achieved the last objective; to maintain a steady swinging of the lower limb as desired in presence of spasticity and muscle fatigue.

Discrete-Time Cycle-to-Cycle Fuzzy Logic Control of FES-Induced Swinging Motion 557

closed-loop control approach using fuzzy logic based cycle-to-cycle control for FES-induced motion control has been proposed. This control technique also emphasizes to overcome some drawbacks of the trajectory based closed-loop FES control. The objectives of this controller have been set to achieve full knee extension angle, to reach target extension angle thus maintain a steady swinging motion and to compensate for muscle fatigue and spasticity. The performance of the controller to achieve these objectives has been assessed through simulation study and validated through experimental work. The controller has been proved to achieve all these objectives. Besides its suitability in generating the target joint orientation, one of the attractions of the cycle-to-cycle control in FES application is the absence of any reference trajectory. Cycle-to-cycle control is easy to implement in practice. Additionally, this method can compensate for non-linearity and time-variance of response of electrically stimulated musculoskeletal system. This controller may be suitable not only for swinging but also for other FES control applications involving movement of cyclical

*Dept of Mechatronics and Robotics, Faculty of Electrical & Electronic Engineering,* 

*Department of Automatic Control and System Engineering, University of Sheffield,* 

electrical stimulation," J. Biomed. Eng., vol. 12, pp. 150–156, 1990.

B.S.K.K.Ibrahim greatly acknowledges the financial supports of Higher Education Ministry

[1] M. Levy, J. Mizrahi, and Z. Susak, "Recruitment force and fatigue characteristicsof quadriceps muscles of paraplegics isometrically activated by surface functional

[2] CL Lynch and MR Popovic Closed-Loop Control for FES: Past Work and Future Directions, 10th Annual Conference of the International FES Society, July 2005 –

[3] P.E. Crago,N. Lan, P. H. Veltink, J.J. Abbas and C.Kantor. New control strategies for neuroprosthetic systems, Journal Rehabilitation Res Device 33(2):158-72,1996. [4] J. J. Abbas, "Feedback control of coronal plane hip angle in paraplegic subjects using functional neuromuscular stimulation," IEEE Trans. Biomed. Eng., vol. 38, pp. 687–698,

*University Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia* 

of Malaysia and University Tun Hussein Onn, Malaysia.

M. O. Tokhi, M. S. Huq and S. C. Gharooni

nature.

**Author details** 

B. S. K. K. Ibrahim

*United Kingdom* 

**5. References** 

1991.

**Acknowledgement** 

Montreal, Canada.

**Figure 15.** Controlled swinging leg with presence of fatigue (Experimental work)

**Figure 16.** Controlled swinging leg with influence of fatigue and voluntary activation (Experimental work)

#### **4. Conclusion**

FES induced movement control is a difficult task due to the highly time-variant and nonlinear nature of the muscle and segmental dynamics. The great merit of a musculoskeletal model of knee joint is to serve for control development. In this study, a new closed-loop control approach using fuzzy logic based cycle-to-cycle control for FES-induced motion control has been proposed. This control technique also emphasizes to overcome some drawbacks of the trajectory based closed-loop FES control. The objectives of this controller have been set to achieve full knee extension angle, to reach target extension angle thus maintain a steady swinging motion and to compensate for muscle fatigue and spasticity. The performance of the controller to achieve these objectives has been assessed through simulation study and validated through experimental work. The controller has been proved to achieve all these objectives. Besides its suitability in generating the target joint orientation, one of the attractions of the cycle-to-cycle control in FES application is the absence of any reference trajectory. Cycle-to-cycle control is easy to implement in practice. Additionally, this method can compensate for non-linearity and time-variance of response of electrically stimulated musculoskeletal system. This controller may be suitable not only for swinging but also for other FES control applications involving movement of cyclical nature.

## **Author details**

556 Fuzzy Controllers – Recent Advances in Theory and Applications

muscle fatigue.

(Experimental work)

**4. Conclusion** 

16. It can be seen that spasticity caused uncontrolled movement. The controller automatically stopped the stimulation due to spasticity. Once there was no motion of the knee, the controller started the stimulation with widest stimulation burst time to compensate for muscle fatigue. After 5s the knee angle reached the desired extension angle and swinging motion was maintained. Therefore, this controller achieved the last objective; to maintain a steady swinging of the lower limb as desired in presence of spasticity and

**Figure 15.** Controlled swinging leg with presence of fatigue (Experimental work)

**Figure 16.** Controlled swinging leg with influence of fatigue and voluntary activation

FES induced movement control is a difficult task due to the highly time-variant and nonlinear nature of the muscle and segmental dynamics. The great merit of a musculoskeletal model of knee joint is to serve for control development. In this study, a new B. S. K. K. Ibrahim

*Dept of Mechatronics and Robotics, Faculty of Electrical & Electronic Engineering, University Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia* 

### M. O. Tokhi, M. S. Huq and S. C. Gharooni

*Department of Automatic Control and System Engineering, University of Sheffield, United Kingdom* 

## **Acknowledgement**

B.S.K.K.Ibrahim greatly acknowledges the financial supports of Higher Education Ministry of Malaysia and University Tun Hussein Onn, Malaysia.

## **5. References**

	- [5] M.S.Huq, Analysis and control of hybrid orthosis in therapeutic treadmill locomotion for paraplegia, PhD Thesis. The University of Sheffield, Sheffield, UK, 2009.

**Chapter 0**

**Chapter 24**

**Three Types of Fuzzy Controllers**

**Applied in High-Performance Electric**

José Luis Azcue, Alfeu J. Sguarezi Filho and Ernesto Ruppert

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48388

**1. Introduction**

fuzzy controllers.

**Drives for Three-Phase Induction Motors**

The electric drives are very common in industrial applications because they provide high dynamic performance. Nowadays exist a wide variety of schemes to control the speed, the electromagnetic torque and stator flux of three-phase induction motors. However, control remains a challenging problem for industrial applications of high dynamic performance, because the induction motors exhibit significant nonlinearities. Moreover, many of the parameters vary with the operating conditions. Although the Field Oriented Control (FOC) [16] schemes are attractive, but suffer from a major disadvantage, because they are sensitive to motor parameter variations such as the rotor time constant, and an incorrect flux estimation at low speeds. Another popular scheme for electric drives is the direct torque control (DTC) scheme [15][8], and an another DTC scheme based on space vector modulation (SVM) technique that reduces the torque ripples. This scheme does not need current regulators because its control variables are the electromagnetic torque and the stator flux. In this chapter we use the DTC-SVM scheme to analyze the performance of our proposed

In the last decade, there was an increasing interest in combining artificial intelligent control tools with conventional control techniques. The principal motivations for such a hybrid implementation were that fuzzy logic issues such as uncertainty (or unknown variations in plant parameters and structure) can be dealt with more effectively. Hence improving the robustness of the control system. Conventional controls are very stable and allow various design objectives such as steady state and transient characteristics of a closed loop system.

However, fuzzy controllers, unlike conventional PI controllers do not necessarily require the accurate mathematic model of the process to be controlled; instead, it uses the experience and knowledge about the controlled process to construct the fuzzy rules base. The fuzzy logic

and reproduction in any medium, provided the original work is properly cited.

©2012 Azcue et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

Several [5][6] works contributed to the design of such hybrid control schemes.

cited.

