Rule base for fuzzy controller (I), output

## 1. Introduction

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 fuzzy controllers.

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

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 controllers are a good alternative for motor control systems since they are well known for treating with uncertainties and imprecisionï¿½’s. For example, in [1] the PI and fuzzy logic controllers are used to control the load angle, which simplifies the induction motor drive system. In [7], the fuzzy controllers are used to dynamically obtain the reference voltage vector in terms of torque error, stator flux error and stator flux angle. In this case, both torque and stator flux ripples are remarkably reduced. In [10], the fuzzy PI speed controller has a better response for a wide range of motor speed and in [3] a fuzzy self-tuning controller is implemented in order to substitute the unique PI controller, present in the DTC-SVM scheme. In this case, performance measures such as settling time, rise time and ITAE index are lower than the DTC-SVM scheme with PI controller.

The fuzzy inference system can be used to modulate the stator voltage vector applied to the induction motor [18]. In this case, unlike the cases mentioned above, the quantity of available vectors are arbitrarily increased, allowing better performance of the control scheme and lower levels of ripple than the classic DTC. However, it requires the stator current as an additional input, increasing the number of input variables. In this chapter we design and analyze in details three kinds of fuzzy controllers: the PI fuzzy controller (PI-F), the PI-type fuzzy controller (PIF) and the self-tuning PI-type fuzzy controller (STPIF). All of these fuzzy controllers are applied to a direct torque control scheme with space vector modulation technique for three-phase induction motor. In this DTC-SVM scheme, the fuzzy controllers generate corrective control actions based on the real torque trend only while minimizing the torque error.

## 2. Background

### 2.1. The three-phase induction motor dynamical equations

By the definitions of the fluxes, currents and voltages space vectors, the dynamical equations of the three-phase induction motor in stationary reference frame can be put into the following mathematical form [17]:

Where

The electromagnetic torque

Where

The three-phase induction motor model was implemented in MATLAB/Simulink as is shown in [2].

### 2.2. The principle of direct torque control

In the direct torque control if the sample time is short enough, such that the stator voltage space vector is imposed to the motor keeping the stator flux constant at the reference value. The rotor flux will become constant because it changes slower than the stator flux. The electromagnetic torque (6) can be quickly changed by changing the angle

For simplicity, let us assume that the stator phase ohmic drop could be neglected in (1). Therefore

Thus the stator flux space vector moves by

## 3. Direct torque control scheme with space vector modulation technique

In Fig. 1, we show the block diagram for the DTC-SVM scheme [14] with a Fuzzy Controller, the fuzzy controller will be substitute for the three kind of proposed Fuzzy Controller one for time. The DTC-SVM scheme is an alternative to the classical DTC schemes [15], [8] and [9]. In this one, the load angle

### 3.1. Flux reference calculation

In stationary reference frame, the stator flux reference

In this chapter we consider the magnitude of stator flux reference as a constant. Therefore, we can use the relation presented in equation (8) to calculate the stator flux reference vector.

Moreover, if we consider the stator voltage

### 3.2. Stator voltage calculation

The stator voltage calculation uses the DC link voltage (

### 3.3. Electromagnetic torque and stator flux estimation

As drawn by Fig. 1 the electromagnetic torque and the stator flux estimation depend on the stator voltage and the stator current space vectors,

The problem with this kind of estimation is that for low speeds the back electromotive force (emf) depends strongly of the stator resistance, to resolve this problem is used the current model to improve the flux estimation as in [13]. The rotor flux

Notice that

In the current model the stator flux is represented by:

Where

With the aim to correct the errors associated with the pure integration and the stator resistance measurement, the voltage model is adapted through the PI controller.

The estimator scheme shown in the Fig. 2 works with a good performance in the wide range of speeds.

Where LPF means low pass filter. In the other hand, when equations (14) and (16) are replaced in (5) we can estimated the electromagnetic torque

## 4. Design of fuzzy controllers

### 4.1. The PI fuzzy controller (PI-F)

The PI fuzzy controller combines two simple fuzzy controllers and a conventional PI controller. Note that fuzzy controllers are responsible for generating the PI parameters dynamically while considering only the torque error variations. The PI-F block diagram is shown in Fig. 3, this controller is composed of two scale factors

Then, considering that the fuzzy controller output is a normalized value

However, for different reference values the range for the proportional gain values is chosen as

Due to nonlinearities of the system and in order to avoid overshoots for large reference torque

#### 4.1.1. Membership Functions (MF)

In the Fig. 3, the first fuzzy controller receives as inputs the errors

The output has two fuzzy sets whose linguistic values are associated with them are S-small and B-Big, respectively. Both have trapezoidal membership functions as shown in Fig.5. The universe of discourse of the fuzzy sets is defined over the closed interval

Similarly, the second fuzzy controller has the same fuzzy sets for its two inputs, however, its output is defined by three constant values defined as

#### 4.1.2. Scaling Factors (SF)

The PI-F controller has two scaling factors,

The scale factor ensures that both inputs are within the universe of discourse previously defined. The parameters

#### 4.1.3. The rule bases

The rules are based on simulation that we conducted of various control schemes. Fig.6 shows an example for one possible response system. Initially, the error is positive around the point

To produce a large signal control, the PI controller should have a large gain

The rule base for the first fuzzy controller is in Table 1, also, the rule base for the second fuzzy controller is in Table 2.

N | ZE | P | |

N | B | B | B |

ZE | S | B | S |

P | B | B | B |

N | ZE | P | |

N | S | S | S |

ZE | B | M | B |

P | S | S | S |

Fig. 7 shows the control surface for the first and second fuzzy controllers. This figure clearly shows the non-linear relationship between (

### 4.2. The PI-type fuzzy controller (PIF) and The self-tuning PI-type fuzzy controller (STPIF)

The PI-type fuzzy controller (PIF) is a fuzzy controller inspired by a digital PI controller, which is depicted in Fig. 8. It is composed by two input scale factors "

This controller has a single input variable, which is the torque error "

In (23),

Fig.9 shows the self-tuning PI-type fuzzy controller (STPIF) block diagram, its main difference with the PIF controller is the gain tuning fuzzy controller (GTF) block.

#### 4.2.1. Membership Functions (MF)

The MF for PIF controller are shown in Fig. 10(a). This MF for input variables "

The MF’s for GTF controller are shown in Fig. 10(a) and in Fig. 10(b) for input and output variables respectively. Input variables "

Most of the MF variables have triangular shape [Fig. 10] with 50% overlapping neighbor functions, except the extremes which are trapezoidal. The linguistic variables are referred to as: NL-Negative Large, NM-Negative Medium, NS-Negative Small, ZE-Zero, VS-Very Small, S-Small, SL-Small Large and so on as shown in Table 3 and in Table 4.

#### 4.2.2. Scaling factors

The two inputs SF "

#### 4.2.3. The rule bases

The incremental change in the controller output

Where

Where

NL | NM | NS | ZE | PS | PM | PL | |

NL | NL | NL | NL | NM | NS | NS | ZE |

NM | NL | NM | NM | NM | NS | ZE | PS |

NS | NL | NM | NS | NS | ZE | PS | PM |

ZE | NL | NM | NS | ZE | PS | PM | PL |

PS | NM | NS | ZE | PS | PS | PM | PL |

PM | NS | ZE | PS | PM | PM | PM | PL |

PL | ZE | PS | PS | PM | PL | PL | PL |

NL | NM | NS | ZE | PS | PM | PL | |

NL | VL | VL | VL | L | SL | S | ZE |

NM | VL | VL | L | L | ML | S | VS |

NS | VL | ML | L | VL | VS | S | VS |

ZE | S | SL | ML | ZE | ML | SL | S |

PS | VS | S | VS | VL | L | ML | VL |

PM | VS | S | ML | L | L | VL | VL |

PL | ZE | S | SL | L | VL | VL | VL |

#### 4.2.4. Gain tuning fuzzy

The purpose of the GTF controller is update continuous the value of

The GTF controller rule base is based on knowledge about the three-phase IM control, using a DTC type control according to the scheme proposed in [14], in order to avoid large overshoot and undershoot, e.g., when "

On the other hand, when "

The nonlinear relationship between "

The inference method used in PIF and GTF controllers is the Mamdani’s implication based on max-min aggregation. We use the center of area method for defuzzification.

## 5. Simulation results

We have conducted our simulation with MATLAB simulation package, which include Simulink block sets and fuzzy logic toolbox. The switching frequency of the pulse width modulation (PWM) inverter was set to be 10kHz, the stator reference flux considered was 0.47 Wb. In order to investigate the effectiveness of the three proposed fuzzy controllers applied in the DTC-SVM scheme we performed several tests.

We used different dynamic operating conditions such as: step change in the motor load (from 0 to 1.0 pu) at 90 percent of rated speed, no-load speed reversion (from 0.5 pu to -0.5 pu) and the application of a specific load torque profile at 90 percent of rated speed. The motor parameters used in the tests are given in Table 5.

Fig. 12, shows the response of the speed and electromagnetic torque when speed reversion for DTC-SVM with PI-F controller is applied. Here, the rotor speed changes its direction at about 1.8 seconds. Fig. 13 shows the stator and rotor current sinusoidal behavior when applying reversion.

Fig. 14 and Fig. 15 show the torque and currents responses respecively, when step change is applied in the motor load for DTC-SVM with the PI-F controller. This speed test was established at 90 percent of rated speed.

In Fig. 16, we demonstrate the speed response when applying a speed reversion for DTC-SVM with PIF controller. In this case the speed of the rotor changes its direction at about 1.4 seconds. Fig. 17 shows the electromagnetic torque behavior when the reversion is applied.

Fig. 18 and Fig. 19 show the response of the electromagnetic torque and phase

Fig. 20 shows the behaviors of the electromagnetic torque, phase

Fig. 21 presents the results when a specific torque profile is imposed to DTC-SVM scheme with STPIF controller. In this case the electromagnetic torque follow the reference.

Fig. 22 illustrates the response of the electromagnetic torque for the DTC-SVM scheme with STPIF controller, when applying step change in the motor load. In this test we used the rise time

Rated voltage (V) | 220/60Hz |

Rated Power (W) | 2238 |

Rated Torque (Nm) | 11.9 |

Rated Speed (rad/s) | 179 |

0.435, 0.816 | |

0.002, 0.002 | |

0.0693 | |

0.089 | |

P | 2 |

## 6. Conclusion

In this chapter we have presented the DTC-SVM scheme that controls a three-phase IM using three different kinds of fuzzy controllers. These fuzzy controllers were used in order to determinate dynamically and on-line the load angle between stator and rotor flux vectors. Therefore, we determine the electromagnetic torque necessary to supply the motor load. We have conducted simulations with different operating conditions. Our simulation results show that the all proposed fuzzy controllers work appropriately and according to the schemes reported in the literature. However, the STPIF controller achieves a fast torque response and low torque ripple in a wide range of operating conditions such as: sudden change in the command speed and step change of the load.

## Acknowledgement

The authors are grateful to FAPESP and CAPES for partially financial support.