**Fuzzy Adaptive Model Following Speed
Control for Vector Controlled Permanent Magnet Synchronous Motor**

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Abdelkader MEROUFEL, Ahmed MASSOUM, Baghdad BELABES

*Intelligent
Control & Electrical Power Systems Laboratory (ICEPS).*

*Faculty of
engineer sciences. Department of electrical engineering,*

*Djillali** **Liabes**
**University**, Sidi Bel Abbes, **Algeria*

E-mail:
__ameroufel@yahoo.fr__

**Abstract**

In this paper a hybrid controller combining a linear model following
controller (LMFC) and fuzzy logic control (FLC) for speed vector controlled
permanent magnet synchronous motor (PMSM) is described on this study. The FLC is introduced at the adaptive mechanism level. First, an LMFC system is designed to allow the
plant states to be controlled to follow the states produced by a reference
model. In the nominal conditions, the model following is perfect and the
adaptive mechanism based on the fuzzy logic is idle. Secondly, when parameter
variations or external disturbances occur, an augmented signal will be
generated by FLC mechanism to preserve the
desired model following control performance. The effectiveness and robustness
of the proposed controller is demonstrated by some simulation results*.*

**Keywords**

PMSM, Flux Oriented Control, FLC, LMFC and Adaptive Model Reference System

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**Introduction**

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With the development of the technology of power electronics control, rare earth magnetic materials and motor design, PMSM get wide applications in many control systems [1,2].They are perfect because the control system is usually less complex then that of field oriented induction motor drives. In typical PMSM drives, classical PI controllers have been used together with vector control method for speed control. However, the performances depend heavily on the motor parameters [3] which are time varying due to temperature rise and changes in motor drive operating conditions. Thus, it is desirable to have a robust controller for the drive system to reduce parameter sensitivity [4]. Adaptive control is an efficient technique for dealing with large parameter variations. The control input is designed to drive the controlled plant to track the response produced by the reference model [5,7]. Various control algorithms developed require the system states, thus they are not easy to implement [6]. To overcome this problem and to enhance the flexibility of changing control algorithm, a FLC is used to implement the adaptation mechanism. The main advantage of FLC resides in the fact that no mathematical modeling is required for the design of the controller. The FLC uses a control rules set that is based essentially on the knowledge of the system behavior and the experience of the control engineer. It has been pointed out that fuzzy controllers can provide high performance with reduced design and implementation complexity [7]. Then, in the proposed hybrid controller first LMFC is designed to allow the plant output to be controlled to follow the reference model output [8,9]. In the nominal conditions, the model following is perfect. But when parameters variations or external disturbance occur, an augmented signal will be generated automatically by the FLC adaptive mechanism which uses the error between plant output and the reference model output as input. The FLC adaptive mechanism output is added to LMFC system [9,10] in order to preserve the desired model following control performance. Under the proposed control Simulink scheme the decoupling control of torque and direct current in the field oriented mechanism is guaranteed and the robust control performance is obtained by the proposed hybrid controller.

This paper presents a theoretical study on an adaptive FLC for vector controlled PMSM drive using model reference adaptive approach. In the proposed controller, FLC is used to implement the adaptation mechanism.

Firstly, the model uncertainty of the PMSM is analyzed, and then vector control technique is presented and applied to drive the motor fed by PWM voltage source inverter. Secondly, the LMFC law is introduced for speed vector controlled PMSM, then FLC principle is proposed for the adaptation mechanism and its application to the speed control of adaptive model following controller. In order to simplify the realization, the controller is designed on the basis of the order-reduced model of the PMSM system. Finally, the control performance of the hybrid controller is evaluated by simulation under Matlab/Simulink software for different operating conditions. The results show that this method can control the PMSM system with uncertainty and parameter variations more effectively.

**Control of PMSM**** **

*Mathematical Model of the PMSM** *

The electrical and mechanical equations of the PMSM in the rotor (dq) reference frame are as follows:

** _{} **(1)

The mechanical equation can be written as:

_{} (2)

Where _{}is
the stator resistance, (_{}) are stator inductances in frame (d,q),
_{} is the
rotor speed, (_{})
are stator flux, _{}is the rotor flux, (_{}are respectively stator currents and
stator voltages in the frame (d,q) ,_{} is the electromagnetic torque, _{}is the load torque.

_{} are the rotor
moment inertia and the friction coefficient.

*Current Controller and Decoupling Compensation*

If a voltage source PWM inverter is used, the stator currents need to be controlled to track the command currents. As can be seen from (1), the dynamics of the stator currents with stator voltages as input are coupled and nonlinear. However, if the stator voltages commands are given in the form

_{} (3)

Where the emfs compensation are

_{}

Then the stator currents dynamics reduce to

_{} (4)

Since the current dynamics in (4) are linear and decoupled, PI controllers can be used for current tracking

_{} (5)

Figure 1 shows the block diagram of the decoupling system

*Figure
1. **Decoupling system with emf* *compensation*

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*Vector Control of the PMSM*

The objective of
the vector control of PMSM is to allow the motor to be controlled just like a
separately excited DC motor. So, the direct ‘d’ axis is aligned with permanent
magnet flux linkage phase and the direct current ‘_{}’ is forced to be zero. Then 1(b) can be
written as follows

_{} (6)

And the electromagnetic torque is

_{} (7)

Note that the electromagnetic torque equation is similar to that of DC motor

**Pwm Inverter**

Pulse Width Modulation (PWM) technique is used to generate the required
voltage or current to feed the motor or phase signals. This method is
increasingly used for AC drives with the condition that the harmonic current is
small as large as possible. Generally, the PWM schemes generate the switching
position patterns by comparing the three-phase sinusoidal wave forms with a
triangular carrier. The inverter model is represented by the relationship between
output phase voltages_{} and the control logic signals (_{}) as follows:

_{} (8)

Where_{}:
rectified voltage, _{}: logic signals

**LMFC for Vector Controlled PMSM**

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*LMFC Theory*

Suppose that the plant and the chosen reference model are expressed as [3-5]

_{} (9)

_{} (10)

Where _{},
_{},_{}, _{}, _{}, _{}, and _{}are constant matrices of
appropriate dimensions. The pairs_{} and _{} are stabilizable and _{}is a stable matrix.

The objective is to find the control input _{} such that the plant states can track
those of the reference model. Then the resulting _{}will fellow _{} automatically. For easy
implementation, the control input _{} is chosen to be

_{} (11)

Where _{} is
the error between the system output and the model output. Define the error
vector

_{} (12)

Then from (9), (10) and (11), we can obtain the following equation:

_{} (13)

Equation (13) shows that if _{} are chosen to let

-_{}be a Hurwitz matrix and (14)

-_{} (15)

-_{} (16)

Where_{}
is the left pseudo inverse matrix of _{}, then the error system of (13) will be
asymptotically stable and the output of the controlled plant will follow that
of the reference model.

**MRAFLC for PMSM**

The linear model following control system proposed above can lead to
perfect model following characteristics only when the plant is invariant. Thus,
an adaptation signal_{} is added to the control law (11). The
added signal is generated from the adaptive fuzzy controller mechanism and it
is included to the LMFC system to reduce the model following error due to the
uncertainties in the plant. A block diagram of the proposed hybrid controller
is shown in figure 2

*Figure 2.**
Proposed adaptive fuzzy controller with LMFC*

* *

The error between the model output _{} and the actual speed _{}and its change are
calculated every sampling period as

_{} (17)

The error _{} and
the change in error _{} will be processed by the fuzzy rule
based adaptation system to produce a correction term _{} which is added to the LMFC output
_{}. The
hybrid controller output is thus modified so that the closed loop system
behaves like the reference model. The control signal is a sum of two terms

_{} (18)

Where

_{} (19)

_{} is
the fuzzy adaptive mechanism output

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**Principle of FLC**

The design of FLC dos not requires mathematical modelling. The formulation of the control rules is based on the knowledge of the PMSM drive and the experience of the control engineer.

*Fuzzy Logic Controller Structure *

The FLC has three functional blocks as shown in figure 3

*Figure 3**. FLC internal structure*

In the fuzzification block, the inputs and output crisp variables are converted into fuzzy variables ‘e’, ‘de’ and ‘du’ using the triangular and the trapezoidal membership functions shown in figure 4 (a)

*Figure 4**.
(a) Membership functions (b) Control surface*

Each universe of discourse is divided into three fuzzy sets: Negative (N), Zero (Z) and Positive (P). The fuzzy variable ‘e’ and ‘de’ produced the fuzzification block are then processed by an inference mechanism that executes a set of control rules contained in (3x3) table as shown in table 1.

Table 1 Fuzzy control rules for ‘du’

The fuzzy rules are expressed under the IF-THEN form. The crisp output of the FLC is obtained by using Max-Min inference algorithm and the center of gravity defuzzification approach.

*FLC Design*

The fuzzy controller behaviour depends on the membership functions, their
distribution and the rules that influence the fuzzy variable in the system.
There is no formal method to determine accurately the parameters of the
controller. Tuning the FLC is an iterative process requiring trial several
combinations of membership functions and control rules. The adjustment can be
done by observing the response of the system regulator and modifying the fuzzy
sets in the universes of discourse of the input variables (_{}
and_{}) and output variable (_{})
until satisfactory response is obtained. The control surface 4 (b), a three
dimensional graphic showing the output variable corresponding to all
combinations of values of the inputs can be used to facilitate the FLC tuning.
The number of rules can be reduced in order to optimize the inference engine
execution speed. In this paper, a trial and error approach is used to determine
and adjust the weighting factors_{} [6, 8].

**Model Reference Adaptive Fuzzy Logic Controller**

The reference model is used to specify the desired performance that satisfies design specifications. A fuzzy logic adaptation loop is added in parallel to the LMFC feedback loop [8-10]. In the nominal case, the model following is perfect and the fuzzy controller adaptation loop is idle. When parameter change an adaptation signal produced by adaptation mechanism will be added to the output signal of the LMFC to preserve the desired model following control performance [8-10]. Figure 5 shows a Simulink block diagram of the proposed hybrid controller for vector control PMSM. The reference model chosen is first order transfer function with time constant set at 0.7s.

*Figure 5.**
Simulink model of proposed hybrid controller for vector controlled*

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**Simulation Results**

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The control performance of the proposed scheme in figure 5 is evaluated by simulation using Matlab/Simulink software. The parameters of the PMSM are as follows:

_{}

In order to valid the adaptive control law method for a wide operating domain, we use the reference profiles shown in figure 6 as command input. The robustness is evaluated by using increasing inertia (3*J), stator resistance augmented +50% and variation load 10Nm.

*Figure
6**.:Reference profile inputs*

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*Figure 7:**
Speed responses for vector control of PMSM*

*(a) and (b) LMFC,
(c) hybrid controller*

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The simulation results without and with the proposed controller in the above three cases are shown in fig.7. When the moment of inertia increases (3*J) the LMFC response becomes oscillatory. Whoever, the robust control performance of the hybrid controller in the command tracking is obvious.

*Figure 8.**
Responses of MRAFLC for vector control of PMSM*

*under abruptly
step load variation*

*Figure 9.**
Responses of MRAFLC for vector control of PMSM under abruptly step load
variation, increasing inertia 3**J

*Figure 10**.
Responses of MRAFLC for vector controlled PMSM under abruptly step load
variation, augmented inertia 3*J, and increasing stator resistance +50%*

Fig.8, 9 and 10 show the robustness of the speed response in the case of external and internal disturbances. The system output tracks very closely the reference model even with increasing inertia, augmented stator resistance and load variations. The results prove the effectiveness of the fuzzy adaptive mechanism facing to the different perturbations.

**Conclusion**

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A hybrid controller combining the advantages of fuzzy logic control and model reference adaptive control for speed vector controlled PMSM fed by voltage source inverter has been proposed in this paper. The proposed controller is insensitive to the external and internal system parameter variations and this proves its robustness. The results obtained show that

The decoupling is maintained under internal and external disturbances.

The combination of LMFC and FLC permit to ovoid the problem of flux orientation and the uncertainties in the model representatively

This strategy of control gives a stable system with a satisfactory performance either with or without load variation. The proposed scheme is effective only during transients since the parameters of the speed LMFC controller are not upgraded by the adaptation mechanism.

The simulation results have been confirmed the efficiency of the proposed adaptation scheme in maintaining good performance under external and internal disturbances.

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