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| ARTICLE |
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| Year : 2011 | Volume
: 57
| Issue : 5 | Page : 443-451 |
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Fuzzy Tracking Control for Indirect Field-oriented Induction Machine Using Integral Action Performance
Moez Allouche1, Mohamed Chaabane1, Mansour Souissi1, Driss Mehdi2
1 University of Sfax, National Schools of Engineers of Sfax, Laboratory of Sciences and Techniques of Automatic and Computer Engineering, B.P. 1173, 3038, Tunisia
2 Laboratory of Automatic Informatics Industrial, L.A.I.I, Superior School of Engineer of Poitiers, 40 Avenue du Recteur Pineau, 86022 Poitiers, France
| Date of Web Publication | 24-Nov-2011 |
Correspondence Address:
Moez Allouche
University of Sfax, National Schools of Engineers of Sfax, Laboratory of Sciences and Techniques of Automatic and Computer Engineering, B.P. 1173, 3038
Tunisia
 DOI: 10.4103/0377-2063.90164
 Abstract | | |
This paper deals with the synthesis of fuzzy controller applied to the induction motor with a guaranteed H∞ tracking performance. First, the Takagi-Sugeno fuzzy model is employed to approximate a nonlinear system in the synchronous d-q frame rotating with electromagnetic field-oriented. Next, a fuzzy observer-based fuzzy tracking controller is designed to stabilize the induction motor and guaranteed a minimum disturbance attenuation level for the closed-loop system. The rotor flux is unavailable for measurement and it is estimated by a fuzzy observer. An integral action is added to the new parallel distributed compensation fuzzy controller related to the tracking to avoid static errors. The gains of fuzzy control and fuzzy observer are obtained by solving a set of Linear Matrix Inequality. Finally, simulation result is given to demonstrate the controller's effectiveness. Keywords: H8734; performance, Linear matrix inequality, Parallel distributed compensation, Takagi-sugeno fuzzy model, Tracking control
How to cite this article:
Allouche M, Chaabane M, Souissi M, Mehdi D. Fuzzy Tracking Control for Indirect Field-oriented Induction Machine Using Integral Action Performance. IETE J Res 2011;57:443-51 |
How to cite this URL:
Allouche M, Chaabane M, Souissi M, Mehdi D. Fuzzy Tracking Control for Indirect Field-oriented Induction Machine Using Integral Action Performance. IETE J Res [serial online] 2011 [cited 2013 Jun 19];57:443-51. Available from: http://www.jr.ietejournals.org/text.asp?2011/57/5/443/90164 |
| 1. Introduction | |  |
Induction motors have been widely applied as the electromechanical actuators because of their ruggedness, easy maintenance, and low cost. Control of induction motor is well known to be difficult owing to the fact that their dynamics are highly nonlinear and coupled, not all state variables (rotor flux) are available for measurement and electrical parameters drift with temperature. In the last decades, the problem of tracking control for induction motor has been dealt with several approaches, and many studies have been developed around this subject. For examples, in the study by Wal and Chang [1] , an adaptive observation system based on model adaptive reference system is proposed in which the rotor time constant is estimated and the decoupled performance was guaranteed. Marino et al. [2],[3] proposed a direct adaptive controller for speed regulation, in which the motor model is input-output decoupled by a feedback-linearizing technique, while the load torque and rotor resistance are adapted in time. However, the asymptotic convergence of the system states and the estimated parameters is unsure. H∞ control schemes for induction motor have been introduced in the studies by Chiaverini and Fusco and Prempaina et al. [4],[5] , to eliminate the effects of the external disturbance and to achieve tracking purpose. Moreover, based on the input-output linearization approach, a mixed H2/H∞ technique with a reference model is used [6] to ensure the tracking error performance and the load torque disturbance rejection. A robust stability analysis method through parameter-dependant Lyapunov function is presented, in which the robust stability of the overall uncertain linearized closed-loop system is tested. Sliding mode technique has been widely used in induction motor control and proved a small tracking error [7],[8],[9] , but chattering phenomenon is inevitable which degrades performances of the system and may even lead to instability. Recently, substantial research efforts have been devoted to intelligent controllers such as artificial neural networks and fuzzy logic to deal with the problems of nonlinearity and uncertainly of system parameters. Among various fuzzy modeling methods, the well-known Takagi-Sugeno (T-S) fuzzy model is considered as a popular and powerful tool in approximating a complex nonlinear system [10],[11] . They correspond to a collection of linear models blended together with scalar-positive nonlinear functions satisfying a convex sum property. Based on the T-S fuzzy model, a variety of approaches have been proposed for tracking problem for nonlinear systems. Tseng et al. [12] proposed a fuzzy tracking control design for nonlinear system through T-S fuzzy model. Using a variable structure control technique, an output tracking control problem for nonlinear system of both parameter perturbations and external disturbances is studied [13] . Chong et al. [14] presented an H∞ output tracking control for nonlinear time-delays systems. Guerra and Toulotte and Khiar, et al [15],[16], proposed a new parallel distributed compensation (PDC) fuzzy controller in which an integrator action is added. However, the design controller supposed that all the states are available for feedback.
The aim of this paper is to design a fuzzy observer-based fuzzy tracking controller for induction motor in which an integrator action is added. The rotor flux is unavailable for measurement and it is estimated by a fuzzy observer. The fuzzy tracking control design problem is parameterized in terms of a linear matrix inequality (LMI) problem which can be solved very efficiently using the convex optimization techniques.
This paper is organized as follows: In section 2, an open-loop control strategy is presented which includes a physical model of the induction motor. The nonlinear induction motor is represented by an equivalent T-S type fuzzy model in Section 3. Section 4 is reserved to the fuzzy observer design. The synthesis of the fuzzy controller with H∞ performance is formulated in section 5. In section 6, simulation result is given to highlight the effectiveness of the proposed control law. The last section gives a conclusion on the main works developed in this paper.
| 2. Open-Loop Control Strategy | | |
2.1 Physical Model of Induction Motor
Under the assumptions of linearity of the magnetic circuit, the electromagnetic dynamic model of the induction motor in the synchronously d-q reference frame can be described as follows:
In which, wm is rotor speed, ws is the electrical speed of stator, (Ψrd, Ψrq) are the rotor fluxes, (isd, isq) are the stator currents, and (usd, usq) are the stator voltages. The load torque Cr is a step disturbance. The motor parameters are moment of inertia J, rotor and stator resistances (Rs, Rr) inductances, (Ls, Lr) mutual inductance M, friction coefficient f, and number of poles pairs np.
2.2 Open-loop Control
In this paragraph, we explore the structure of the open-loop control in the goal to generate the reference signals. By replacing the state variables of the motor  by their reference signals  in (1), we obtain the following reference model [17] :
The two last equations in (2) lead to the stator current reference:
According to (2), the electrical speed reference of the stator is:
From the first two equations in (2), we obtain the expression of the open-loop control:
| 3. Takagi-Sugeno Fuzzy Model of Induction Motor | | |
The T-S fuzzy model is employed here to approximate the induction model system in the synchronous d-q frame rotating with electromagnetic field-oriented. The principle behind field-oriented control is that the machine flux and torque are controlled independently, in a similar manner to a separately exited DC machine. Consequently, the rotor flux vector (Ψrd, Ψrq is aligned to the d-axis and the following results can be obtained:
In the other part, the electrical speed of the stator defined in the synchronous d-q frame rotating with electromagnetic field-oriented is obtained from (4) with replacing wmc by wm and isqc by the current measurement as follows:
Substituting the electrical speed of the stator (7) in the physical model (1) leads to the following equivalent state space:
Where, v(t) denotes the measurement noises that envelop the stator currents and the rotor speed.
Considering the sector of nonlinearities of the terms of  the matrix A(x(t)) with j = 1, 2, 3:
Thus, we can transform the nonlinear terms under the following shape:
Where,
The fuzzy model is described by fuzzy If-Then rules and will be employed here to deal with the control design problem for the induction motor. The i th rule of the fuzzy model for the nonlinear system is of the following form:
Rule Ri
The global fuzzy model is inferred as follows:
Before designing the fuzzy observer-based fuzzy tracking controller, some assumptions are made as follows:
A.1 The reference signal xr(t) is smooth and bounded
A.2 The external disturbance w(t) is bounded
A.3 The stator current and rotor speed are measurable
Define the tracking error as
According to the previous assumptions, by differencing (17), we get
| 4. Fuzzy Observer Design | | |
In this section, we will design the fuzzy observer to estimate the immeasurable states of rotor flux. According to the fuzzy model (12), the structure of the fuzzy observer is given as follows:
Rule Ri
Where,  and  denote the estimations of x(t) and y(t). Li is an observer gain to be determined.
The inferred output of the observer is:
Define the estimation error state as:
By differencing (21), we obtain the error dynamics:
| 5. Fuzzy Controller Design | | |
In this section, we firstly establish matrix inequality conditions for the design of fuzzy controllers so that the H∞ tracking performance index p is achieved. Then, we give an LMI-procedure to solve the matrix inequalities. The fuzzy controller is assumed to incorporate information from  and xr(t), it shares the same fuzzy sets with the fuzzy model. In order to avoid static errors, an integral action is added to the new PDC fuzzy controller related to the tracking error. In this case, the fuzzy controller is designed as [18] :
Then,
Hence, the overall fuzzy controller is given by:
From (17), we get:
Equation (21) gives:
Subtracting (26) from (25), we get
Then, relation (24) becomes
Substituting (28) into (18) yields
After manipulation, the augmented system can be expressed as the following form:
It is known that the effect of  will deteriorate the control performance of the fuzzy control system and even lead to instability of the control system. Therefore, H∞ control is the most important control design to efficiently eliminate the effect of  on the control system. If we consider the initial condition, the H∞ performance is defined as follows [12] :
The objective is to determine the Ki and Li for the augmented model (30) such as the H∞ performance is guaranteed.
Theorem 1
If there exists symmetric and positive definite matrices  and a prescribed positive constant p2 such that the following matrix inequality are satisfied
Then, the H∞ tracking control performance for the prescribed p2 is guaranteed and the quadratic stability for the closed-loop system is assured.
Theorem 1 provides a matrix inequality condition for the considered H∞ tracking performance (31). However, it does not give the methods of obtaining the solution of controller gains and observer gains. To facilitate the resolution, the matrix variable  is chosen diagonal with respect to appropriate matrix blocks [12],[14] :
By substituting (33) into (32), we obtain:
We consider the change of variable, Zi = P1Li . By the Schur complement, (34) is equivalent to
S 44 and S 55 contain design variables, P2 and K i we need to seek methods to decouple them. There are no effective algorithms for solving them simultaneously. However, we can solve them by the following two-step procedure. First, we can find P2 and Ki from the diagonal block S44 then, P1, P3 and Li from the inequality (35).
After congruence (35) with diag  considering the change of variable  and using the Schur complement, S44 is equivalent to the following LMI:
The parameters  are obtained by solving LMIs in (36). In the second step, by substituting P2 and Ki into (35), the last condition become a standard LMI and we can easily solve P1, P3, and Li .
| 6. Simulation Results | | |
Numerical simulations have been presented to prove the performance of the proposed fuzzy controllers both in terms of tracking performance and in terms of load torque rejection capability [Figure 1]. The characteristics of the induction motor parameters are listed as below:
The closed-loop simulation results are shown in [Figure 2](a-h). Solving LMIs (35) and (36) by LMI optimization algorithm, controller gains and observer gains matrices can be obtained as:
 | Figure 2: Simulation results for a step of speed change (100 rad/s toward 150 rad/s).
Click here to view |
The speed reference goes from zero to 100 rad/ sec at origin of the times and grows up to 150 rad/sec at 10 seconds. From time to , a constant load torque of 5 N.m value is applied. The simulation results illustrate the performances of the developed approach in terms of tracking error and disturbance rejection. Indeed, [Figure 2](a-g) shows the convergence of the estimated states toward the real states and then toward the desired trajectory and this proves the observer's convergence. In addition, a less tracking error for speed and rotor flux is observed in [Figure 2](a-d), despite the load torque applied, and this demonstrates that the disturbance rejection is guaranteed. [Figure 2](c-d) shows that when the rotor speed reference changes of value, the d-axis rotor flux undergo a weak fluctuation and remain close to its T-S reference value, while the q-axis rotor flux remains null in [Figure 2](g).
| 7. Conclusion | | |
In this study, an integral fuzzy control scheme for an induction motor through T-S fuzzy model approach has been proposed. Based on the T-S fuzzy approach, a fuzzy observer-based tracking controller is designed to reduce the tracking error and to guaranty the disturbance rejection. The rotor flux is unavailable for measurement and it is estimated by a fuzzy observer. To solve the tracking control problem, an easy and systematic algorithm based on LMI optimization techniques is presented. The efficiency of the proposed controller is demonstrated through numerical simulations.
| 8. Appendix | | |
The premise variables are bounded as:
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| Authors | | |
 Moez Allouche was born in Sfax, Tunisia, on November 15, 1972. He received the Master degree from National Engineering School of Sfax, Tunisia, in 2006. Also, he obtained the Ph. D degree in Electrical Engineering from the National Engineering School of Sfax, Tunisia, in 2010. His currents research interests robust control, fuzzy logic control, D-stability analysis.
 Mohamed Chaabane was born in Sfax, Tunisia, on August 26, 1961. He received the Ph. D degree in Electrical Engineering from the University of Nancy, French in 1991. He was associate professor at the University of Nancy and is a researcher at Center of Automatic Control of Nancy (CRAN) from 1988-1992. Actually he is a professor in ENI-Sfax and editor in chief of the International Journal on Sciences and Techniques of Automatic Control and Computer Engineering IJSTA. Since 1997, he is holding a research position at Automatic Control Unit, ENIS. The main research interests are in the filed of robust control, delay systems, descriptor systems and applications of theses techniques to fed-batch processes and agriculture systems. Currently, he is an associate editor of International Journal on Sciences and Techniques of Automatic Control & Computer Engineering, IJ-STA (www.sta-tn.com).
 Mansour Souissi was born in Kerkennah, Tunisia, on April 15, 1955, He received his Ph. D. in Physical Sciences form the University of Tunis in 2002. He is professor in Automatic Control at preparatory Institute of Engineers of Sfax, Tunisai. Since 2003, he is holding a research position at Automatic Control Unit, National School of Engineers of Sfax, Tunisia. His current research interests robust control, optimal control, fuzzy logic, linear matrix inequalities and applications of these techniques to agriculture systems. Dr. Souissi is a member of the organization committees if several national and international conferences (STA, CASA).
 Driss Mehdi is currently a professor at Institute of Technology of Poitiers, France. He is the headmaster of research group "Multivariable Systems and Robust Control" in Laboratory of Automatic and Industrial Informatics of Poitiers. His main fields of research include robust control, delay system, robust root-clustering, descriptor systems, control of industrial processes, etc.
[Figure 1], [Figure 2]
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