Discrete Dynamics in Nature and Society

Volume 2010 (2010), Article ID 269283, 13 pages

http://dx.doi.org/10.1155/2010/269283

## Robust Adaptive Fuzzy Control of Chaos in the Permanent Magnet Synchronous Motor

^{1}Institute of Complexity Science, Qingdao University, Qingdao 266071, China^{2}State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China

Received 5 May 2010; Accepted 12 July 2010

Academic Editor: Recai Kilic

Copyright © 2010 Jinpeng Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An adaptive fuzzy control method is developed to control chaos in the permanent magnet synchronous motor drive system via backstepping. Fuzzy logic systems are used to approximate unknown nonlinearities, and an adaptive backstepping technique is employed to construct controllers. The proposed controller can suppress the chaos of PMSM and track the reference signal successfully. The simulation results illustrate its effectiveness.

#### 1. Introduction

Permanent magnet synchronous motors (PMSMs) are intensively used in industrial applications due to their high speed, high efficiency, high power density, and large torque to inertia ratio. Then, it is still a challenging problem to control the PMSM to get the perfect dynamic performance, because the dynamic model of PMSM is nonlinear, multivariable and even experiencing Hopf bifurcation, limit cycles, and chaotic attractors with systemic parameters falling into a certain area [1]. The chaotic behavior in PMSM is undesirable since it can extremely destroy the stabilization of the motor or even induce drive system collapse. Chaos in the PMSM and its control have been an active research area in the field of nonlinear control of electric motors [2]. Up to now, some control methods, such as OGY method [3], feedback linearization [4], time delay feedback control [5–7], sliding model control [8], adaptive control method [9, 10], backstepping method [11–14], and dynamic surface control [9] are successfully used to control or suppress chaos in PMSM. However, the existing control methods also have some disadvantages. The OGY method requires a variable system parameter which is usually unavailable in the control of the PMSM. The employed method of feedback linearization requires the exact mathematical model; so the controller requires the desired dynamics to replace the system at the axis stator currents. The time delay feedback control was successfully implemented to control the PMSM, but it is difficult to determine the time delay for TDFC method given a special target and is not suitable when the desired target is not the equilibrium or an unstable periodic orbit of the system. Chattering phenomenon and high heat loss in electrical power circuits are the drawbacks of the sliding mode control.

Backstepping is a newly developed technique to control the nonlinear systems with parameter uncertainty, particularly those systems in which the uncertainty does not satisfy matching conditions. Though the conventional backstepping is successfully applied to the control of PMSM drivers recently, it usually makes the designed controllers' structure to be very complex.

In recent years, fuzzy logic control (FLC) [15–17] has been found one of the most popular and conventional tools in functional approximations. An FLC has strong ability of handling uncertain information and can be easily used in the control of systems which is ill-defined or too complex to have a mathematical model. The essential part of an FLC is a set of the linguistic control rules related by the dual concepts of fuzzy implication and the compositional rule of inference [18]. Classically, fuzzy variables have been adjusted by expert knowledge and trial and error. It provides an effective way to design control system that is one of the important applications in the area of control engineering.

In this paper, an adaptive fuzzy control method is developed to control chaos in the permanent magnet synchronous motor drive systems via backstepping technology. During the controller design process, fuzzy logic systems are employed to approximate the nonlinearities of the chaotic PMSM drive system; the adaptive fuzzy controllers are constructed via backstepping. Compared with the conventional backstepping, the designed fuzzy controller has a simple structure, which can suppress the chaos of PMSM and track the reference signal generated by a reference model quite well.

#### 2. Mathematical Model of Chaotic PMSM Drive System and Preliminaries

The dimensionless mathematical model of PMSM with the smooth air gap can be described as follows [1]: where , and are state variables, which denote angle speed and the axis currents, respectively. and are system operating parameters, which are positive. , , and stand for the axis voltages and load torque,respectively.

In system (2.1), the external inputs are set to zero, namely, [1]. Then, the system (2.1) becomes an unforced system:

The modern nonlinear theory such as bifurcation and chaos has been widely used to study the stability of PMSM driver system. The study found that the PMSM is experiencing chaotic behavior when the operating parameters and fall into a certain regime. For example, the PMSM displays chaos with = 5.45 and = 20. The typical chaotic attractor is shown in Figure 1. These chaotic oscillations can destroy the stabilization of the PMSM drive system. In order to remove or control chaos, we use as the manipulated variable which is desirable for the real application. Then, an adaptive fuzzy control approach is proposed to control chaos in the PMSM drive system via the backstepping technique. For simplicity, the following notations are introduced: and By using these notations, the dynamic model of PMSM driver system can be described by the following differential equations:

The control objective is to design an adaptive fuzzy controller such that the state variable follows the given reference signal and all the closed-loop signals are bounded. To this end, we adopt the singleton fuzzifier, product inference, and the center-defuzzifier to deduce the following fuzzy rules: where and are the input and output of the fuzzy system, respectively and are fuzzy sets in The fuzzy inference engine performs a mapping from fuzzy sets in to fuzzy set in based on the IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point into a fuzzy set in The defuzzifier maps a fuzzy set in to a crisp point in Since the strategy of singleton fuzzification, center-average defuzzification, and product inference is used, the output of the fuzzy system can be formulated as where is the point at which fuzzy membership function achieves its maximum value, and it is further assumed that Let , and then the fuzzy logic system above can be rewritten as If all memberships are taken as Gaussian functions, then the following lemma holds.

Lemma 2.1 (see [19]). *Let be a continuous function defined on a compact set Then for any scalar there exists a fuzzy logic system in the form of (2.6) such that
*

#### 3. Adaptive Fuzzy Controller with the Backstepping Technique

In this section, we will develop an adaptive fuzzy control approach to control chaos in PMSM drive system via the backstepping. The backstepping design procedure contains steps. At each design step, a virtual control function will be constructed by using an appropriate Lyapunov function. At the last step, the real controller is constructed to control the system.

*Step 1. *For the reference signal , define the tracking error variable as . From the first differential equation of (2.3), the error dynamic system is given by

Choose Lyapunov function candidate as then the time derivative of is computed by
Construct the virtual control law as
where is a positive constant. By using (3.2), (3.1) can be rewritten in the following form:
with being a design parameter and

*Step 2. *Differentiating gives

Now, choose the Lyapunov function candidate as . Obviously, the time derivative of is given by
In the realistic model of PMSM, limited to the work conditions, the parameter is unknown. So it cannot be used to construct the control signal. Thus, let be the estimation of . The corresponding adaptation laws will be specified later. The virtual control is constructed as
where is a positive design parameter and . Adding and subtracting in the bracket in (3.5) shows that
with

*Step 3. *Differentiating results in the following differential equation:
Choose the Lyapunov function candidate as Furthermore, differentiating yields
where

Notice that containing the derivative of therewithal, the unknown parameter appears in the expression of This will make the classical adaptive backstepping design become very complex and troubled, and the designed control law will have a complex structure. To avoid this trouble and simplify the control signal structure, we will employ the fuzzy logic system to approximate the nonlinear function As shown later, the design procedure of becomes simple, and is of a simple structure.

According to Lemma 2.1, for any given there exists a fuzzy logic system such that where is the approximation error and satisfies Consequently, a straightforward calculation produces the following inequality: Thus, it follows immediately from substituting (3.6) into (3.9) that At this present stage, the control law is designed as where is the estimation of the unknown constant which will be specified later. Define Furthermore, using equality (3.14), it can be verified easily that

Introduce variables and as and choose the Lyapunov function candidate as where , are positive constants. By differentiating and taking (3.15)–(3.17) into account, one has According to (3.18), the corresponding adaptive laws are chosen as follows: where for and are positive constants.

#### 4. Stability Analysis

To address the stability analysis of the resulting closed-loop system, substitute (3.19) into (3.18) to obtain that For the term one has Similarly, holds. Consequently, by using these inequalities, (4.1) can be rewritten in the following form: where and Furthermore, (4.2) implies that As a result, all , and belong to the compact set Namely, all the signals in the closed-loop system are bounded. Especially, from (4.3) we have From the definitions of and , it is clear that to get a small tracking error by taking sufficiently large and and small enough after giving the parameters and

#### 5. Simulation

In order to illustrate the effectiveness of the proposed results, the simulation will be conducted to control chaos in PMSM drive system with two sets. First we tested the chaotic PMSM drive system with which are shown in Figures 2, 3, and 4. Then the proposed adaptive fuzzy approach in this paper is used to control the chaotic PMSM system, which are shown in Figures 5, 6, 7, and 8. The control parameters are chosen as follows:

And the fuzzy membership functions are: Give the reference signals and the simulation is carried out for the PMSM drive system. Compared two sets figures, it is seen clearly that the proposed controller can suppress the chaos in PMSM drive system and good tracking performance has been achieved successfully.

#### 6. Conclusion

Based on backstepping technique, an adaptive fuzzy control scheme is proposed to control chaos in the permanent magnet synchronous motor drive systems. The proposed controllers guarantee that the tracking error converges to a small neighborhood of the origin, and all the closed-loop signals are bounded. The simulation results are provided to demonstrate the effectiveness and feasibility of the proposed method.

#### Acknowledgments

This paper is partially supported by the Natural Science Foundation of China (60674055, 60774027, 60774047), the Natural Science Foundation of Shandong Province (ZR2009GM034), the State Key Laboratory of Rail Traffic Control and Safety (Beijing Jiaotong University) (RCS2008ZZ004), the cultivating plan of excellent degree thesis (Qingdao University), and Shandong Province Domestic Visitor Foundation.

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