Controlling Chaos in Permanent Magnet Synchronous Motor Control System via Fuzzy Guaranteed Cost Controller

and Applied Analysis 3 In the system 2.2 , the external inputs are set to zero, namely, T̃L ũd ũq 0. Then, the system 2.2 becomes dĩd dt − ĩd ĩqw̃, dĩq dt − ĩq − ĩdw̃ γw̃, dw̃ dt σ ( ĩq − w̃ ) , 2.3


Introduction
The permanent magnet synchronous motor PMSM is an important role in industrial applications due to its simple structure, high efficiency, high power density, and low maintenance cost 1-3 . However, the dynamic characteristics and stability analysis of PMSM has emerged as a new and attractive research field, such as bifurcation, chaos, and limit cycle dynamic behaviors 4-9 . Moreover, many profound theories and methodologies [10][11][12][13][14][15][16] have been developed to deal with this issue. In 4 , the adaptive dynamic surface control DSC of PMSM has been presented. In 5, 6 , the authors had derived some feedback control design methods for stability of PMSM in their results. Some control methods had studied to stabilize the PMSM systems, such as sliding mode control SMC 7 , differential geometry method 8 , passivity control 9, 10 , sensorless control 11-14 , Lyapunov exponents LEs placement 15 , and fuzzy control 16 .
Takagi-Sugeno T-S fuzzy concept was introduced by the pioneering work of Takagi and Sugeno 17 and has been successfully and effectively used in complex nonlinear systems 18 . The main feature of T-S fuzzy model is that a nonlinear system can be approximated by a set of T-S linear models. The overall fuzzy model of complex nonlinear systems is achieved by fuzzy "blending" of the set of T-S linear models. Therefore, the controller design and the stability analysis of nonlinear systems can be analyzed via T-S fuzzy models and the so-called parallel distributed compensation PDC scheme 19, 20 . This paper contributes to the development of the state-feedback control design for PMSM. Based on Lyapunov stability theory and LMI technique, the stability conditions of PMSM are analyzed. Finally, an example is given to illustrate the usefulness of the obtained results.

Problem Formulation and Main Results
Based on d-q axis, the dynamics of permanent synchronous motor plant can be described by the following differential equation 15 : where i d , i q , and w are state variables, which denote d, q axis stator currents, and w is motor angular speed, respectively. T L , u d , and u q are the external load torque, the direct-and quadrature-axis stator voltage components, respectively. J is the polar moment of inertia, β is the viscous damping coefficient, R 1 is the stator winding resistance, and L d and L q are the direct-and quadrature-axis stator inductors, respectively. ψ r is the permanent magnet flux, and n p is the number of pole pairs. By applying the affine transformation x λ x, β/ n p τψ r , τ L q /R, and λ diag λ d λ q λ w diag bk k 1/τ , system 2.1 can be transformed as follows: where u d u d /kR, γ −ψ r /kL q , u q u q /kR, σ βτ/J, and T L τ 2 T L /J.
Abstract and Applied Analysis 3 In the system 2.2 , the external inputs are set to zero, namely, T L u d u q 0. Then, the system 2.2 becomes To investigate the control design of system 2.4 , let the system's state vector x t x 1 x 2 x 3 T and the control input vector be u t . Then, the state equations of PMSM can be represented as follows:ẋ The continuous fuzzy system was proposed to represent a nonlinear system 17 . The system dynamics can be captured by a set of fuzzy rules which characterize local correlations in the state space. Each local dynamic described by the fuzzy IF-THEN rule has the property of linear input-output relation. Based on the T-S fuzzy model concept, the nonlinear PMSM system can be expressed as follows.
Model rule i: where z 1 t , z 2 t , . . . , z r t are known premise variables, M ij , i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , r} is the fuzzy set, and m is the number of model rules; x t is the state vector, and u t is input vector. The matrices A i and B are known constant matrices with appropriate dimensions. Given a pair of x t , u t , the final outputs of the fuzzy system are inferred as follows: In this paper, we assume that w i z t ≥ 0, i ∈ {1, 2, . . . , m}, and m i 1 w i z t > 0. Therefore, we have η i z t ≥ 0, i ∈ {1, 2, . . . , m} and m i 1 η i z t 1, for all t ≥ 0. To derive the main results, we first introduce the cost fuction of system 2.4 as follows: where S 1 and S 2 are two given positive definite symmetric matrices with appropriate dimensions. Associated with cost function 2.8 , the fuzzy guaranteed cost control is defined as follows.
Definition 2.1 see 21 . Consider the T-S fuzzy PMSM system 2.6 ; if there exist a control law u t and a positive scalar J * such that the closed-loop system is stable and the value of cost function 2.8 satisfies J ≤ J * , then J * is said to be a guaranteed cost and u t is said to be a guaranteed cost control law for the T-S fuzzy PMSM system 2.6 . This paper aims at designing a guaranteed cost control law for the asymptotic stabilization of the T-S fuzzy PMSM system 2.6 . To achieve this control goal, we utilize the concept of PDC 17 scheme and select the fuzzy guaranteed cost controller via state feedback as follows.
Control rule j: If z 1 t is M j1 and · · · z r t is M jr , then where K j , j ∈ {1, 2, . . . , m} are the state feedback gains. Hence, the overall state feedback control law is represented as follows: Before proposing the main theorem for determining the feedback gains K j j 1, 2, . . . , m , a lemma is introduced.  where V x t is a legitimate Lyapunov functional candidate, and P is positive definte symmetric matrices . By the system 2.6 with m i 1 η i z t 1, the time derivatives of V x t , along the trajectories of system 2.6 with 2.8 and 2.10 , satisfẏ

Lemma 2.2 see 22 Schur complement . For a given matrix
2.14 6 Abstract and Applied Analysis In order to guaranteeV x t − m i 1 m j 1 η i z t η j z t x T t S 1 K T j S 2 K j x t < 0, we need to satisfy Φ ij < 0. By Lemma 2.2, premultiplying, and postmultiplying the Φ ij in 2.14 by P −1 > 0, Φ ij < 0 are equivalent to Φ ij < 0 in 2.11 , then we can obtain the following:

2.15
From the inequality 2.15 ,V x t < 0, we conclude that system 2.6 with 2.8 is asymptotically stable. Integrating 2.12 from 0 to ∞, we have Since that the system 2.6 with 2.8 is asymptotically stable, we can obtain the following results: Consequently, J ∞ 0 x T s · S 1 · x s u T s · S 2 · u s ds ≤ x T 0 Px 0 V x 0 J * . This completes the proof.

Numerical Simulation and Analysis
In this section, a numerical example is presented to demonstrate and verify the performance of the proposed results. Consider a PMSM as given in 2.1 with the following parameters 23 : L d L q L 14.25 mH, R 1 0.9 Ω, ψ r 0.031 Nm/A, n p 1, J 4.7 × 10 −5 kg m 2 , β 0.0162 N/rad, γ 20, and σ 5.46.
From the simulation result, we can get that x 3 t is bounded in interval −12 12 . By solving the equation, M 1 and M 2 are obtained as follows: 3.1 M 1 and M 2 can be interpreted as membership functions of fuzzy sets. Using these fuzzy sets, the nonlinear system with time-varying delays can be expressed by the following T-S fuzzy models.
By the theorem, the stabilizing fuzzy control gains are given by K 1 K 2 3.968 19.902 77.990 .
Consequently, the minimal guaranteed cost is J * 5.443 × 10 −11 . The simulation was done with a four-order Runge-Kutta integration algorithm in MATLAB 7 with a step size of 0.0001. The simulation results with initial conditions x 0 13.5 − 5 − 5 T are shown in Figures 1-2. The chaotic attractor of PMSM system is given in Figure 1. The frequency power spectrum of the PMSM system variables is illustrated in Figure 2. The system state responses trajectory of controller design is shown in Figure 3. Figure 4 depicts the time responses of the control input of u t . When t 20 sec, it is obvious that the feedback control gain can guarantee stability of PMSM systems. From the simulation results, it is shown that the proposed controller works well to guarantee stability.

Conclusion
We have presented the solutions to the guaranteed cost control of chaos problem via the Takagi-Sugeno fuzzy control for PMSM system. Based on Lyapunov stability theory and LMI technique, the guaranteed cost control gains can be easily obtained through a convex  optimization problem. Finally, a numerical example shows the validity and superiority of the developed result. The future work will extend the proposed method to the underlying systems with noise and disturbances effects, like noise or disturbances, uncertainties effects, and robustness to time-varying delays. Also, the future application in the experiment will be included.