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

Abstract

This paper investigates the guaranteed cost control of chaos problem in permanent magnet synchronous motor (PMSM) via Takagi-Sugeno (T-S) fuzzy method approach. Based on Lyapunov stability theory and linear matrix inequality (LMI) technique, a state feedback controller is proposed to stabilize the PMSM systems. An illustrative example is provided to verify the validity of the results developed in this paper.

1. 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 [10–16] 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.

2. 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]: 𝑑𝑖_𝑑=𝑢𝑑𝑡^𝑑−𝑅₁𝑖_𝑑+𝑤𝐿_𝑞𝑖_𝑞𝐿_𝑑,𝑑𝑖_𝑞=𝑢𝑑𝑡^𝑞−𝑅₁𝑖_𝑞+𝑤𝐿_𝑞𝑖_𝑞−𝑤𝜓_𝑟𝐿_𝑞,𝑑𝑤=𝑛𝑑𝑡^𝑝𝜓_𝑟𝑖_𝑞+𝑛_𝑝𝐿_𝑑−𝐿_𝑞𝑖_𝑑𝑖_𝑞−𝑇_𝐿−𝛽𝑤𝐽, (2.1) 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, 𝑅₁ 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=𝜆𝑥, 𝑡=𝜏𝑡̃, 𝑥=[𝑖_𝑑𝑖_𝑞𝑤]^𝑇, 𝑥=[𝑖_𝑑̃𝑖_𝑞̃𝑤]^𝑇, 𝑏=𝐿_𝑞/𝐿_𝑑, 𝑘=𝛽/(𝑛_𝑝𝜏𝜓_𝑟), 𝜏=𝐿_𝑞/𝑅, and 𝜆=diag[𝜆_𝑑𝜆_𝑞𝜆_𝑤]=diag[𝑏𝑘𝑘1/𝜏], system (2.1) can be transformed as follows: 𝑑𝑖_𝑑̃̃𝑖𝑑𝑡=−_𝑑+𝑖_𝑞̃𝑤+𝑢_𝑑,𝑑𝑖_𝑞̃𝑖̃𝑑𝑡=−_𝑞−𝑖_𝑑̃𝑤+𝛾𝑤+𝑢_𝑞,𝑑𝑤̃𝑖𝑑𝑡=𝜎_𝑞−𝑤−𝑇_𝐿, (2.2) where 𝑢_𝑑=𝑢_𝑑/𝑘𝑅, 𝛾=−𝜓_𝑟/𝑘𝐿_𝑞, 𝑢_𝑞=𝑢_𝑞/𝑘𝑅, 𝜎=𝛽𝜏/𝐽, and 𝑇_𝐿=𝜏²𝑇_𝐿/𝐽.

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 𝑑𝑖_𝑑̃̃𝑖𝑑𝑡=−_𝑑+𝑖_𝑞̃𝑑𝑖̃𝑤,_𝑞𝑖̃𝑑𝑡=−_𝑞−𝑖_𝑑̃𝑤+𝛾𝑤,𝑑𝑤̃𝑖𝑑𝑡=𝜎_𝑞−𝑤, (2.3) or 𝑥̇₁(𝑡)=−𝑥₁(𝑡)+𝑥₂(𝑡)𝑥₃(𝑡),𝑥̇₂(𝑡)=−𝑥₂(𝑡)−𝑥₁(𝑡)𝑥₃(𝑡)+𝛾⋅𝑥₃(𝑡),𝑥̇₃(𝑡)=𝜎⋅𝑥₁(𝑡)−𝑥₃(𝑡), (2.4) where 𝑥₁=𝑖_𝑑̃, 𝑥₂=𝑖_𝑞̃, 𝑥₃=𝑤.

To investigate the control design of system (2.4), let the system’s state vector 𝑥(𝑡)=[𝑥₁𝑥₂𝑥₃]^𝑇 and the control input vector be 𝑢(𝑡). Then, the state equations of PMSM can be represented as follows: 𝑥̇(𝑡)=𝐴(𝑥(𝑡))𝑥(𝑡)+𝐵𝑢(𝑡), (2.5) where 𝑥(𝑡)=[𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)]^𝑇, 𝐴(𝑥(𝑡))=^{−1𝑥3(𝑡)0}_{−𝑥3(𝑡)−1𝛾0𝜎−𝜎}.

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:

If 𝑧₁(𝑡) is 𝑀_𝑖1 and ⋯𝑧_𝑟(𝑡) is 𝑀_𝑖𝑟, then 𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡), (2.6) where 𝑧₁(𝑡),𝑧₂(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: (𝑡)=∑𝑥̇^𝑚𝑖=1𝑤_𝑖(𝑧(𝑡))⋅𝐴_𝑖𝑥(𝑡)𝐵_𝑖𝑢(𝑡)∑^𝑚_𝑖=1𝑤_𝑖(𝑧(𝑡))=^𝑚_𝑖=1𝜂_𝑖(𝑧(𝑡))⋅𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑢(𝑡), (2.7) where 𝑧(𝑡)=[𝑧₁(𝑡)𝑧₂(𝑡)⋯𝑧_𝑟(𝑡)], 𝑤_𝑖(𝑧(𝑡))=∏^𝑟_𝑗=1𝑀_𝑖𝑗(𝑧_𝑗(𝑡)), 𝜂_𝑖(𝑧(𝑡))=𝑤_𝑖(𝑧(𝑡))/∑^𝑚_𝑖=1𝑤_𝑖(𝑧(𝑡)). The term 𝑀_𝑖𝑗(𝑧_𝑗(𝑡)) is the grade of membership of 𝑧_𝑗(𝑡) in 𝑀_𝑖𝑗. In this paper, we assume that 𝑤_𝑖(𝑧(𝑡))≥0,𝑖∈{1,2,…,𝑚}, and ∑^𝑚_𝑖=1𝑤_𝑖(𝑧(𝑡))>0. Therefore, we have 𝜂_𝑖(𝑧(𝑡))≥0,𝑖∈{1,2,…,𝑚} and ∑^𝑚_𝑖=1𝜂_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: 𝐽=^∞₀𝑥^𝑇(𝑠)⋅𝑆₁⋅𝑥(𝑠)+𝑢^𝑇(𝑠)⋅𝑆₂⋅𝑢(𝑠)𝑑𝑠, (2.8) where 𝑆₁ and 𝑆₂ 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 𝑧₁(𝑡) is 𝑀_𝑗1 and ⋯𝑧_𝑟(𝑡) is 𝑀_𝑗𝑟, then 𝑢(𝑡)=−𝐾_𝑗𝑥(𝑡),𝑡≥0,𝑗={1,2,…,𝑚}, (2.9) where 𝐾_𝑗, 𝑗∈{1,2,…,𝑚} are the state feedback gains. Hence, the overall state feedback control law is represented as follows: 𝑢(𝑡)=−^𝑚_𝑗=1𝜂_𝑗(𝑧(𝑡))⋅𝐾_𝑗𝑥(𝑡),𝑡≥0. (2.10)
Before proposing the main theorem for determining the feedback gains 𝐾_𝑗(𝑗=1,2,…,𝑚), a lemma is introduced.

Lemma 2.2 (see [22] (Schur complement)). For a given matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22} with 𝑆₁₁=𝑆^𝑇₁₁, 𝑆₂₂=𝑆^𝑇₂₂, then the following conditions are equivalent:(1)𝑆<0, (2)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

Now we present a asymptotic stabilization condition for T-S fuzzy PMSM system (2.6).

Theorem 2.3. If there exist some positive definite symmetric matrices 𝑃 and matrices 𝐾_𝑗, 𝑗∈{1,2,…,𝑚} such that the following LMI condition holds for all 𝑖,𝑗∈{1,2,…,𝑚}: Φ_𝑖𝑗=𝐴_𝑖𝑃+𝑃𝐴^𝑇_𝑖−𝐵_𝑖𝐾_𝑗−𝐾^𝑇_𝑗𝐵^𝑇_𝑖𝑃𝐾^𝑇_𝑗∗−𝑆₁⁻¹0∗∗−𝑆₂⁻¹<0, (2.11) then system (2.6) is asymptotically stabilizable by controller (2.10). The stabilizing feedback control gain is given by 𝐾_𝑗=𝐾_𝑗𝑃⁻¹, and the system performance (2.8) is bounded by 𝐽≤𝐽^∗=𝑥^𝑇(0)𝑃𝑥(0), (2.12) where 𝑃=𝑃⁻¹.

Proof. Define the Lyapunov functional 𝑉𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡), (2.13) 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=1𝜂_𝑖(𝑧(𝑡))=1, the time derivatives of 𝑉(𝑥_𝑡), along the trajectories of system (2.6) with (2.8) and (2.10), satisfy 𝑉̇𝑥_𝑡−^𝑚_𝑖=1^𝑚_𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)𝑆₁+𝐾^𝑇_𝑗𝑆₂𝐾_𝑗𝑥(𝑡)=^𝑚_𝑖=1^𝑚_𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃−𝐾^𝑇_𝑗𝐵^𝑇_𝑖𝑃−𝑃𝐵_𝑖𝐾_𝑗𝑥(𝑡)=^𝑚_𝑖=1^𝑚_𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃−𝐾^𝑇_𝑗𝐵^𝑇_𝑖𝑃−𝑃𝐵_𝑖𝐾_𝑗+𝑆₁+𝐾^𝑇_𝑗𝑆₂𝐾_𝑗𝑥(𝑡)≤^𝑚_𝑖=1^𝑚_𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)Φ_𝑖𝑗𝑥(𝑡). (2.14) In order to guarantee 𝑉̇(𝑥_𝑡)−∑^𝑚_𝑖=1∑^𝑚_𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)(𝑆₁+𝐾^𝑇_𝑗𝑆₂𝐾_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: 𝑉̇𝑥_𝑡≤−^𝑚_𝑖=1^𝑚_𝑗=1𝜂_𝑖(𝑧(𝑡))𝜂_𝑗(𝑧(𝑡))𝑥^𝑇(𝑡)𝑆₁+𝐾^𝑇_𝑗𝑆₂𝐾_𝑗𝑥(𝑡)=−𝑥^𝑇(𝑡)⋅𝑆₁⋅𝑥(𝑡)+𝑢(𝑡)⋅𝑆₂⋅𝑢(𝑡)<0. (2.15)
From the inequality (2.15), 𝑉̇(𝑥_𝑡)<0, we conclude that system (2.6) with (2.8) is asymptotically stable. Integrating (2.12) from 0 to ∞, we have 𝑉̇^∞₀𝑥_𝑠𝑑𝑠=lim_𝑡→∞𝑉𝑥_𝑡−𝑉𝑥₀≤−^∞₀𝑥^𝑇(𝑠)⋅𝑆₁⋅𝑥(𝑠)+𝑢^𝑇(𝑠)⋅𝑆₂⋅𝑢(𝑠)𝑑𝑠. (2.16) Since that the system (2.6) with (2.8) is asymptotically stable, we can obtain the following results: lim_𝑡→∞𝑉𝑥_𝑡=0. (2.17) Consequently, 𝐽=∫^∞₀[𝑥^𝑇(𝑠)⋅𝑆₁⋅𝑥(𝑠)+𝑢^𝑇(𝑠)⋅𝑆₂⋅𝑢(𝑠)]𝑑𝑠≤𝑥^𝑇(0)𝑃𝑥(0)=𝑉(𝑥₀)=𝐽^∗.

This completes the proof.

3. 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=14.25mH, 𝑅₁=0.9Ω, 𝜓_𝑟=0.031Nm/A, 𝑛_𝑝=1, 𝐽=4.7×10⁻⁵kgm², 𝛽=0.0162N/rad, 𝛾=20, and 𝜎=5.46.

From the simulation result, we can get that 𝑥₃(𝑡) is bounded in interval [−1212]. By solving the equation, 𝑀₁ and 𝑀₂ are obtained as follows: 𝑀₁𝑥₃(𝑡)=12𝑥1+³(𝑡)𝑑,𝑀₂𝑥₃(𝑡)=1−𝑀₁𝑥₃(𝑡)=12𝑥1−³(𝑡)𝑑. (3.1)𝑀₁ and 𝑀₂ 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.

Rule 1. IF 𝑥₃(𝑡) is 𝑀₁, then 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐵₁𝑢(𝑡). (3.2)

Rule 2. IF 𝑥₃(𝑡) is 𝑀₂, then 𝑥̇(𝑡)=𝐴₂𝑥(𝑡)+𝐵₂𝑢(𝑡), (3.3) where 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)^𝑇,𝐴₁=𝐴−1−12012−1205.460−5.46₂,=−1120−12−1205.460−5.46,𝐵₁=𝐵₂=001. (3.4)(3.5) By the theorem, the stabilizing fuzzy control gains are given by 𝐾₁=𝐾₂=[3.96819.90277.990].

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.

Figure 1 

The chaotic attractor of the PMSM system.

(a)

(b)

(c)

Figure 2 

Power spectrum of the state variables 𝑥1(𝑡), 𝑥2(𝑡), and 𝑥3(𝑡) of the PMSM system.

Figure 3 

The state responses of the controlled PMSM system.

Figure 4 

Time response of control input 𝑢(𝑡).

4. 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.

Acknowledgments

The author thanks the National Science Council of Taiwan for supporting this work under Grant NSC 101-2622-E-269-012-CC3. The author also wishes to thank the anonymous reviewers for providing constructive suggestions.