Adaptive Fuzzy Tracking Control for a Permanent Magnet Synchronous Motor via Backstepping Approach

Abstract

The speed tracking control problem of permanent magnet synchronous motors with parameter uncertainties and load torque disturbance is addressed. Fuzzy logic systems are used to approximate nonlinearities, and an adaptive backstepping technique is employed to construct controllers. The proposed controller guarantees the tracking error convergence to a small neighborhood of the origin and achieves the good tracking performance. Simulation results clearly show that the proposed control scheme can track the position reference signal generated by a reference model successfully under parameter uncertainties and load torque disturbance without singularity and overparameterization.

1. Introduction

Permanent magnet synchronous motors (PMSMs) are of great interest for industrial applications requiring dynamic performance 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 PMSM to get the perfect dynamic performance because the motor dynamic model of PMSM is nonlinear and multivariable, the model parameters such as the stator resistance and the friction coefficient are also not be exactly known. The control of PMSM drivers has recently received wide attention and become an active research area. Some advanced control techniques, such as sliding mode control [1, 2], feedback linearization control [3], adaptive control [2, 4], backstepping principles [5–7], and Fuzzy logic control [8–10], are used to the problems of speed or position control of PMSMs.

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 backstepping is successfully applied to the control of PMSM drivers recently, it usually makes the designed controllers' structure to be very complex.

Fuzzy logic control (FLC) has been found one of the most popular and conventional tools in functional approximations. An FLC [11, 12] 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. It provides an effective way to design control system that is one of important applications in the area of control engineering.

In this paper, an adaptive fuzzy control approach is proposed for speed tracking control of PMSM drive system via the backstepping technique. During the controller design process, fuzzy logic systems are employed to approximate the nonlinearities, the adaptive fuzzy controllers are constructed via backstepping. The designed fuzzy controller can track the reference signal quite well even the existence of the parameter uncertainties and load torque disturbance. Compared with the existing controller design schemes via backstepping, the proposed method is very simple and the proposed controller has a simple structure.

2. Mathematical Model of the PMSM Drive System and Preliminaries

In this section, some preparatory knowledge of a PMSM will be introduced. The following assumptions are made in the derivation of the mathematical model of a PMSM [13].

Assumption 2.1. Saturation and iron losses are neglected although it can be taken into account by parameter changes.

Assumption 2.2. The back emf is sinusoidal.

The model of a PMSM can be described in the well known (𝑑−𝑞) frame through the Park transformation as follows. The stator 𝑑,𝑞 equations in the rotor frame are expressed as follows [14]:

𝐽𝑑𝜔𝑑𝑡=𝑇−𝑇_𝐿3−𝐵𝜔=2𝑛_𝑝[(𝐿_𝑑−𝐿_𝑞)𝑖_𝑑𝑖_𝑞+Φ𝑖_𝑞]−𝐵𝜔−𝑇_𝐿,𝐿_𝑑𝑑𝑖_𝑑𝑑𝑡=−𝑅_𝑠𝑖_𝑑+𝑛_𝑝𝜔𝐿_𝑞𝑖_𝑞+𝑢_𝑑,𝐿_𝑞𝑑𝑖_𝑞𝑑𝑡=−𝑅_𝑠𝑖_𝑞−𝑛_𝑝𝜔𝐿_𝑑𝑖_𝑑−𝑛_𝑝𝜔Φ+𝑢_𝑞. (2.1) The denotation of the PMSM parameters is shown in Table 1.

Table 1 

The denotation of the PMSM parameters.

To simplify the previous method mode, the following notations are introduced:

𝑥₁=𝜔,𝑥₂=𝑖_𝑞,𝑥₃=𝑖_𝑑,𝑎₁=3𝑛_𝑝Φ2,𝑎₂=3𝑛_𝑝(𝐿_𝑑−𝐿_𝑞)2,𝑏₁𝑅=−^𝑠𝐿_𝑞,𝑏₂𝑛=−^𝑝𝐿_𝑑𝐿_𝑞,𝑏₃𝑛=−^𝑝Φ𝐿_𝑞,𝑏₄=1𝐿_𝑞,𝑐₁𝑅=−^𝑠𝐿_𝑑,𝑐₂=𝑛_𝑝𝐿_𝑞𝐿_𝑑,𝑐₃=1𝐿_𝑑. (2.2) By using these notations, the dynamic model of a PMSM motor can be described by the following differential equations:

𝑥̇₁=𝑎₁𝐽𝑥₂+𝑎₂𝐽𝑥₂𝑥₃−𝐵𝐽𝑥₁−𝑇_𝐿𝐽,𝑥̇₂=𝑏₁𝑥₂+𝑏₂𝑥₁𝑥₃+𝑏₃𝑥₁+𝑏₄𝑢_𝑞,𝑥̇₃=𝑐₁𝑥₃+𝑐₂𝑥₁𝑥₂+𝑐₃𝑢_𝑑. (2.3) The control objective is to design an adaptive fuzzy controller such that the state variable 𝑥₁ tracks the given reference signal 𝑥_𝑑 and all signals of the resulting closed-loop system are uniformly ultimately bounded. In this paper, we adopt the singleton fuzzifier, product inference, and the center-defuzzifier to deduce the following fuzzy rules:

𝑅_𝑖∶IF𝑥₁is𝐹^𝑖₁and…and𝑥_𝑛is𝐹^𝑖_𝑛THEN𝑦is𝐵^𝑖(𝑖=1,2,…,𝑁), (2.4) where 𝑥=[𝑥₁,…,𝑥_𝑛]^𝑇∈𝑅^𝑛, and 𝑦∈𝑅 are the input and output of the fuzzy system, respectively, 𝐹^𝑗_𝑖 and 𝐵^𝑖 are fuzzy sets in 𝑅. The fuzzy inference engine performs a mapping from fuzzy sets in 𝑅^𝑛 to fuzzy set in 𝑅 based on the IF-THEN rules in the fuzzy rule base and the compositional rule of inference. The fuzzifier maps a crisp point 𝑥=[𝑥₁,…,𝑥_𝑛]^𝑇∈𝑅^𝑛 into a fuzzy set 𝐴_𝑥 in 𝑅. The defuzzifier maps a fuzzy set in 𝑅 to a crisp point in 𝑅. Since the strategy of singleton fuzzification, center-average defuzzification, and product inference is used, the output of the fuzzy system can be formulated as

∑𝑦(𝑥)=^𝑁𝑗=1𝑊_𝑗∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖(𝑥_𝑖)∑^𝑁_𝑗=1[∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖(𝑥_𝑖,)] (2.5) where 𝑊_𝑗 is the point at which fuzzy membership function 𝜇_𝐵𝑗(𝑊_𝑗) achieves its maximum value, and it is assumed that 𝜇_𝐵𝑗(𝑊_𝑗)=1. Let 𝑝_𝑗(𝑥)=∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖(𝑥_𝑖)/∑^𝑁_𝑗=1[∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖(𝑥_𝑖)] , 𝑆(𝑥)=[𝑝₁(𝑥),𝑝₂(𝑥),…,𝑝_𝑁(𝑥)]^𝑇 and 𝑊=[𝑊₁,…,𝑊_𝑁]^𝑇, then the fuzzy logic system above can be rewritten as

𝑦(𝑥)=𝑊^𝑇𝑆(𝑥). (2.6) If all memberships are taken as Gussian functions, then the following lemma holds.

Lemma 2.3 (see [15]). Let 𝑓(𝑥) be a continuous function defined on a compact set Ω. Then for any scalar 𝜀>0, there exists a fuzzy logic system in the form (2.6) such that sup_𝑥∈Ω|𝑓(𝑥)−𝑦(𝑥)|≤𝜀. (2.7)

3. Adaptive Fuzzy Controller with the Backstepping Technique

For the system (2.3), the backstepping design procedure contains 3 steps. At each design step, a virtual control function 𝛼_𝑖(𝑖=1,2) will be constructed by using an appropriate Lyapunov function 𝑉. At the last step, a real controller is constructed to control the system. In the following, we will give the procedure of the backstepping design.

Step 1. For the reference signal 𝑥_𝑑 , define the tracking error variable as 𝑧₁=𝑥₁−𝑥_𝑑 . From the first subsystem of (2.3), the error dynamic system is given by 𝑧̇₁=(𝑎₁/𝐽)𝑥₂+(𝑎₂/𝐽)𝑥₂𝑥₃−(𝐵/𝐽)𝑥₁−𝑇_𝐿/𝐽−𝑥̇_𝑑.
Choose Lyapunov function candidate as 𝑉₁=(𝐽/2)𝑧²₁, then the time derivative of 𝑉₁ is given by 𝑉̇₁=𝐽𝑧₁𝑧̇₁=𝑧₁(𝑎₁𝑥₂+𝑎₂𝑥₂𝑥₃−𝐵𝑥₁−𝑇_𝐿−𝐽𝑥̇_𝑑). (3.1)
As the parameters 𝐵,𝑇_𝐿, and 𝐽 are unknown, they cannot be used to construct the control signal. Thus, let 𝐵,𝑇_𝐿, and 𝐽̂ be their estimations of 𝐵,𝑇_𝐿, and 𝐽, respectively. The corresponding adaptation laws will be determined later. Now, construct the virtual control law 𝛼₁ as 𝛼₁(𝑍₁1)=𝑎₁(−𝑘₁𝑧₁+𝐵𝑥₁+𝑇_𝐿+𝐽𝑥̂̇_𝑑), (3.2) where 𝑘₁>0 is a design parameter and 𝑍₁=[𝑥₁,𝑥_𝑑,𝑥̇_𝑑,𝐵,𝑇_𝐿,𝐽]̂^𝑇. Defining 𝑧₂=𝑥₂−𝛼₁ and substituting (3.2) into (3.1) yield 𝑉̇₁=−𝑘₁𝑧²₁+𝑎₁𝑧₁𝑧₂+𝑎₂𝑧₁𝑥₂𝑥₃+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_𝐿−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑. (3.3)

Step 2. Differentiating 𝑧₂ and using the second subsystem of (2.3) give 𝑧̇₂=𝑥̇₂−𝛼̇₁=𝑏₁𝑥₂+𝑏₂𝑥₁𝑥₃+𝑏₃𝑥₁+𝑏₄𝑢_𝑞−𝛼̇₁. (3.4)
Now, choose the Lyapunov function candidate as 𝑉₂=𝑉₁+(1/2)𝑧²₂. Obviously, the time derivative of 𝑉₂ is given by 𝑉̇₂=𝑉̇₁+𝑧₂𝑧̇₂=−𝑘₁𝑧²₁+𝑎₂𝑧₁𝑥₂𝑥₃+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_𝐿−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑+𝑧₂(𝑓₂+𝑏₄𝑢_𝑞), (3.5) where 𝛼̇₁=𝜕𝛼₁𝜕𝑥₁𝑥̇₁+¹_𝑖=0𝜕𝛼₁𝜕𝑥_𝑑^(𝑖)𝑥_𝑑^(𝑖+1)+𝜕𝛼₁𝜕𝐵𝐵̇+𝜕𝛼₁𝜕𝑇_𝐿𝑇̇_𝐿+𝜕𝛼₁𝜕𝐽̂𝐽̇̂=𝜕𝛼₂𝜕𝑥₁(𝑎₁𝐽𝑥₂+𝑎₂𝐽𝑥₂𝑥₃−𝐵𝐽𝑥₁−𝑇_𝐿𝐽)+¹_𝑖=0𝜕𝛼₁𝜕𝑥_𝑑^(𝑖)𝑥_𝑑^(𝑖+1)+𝜕𝛼₁𝜕𝐵𝐵̇+𝜕𝛼₁𝜕𝑇_𝐿𝑇̇_𝐿+𝜕𝛼₁𝜕𝐽̂̇̂𝑓𝐽,₂(𝑍₂)=𝑎₁𝑧₁+𝑏₁𝑥₂+𝑏₂𝑥₁𝑥₃+𝑏₃𝑥₁−𝛼̇₁,𝑍₂=[𝑥₁,𝑥₂,𝑥₃,𝑥_𝑑,𝑥̇_𝑑,𝑥̈_𝑑,𝐵,𝑇_𝐿,𝐽]̂^𝑇. (3.6)
Apparently, there are two nonlinear terms in (3.5), that is, 𝑎₂𝑧₁𝑥₂𝑥₃ and 𝑓₂, therewithal, 𝑓₂ contains the derivative of 𝛼̇₁. This will make the classical adaptive backstepping design become very complex and troubled, and the designed control law 𝑢_𝑞 will have the complex structure. To avoid this trouble in design procedure and simplify the control signal structure, we will employ the fuzzy logic system to approximate the nonlinear function 𝑓₂. As shown later, the design procedure of 𝑢_𝑞 becomes simple and 𝑢_𝑞 has the simple structure. According to Lemma 2.3, for any given 𝜀₂>0, there exists a fuzzy logic system 𝑊^𝑇₂𝑆₂(𝑍₂) such that 𝑓₂(𝑍₂)=𝑊^𝑇₂𝑆₂(𝑍₂)+𝛿₂(𝑍₂), (3.7) with 𝛿₂(𝑍) being the approximation error and satisfying |𝛿₂|≤𝜀₂. Consequently, a simple method computing produces the following inequality: 𝑧₂𝑓₂=𝑧₂(𝑊^𝑇₂𝑆₂+𝛿₂1)≤2𝑙²₂𝑧²₂‖𝑊₂‖²𝑆²₂+12𝑙²₂+12𝑧²₂+12𝜀²₂. (3.8) It follows immediately from substituting (3.8) into (3.5) that 𝑉̇₂≤−²_𝑖=1𝑘_𝑖𝑧²_𝑖+𝑎₂𝑧₁𝑥₂𝑥₃+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_𝐿−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑+12𝑙²₂𝑧²₂‖𝑊₂‖²𝑆²₂+12𝑙²₂+12𝑧²₂+12𝜀²₂+𝑧₂𝑏₄𝑢_𝑞. (3.9) The control input 𝑢_𝑞 is designed as 𝑢_𝑞=1𝑏₄(−𝑘₂𝑧₂−12𝑧₂−12𝑙²₂𝑧₂𝜃𝑆²₂), (3.10) where 𝜃 is the estimation of the unknown constant 𝜃 which will be specified later. Using equality (3.10), the derivative of 𝑉₂ becomes as 𝑉̇₂≤−²_𝑖=1𝑘_𝑖𝑧²_𝑖+𝑎₂𝑧₁𝑥₂𝑥₃+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_𝐿−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑+12𝑙²₂𝑧²₂(‖𝑊₂‖²−𝜃)𝑆²₂+12𝑙²₂+12𝜀²₂. (3.11)

Step 3. At this step, we will construct the control law 𝑢_𝑑. To this end, define 𝑧₃=𝑥₃ and choose the following Lyapunov function candidate as 𝑉₃=𝑉₂+(1/2)𝑧²₃. Then the derivative of 𝑉₃ is given by 𝑉̇₃=𝑉̇₂+𝑧₃𝑧̇₃≤−²_𝑖=1𝑘_𝑖𝑧²_𝑖+𝑎₂𝑧₁𝑥₂𝑥₃+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_𝐿−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑+12𝑙²₂𝑧²₂(‖𝑊₂‖²−𝜃)𝑆²₂+12𝑙²₂+12𝜀²₂+𝑧₃(𝑓₃(𝑍₃)+𝑐₃𝑢_𝑑), (3.12) where 𝑓₃(𝑍₃)=𝑎₂𝑧₁𝑥₂+𝑐₁𝑥₃+𝑐₂𝑥₁𝑥₂ and 𝑍₃=[𝑥₁,𝑥₂,𝑥₃,𝑥_𝑑]^𝑇. Similarly, by Lemma 2.3 the fuzzy logic system 𝑊^𝑇₃𝑆₃(𝑍₃) is utilized to approximate the nonlinear function 𝑓₃ such that for given 𝜀₃>0,𝑧₃𝑓₃≤12𝑙²₃𝑧²₃‖𝑊₃‖²𝑆²₃+12𝑙²₃+12𝑧²₃+12𝜀²₃. (3.13) Substituting (3.13) into (3.12) gives 𝑉̇₃=𝑉̇₂+𝑧₃𝑧̇₃≤−²_𝑖=1𝑘_𝑖𝑧²_𝑖+12𝑙²₂𝑧²₂(‖𝑊₂‖²−𝜃)𝑆²₂+³_𝑖=212(𝑙²_𝑖+𝜀²_𝑖)+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_L−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑+12𝑙²₃𝑧²₃‖𝑊₃‖²𝑆²₃+12𝑧²₃+𝑐₃𝑧₃𝑢_𝑑. (3.14) Now design 𝑢_𝑑 as 𝑢_𝑑=−1𝑐₃(𝑘₃𝑧₃+12𝑧₃+12𝑙²₃𝑧₃𝜃𝑆²₃). (3.15) Then, define 𝜃=max{‖𝑊₂‖²,‖𝑊₃‖²}. Then, combining (3.14) with (3.15) results in 𝑉̇₄≤−³_𝑖=1𝑘_𝑖𝑧²_𝑖+³_𝑖=212(𝑙²_𝑖+𝜀²_𝑖)+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_𝐿−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑+³_𝑖=212𝑙²_𝑖𝑧²_𝑖(‖𝑊_𝑖‖²−𝜃)𝑆^𝑇_𝑖(𝑍_𝑖)𝑆_𝑖(𝑍_𝑖)≤−³_𝑖=1𝑘_𝑖𝑧²_𝑖+³_𝑖=212(𝑙²_𝑖+𝜀²_𝑖)+𝑧₁(𝐵−𝐵)𝑥₁+𝑧₁(𝑇_𝐿−𝑇_𝐿)+𝑧₁(𝐽̂−𝐽)𝑥̇_𝑑+³_𝑖=212𝑙²_𝑖𝑧²_𝑖𝑆^𝑇_𝑖(𝑍_𝑖)𝑆_𝑖(𝑍_𝑖)(𝜃−𝜃). (3.16)
At the present stage, to estimate the unknown constants 𝐵,𝑇_𝐿,𝐽, and 𝜃, define the adaptive variables as follows: 𝑇_𝐿=𝑇_𝐿−𝑇_𝐿,̃̂𝐵=𝐵−𝐵,𝐽=𝐽−𝐽,𝜃=𝜃−𝜃. (3.17) In order to determine the corresponding adaptation laws, choose the following Lyapunov function candidate: 𝑉=𝑉₃+12𝑟₁𝑇²_𝐿+12𝑟₂𝐵²+12𝑟₃𝐽²̃+12𝑟₄𝜃², (3.18) where 𝑟_𝑖, 𝑖=1,2,3,4 are positive constant. By differentiating 𝑉 and taking (3.16)–(3.18) into account, one has 𝑉̇≤−³_𝑖=1𝑘_𝑖𝑧²_𝑖+³_𝑖=212(𝑙²_𝑖+𝜀²_𝑖)+𝑧₁𝐵𝑥₁+𝑧₁𝑇_𝐿+𝑧₁𝐽𝑥̃̇_𝑑−³_𝑖=212𝑙²_𝑖𝑧²_𝑖𝜃𝑆^𝑇_𝑖(𝑍_𝑖)𝑆_𝑖(𝑍_𝑖1)+𝑟₁𝑇_𝐿𝑇_𝐿+1𝑟₂𝐵1𝐵+𝑟₃𝐽𝐽̃̂+1𝑟₄𝜃𝜃=−³_𝑖=1𝑘_𝑖𝑧²_𝑖+³_𝑖=212(𝑙²_𝑖+𝜀²_𝑖1)+𝑟₁𝑇_𝐿(𝑟₁𝑧₁+𝑇̇_𝐿)+1𝑟₂𝐵(𝑟₂𝑧₁𝑥₁+̇1𝐵)+𝑟₃𝐽(̃𝑟₃𝑧₁𝑥̇_𝑑+𝐽̇)̂+1𝑟₄𝜃[−³_𝑖=2𝑟₄2𝑙²_𝑖𝑧²_𝑖𝑆^𝑇_𝑖(𝑍_𝑖)𝑆_𝑖(𝑍_𝑖̇)+𝜃]. (3.19) According to (3.19), the corresponding adaptive laws are chosen as follows: 𝑇̇_𝐿=−𝑟₁𝑧₁−𝑚₁𝑇_𝐿,𝐵̇=−𝑟₂𝑧₁𝑥₁−𝑚₂̇̂𝐵,𝐽=−𝑟₃𝑧₁𝑥̇_𝑑−𝑚₃̂̇𝐽,𝜃=³_𝑖=2𝑟₄2𝑙²_𝑖𝑧²_𝑖𝑆^𝑇_𝑖(𝑍_𝑖)𝑆_𝑖(𝑍_𝑖)−𝑚₄𝜃, (3.20) where 𝑚_𝑖 for 𝑖=1,2,3,4 and 𝑙_𝑖 for 𝑖=2,3 are positive constant.

4. Stability Analysis

In this section, the stability analysis of the resulting closed-loop system will be addressed. Substituting (3.20) into (3.19) yields

𝑉̇≤−³_𝑖=1𝑘_𝑖𝑧²_𝑖+³_𝑖=212(𝑙²_𝑖+𝜀²_𝑖𝑚)−¹𝑟₁𝑇_𝐿𝑚𝑇−²𝑟₂𝐵𝑚𝐵−³𝑟₃𝐽𝐽̃̂−𝑚₄𝑟₄𝜃𝜃. (4.1)

For the term −𝑇_𝐿𝑇, one has

−𝑇_𝐿𝑇_𝐿=−𝑇_𝐿(𝑇_𝐿+𝑇_𝐿1)≤−2𝑇²_𝐿+12𝑇²_𝐿. (4.2) Similarly, we have

−𝐵1𝐵≤−2𝐵²+12𝐵²,−𝐽̃̂1𝐽≤−2𝐽²̃+12𝐽²,−𝜃1𝜃≤−2𝜃²+12𝜃². (4.3) Consequently, by using these inequalities, (4.1) can be rewritten in the following form:

𝑉̇≤−³_𝑖=1𝑘_𝑖𝑧²_𝑖−𝑚₁2𝑟₁𝑇²_𝐿−𝑚₂2𝑟₂𝐵²−𝑚₃2𝑟₃𝐽²̃−𝑚₄2𝑟₄𝜃²+³_𝑖=212(𝑙²_𝑖+𝜀²_𝑖𝑚)+¹2𝑟₁𝑇²_𝐿+𝑚₂2𝑟₂𝐵²+𝑚₃2𝑟₃𝐽²+𝑚₄2𝑟₄𝜃²≤−𝑎₀𝑉+𝑏₀, (4.4) where 𝑎₀=min{2𝑘₁/𝐽,2𝑘_2,2𝑘_3,𝑚₁,𝑚₂,𝑚₃,𝑚₄} and 𝑏₀=∑³_𝑖=2(1/2)(𝑙²_𝑖+𝜀²_𝑖)+(𝑚₁/2𝑟₁)𝑇²_𝐿+(𝑚₂/2𝑟₂)𝐵²+(𝑚₃/2𝑟₃)𝐽²+(𝑚₄/2𝑟₄)𝜃². Furthermore, (4.4) implies that

𝑉(𝑡)≤(𝑉(𝑡₀𝑏)−⁰𝑎₀)𝑒^{−𝑎0(𝑡−𝑡0)}+𝑏₀𝑎₀≤𝑉(𝑡₀𝑏)+⁰𝑎₀,∀𝑡≥𝑡₀. (4.5) As a result, all 𝑧_𝑖(𝑖=1,2,3) , 𝑇_𝐿,𝐵,𝐽̃ and 𝜃 belong to the compact set

Ω={(𝑧_𝑖,𝑇_𝐿,𝐵,𝐽,̃𝜃)∣𝑉≤𝑉(𝑡₀𝑏)+⁰𝑎₀,∀𝑡≥𝑡₀}. (4.6) Namely, all the signals in the closed-loop system are bounded. Especially, from (4.5) we have

lim_𝑡→∞𝑧²₁≤2𝑏₀𝑎₀. (4.7) From the definitions of 𝑎₀ and 𝑏₀, it is clear that to get a small tracking error we can take 𝑟_𝑖 large and 𝑙_𝑖 and 𝜀_𝑖 small enough after giving the parameters 𝑘_𝑖 and 𝑚_𝑖.

5. Simulation

To illustrate the effectiveness of the proposed results, the simulation will be done for the PMSM motor with the parameters:

𝐽=0.00379Kgm²,𝑅_𝑠=0.68Ω,𝐿_𝑑𝐿=0.00315H,_𝑞=0.00285H,𝐵=0.001158Nm/(rad/s),Φ=0.1245Wb,𝑛_𝑝=3. (5.1) Then, the proposed adaptive fuzzy controllers are used to control this PMSM motor. Given the reference signal is 𝑥_𝑑=30 and the control parameters are chosen as follows:

𝑘₁=2.5,𝑘₂=𝑘₃=50,𝑟₁=𝑟₂=𝑟₃=𝑟₄𝑚=2.5,₁=𝑚₂=𝑚₃=𝑚₄=0.0005,𝑙₂=𝑙₃=5. (5.2) The fuzzy membership functions are chosen as

𝜇_𝐹1𝑖=exp[−(𝑥+5)²2],𝜇_𝐹2𝑖=exp[−(𝑥+4)²2𝜇],_𝐹3𝑖=exp[−(𝑥+3)²2],𝜇_𝐹4𝑖=exp[−(𝑥+2)²2𝜇],_𝐹5𝑖=exp[−(𝑥+1)²2],𝜇_𝐹6𝑖=exp[−(𝑥−0)²2𝜇],_𝐹7𝑖=exp[−(𝑥−1)²2],𝜇_𝐹8𝑖=exp[−(𝑥−2)²2𝜇],_𝐹9𝑖=exp[−(𝑥−3)²2],𝜇_𝐹𝑖10=exp[−(𝑥−4)²2𝜇],_𝐹𝑖11=exp[−(𝑥−5)²2]. (5.3) The simulation is carried out under the zero initial condition for two cases. In the first case, 𝑇_𝐿=1.5 and in the second case,

𝑇_𝐿=1.5,0≤𝑡≤1,3,𝑡≥1. (5.4) Figures 1 and 2 show the simulation results for case 1, and Figures 3 and 4 show the simulation results for the second case. From these figures, it is seen clearly that the tracking performance has been achieved very well. This means that the proposed controller can track the reference signal satisfactorily even under parameter uncertainties and load torque disturbance.

Figure 1 

The curve of the rotor speed $x_1$.

Figure 2 

The curve of the speed tracking error.

Figure 3 

The curve of the rotor position $x_1$.

Figure 4 

The curve of the speed tracking error.

6. Conclusion

Based on adaptive fuzzy control approach and backstepping technique, an adaptive fuzzy control scheme is proposed to control a permanent magnet synchronous motor. The proposed controllers guarantee that the tracking error converges to a small neighborhood of the origin and all the closed-loop signals are bounded. Simulation results illustrate the effectiveness of the presented method.

Appendix

The Controller of PMSM Based on Conventional Backstepping

The controller of PMSM based on conventional backstepping technique is briefly reviewed here. The detailed design procedure is in [7]. The controllers based on conventional backstepping are shown as follows:

𝑢_𝑞=1𝑏₄(−𝑘₂𝑧₂+𝑏₄(𝐽̇̂𝑎₁(−𝑘₁𝑧₁+𝐹𝑥₁+Γ+𝑥̇_𝑑𝐽̂)+𝑎₁(𝐹̇𝑥₁+Γ̇+𝑥̈_𝑑+𝑘₁𝑥̇_𝑑)+(𝐹−𝑘₁)(𝑥₂+𝑎₂𝑎₁𝑥₂𝑥₃𝐽̂)−𝑎₁(𝐹−𝑘₁)(𝐹𝑥₁+Γ)−(𝑏₁𝑥₂+𝑏₂𝑥₁𝑥₃+𝑏₃𝑥₁𝑢))),_𝑑=1𝑐₃(−𝑘₃𝑧₃−𝑐₁𝑥₃−𝑐₂𝑥₁𝑥₂−𝑎₂𝐽̂𝑐₃𝑧₁𝑥₂). (A.1) Comparing the presented method with the above controller designed via conventional backstepping, it can be seen that the proposed controllers based on adaptive fuzzy backstepping have a very simple structure. And the simulation results illustrate the effectiveness of the presented method in this paper.

Acknowledgments

This work is partially supported by the Natural Science Foundation of China (60674055, 60774027, 60444047), the National 863 High Technology Plan of China (2007AA11Z247), the State Key Laboratory of Rail Traffic Control and Safety (Beijing Jiaotong University) (RCS2008ZZ004, SKL2007K006), the Shan Dong Province Key Laboratory of Industrial Control Technique, and Shandong Province Domestic Visitor Foundation (2007001).