# Nonlinear Dynamic Surface Control of Chaos in Permanent Magnet Synchronous Motor Based on the Minimum Weights of RBF Neural Network

Authors: Shaohua Luo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper is concerned with the problem of the nonlinear dynamic surface control (DSC) of chaos based on the minimum weights of RBF neural network for the permanent magnet synchronous motor system (PMSM) wherein the unknown parameters, disturbances, and chaos are presented. RBF neural network is used to approximate the nonlinearities and an adaptive law is employed to estimate unknown parameters. Then, a simple and effective controller is designed by introducing dynamic surface control technique on the basis of first-order filters. Asymptotically tracking stability in the sense of uniformly ultimate boundedness is achieved in a short time. Finally, the performance of the proposed controller is testified through simulation results.


---

## Body

## 1. Introduction

The permanent magnet synchronous motor is widely used in the industrial applications [1, 2]. But PMSM with nonuniform breath appears to be the chaotic behavior at specific parameters and working conditions. This behavior leads to the intermittent oscillation of torque and speed, irregular current noise of the system, and unstable control performance. Furthermore, it intensively influences the stability and safety of the system [3].For ameliorating the performance of the PMSM system, a large amount of literatures and control methods have been attempted to apply to the motor. For example, to improve the error convergence rate, the nonsingular fast terminal sliding mode control (SMC) [4] which can reach finite-time stability is applied. In [5], a high-order SMC method via backstepping is presented to attain finite-time tracking control regardless of mismatched disturbance. However, the controller appears to be discontinuous phenomenon in dynamic sliding manifolds. In [6], an adaptive backstepping based terminal sliding mode controller for parameter strict-feedback system is proposed, where the finite-time convergence of the error is achieved.Furthermore, the OGY method is a fundamental technology for controlling chaos [7, 8]. Unfortunately, choosing an adjustable parameter usually becomes very difficult in real practice. Control of chaos by using the time-delay feedback control technology is introduced to the real applications, but it suffers from some problems as the control objective must be the equilibrium [9]. The dynamic surface control developed by Swaroop et al. [10] is a control technique by introducing a filter at each recursive step of the backstepping design procedure, so the differentiation items on the virtual function can be avoided. DSC has been pioneered by the work of Swaroop et al. about 10 years ago, but it does not consider nonlinear plant with uncertain time delays and disturbances. Also, nonlinear items are assumed to be completely known and the control gain is equal to one. In 2005, by incorporating DSC into a neural network based adaptive control design framework, Wang and Huang proposed a backstepping based control design for a class of nonlinear systems in strict-feedback form with arbitrary uncertainty [11]. However, uncertain time delays and disturbances are not involved in the model and the control gain is equal to one. Zhang et al. in 2008 proposed adaptive DSC for a class of pure-feedback nonlinear systems with unknown dead zone and perturbed uncertainties using neural networks [12], Na et al. in 2011 [13] presented adaptive neural dynamic surface control for servo systems with unknown dead zone, Li et al. in 2013 presented an adaptive fuzzy DSC output feedback approach for a single-link robotic manipulator coupled to a brushed direct current motor with a nonrigid joint [14], and Tong et al. in 2013 presented an adaptive fuzzy decentralized backstepping output feedback control approach for a class of uncertain large-scale stochastic nonlinear systems without the measurements of the states [15]. In their works, it is assumed that there exists positive constant which satisfies specified constraint condition. But it is very difficult to define the boundedness of unknown control gain in real practice. Incidentally, the PMSM is known to exhibit chaotic behavior under certain conditions. Whether the latter methods can suppress the chaos oscillation in PMSM needs further research since DSC with RBF has been pioneered by the work of Wang et al. In addition, Hu et al. overcame the gain-constrained recursive filtering challenge for a class of time-varying nonlinear stochastic systems with probabilistic sensor delays and correlated noises [16] and solved the recursive filtering problem for a class of discrete-time nonlinear stochastic systems with random parameter matrices, multiple fading measurements, and correlated noises [17]. He further investigated the recursive finite-horizon filtering problem for a class of nonlinear time-varying systems subject to multiplicative noises, missing measurements, and quantisation effects [18, 19]. The issues discussed in these literatures offer significant references to the research of chaos control in PMSM.Inspired by the work above, a new approach to design the nonlinear dynamic surface controller based on the minimum weights of RBF neural network is proposed for permanent magnet synchronous motor with the unknown parameters, disturbances, and chaos. During the controller design process, RBF neural network is employed to approximate unknown nonlinear functions. The main difficulty encountered in the controller design process is how to deal with the unknown control gain in the system. To overcome this difficulty, the adaptive method was also introduced to handle the problem. The proposed controller guarantees a good tracking performance and the boundedness of all the signals in the closed-loop system. Furthermore, the suggested controller contains the minimum weights of RBF neural network. As a result, the computational burden of the scheme is greatly alleviated. This makes our design controller more suitable for practical applications.


## 2. Mathematic Model

It is well known that the PMSM is applied widely in the motor drives, servo systems, and household appliances owing to advantages, for instance, simple structure, high efficiency, high power density, and low manufacturing cost [20]. However, the PMSM is experiencing chaotic behavior when the system parameters are falling into a special area, which can lead to the enormous destruction. The mathematic model of the PMSM based on the d-q axis is given as follows [21, 22]:
(1)dω-dt-=npJ(Ld-Lq)i-di-q+npJψri-q-BJω--1JT-L,di-qdt-=-RLqi-q-LdLqω-i-d-ψrLqω-+1Lqu-q,di-ddt-=-RLdi-d+LqLdω-i-q+1Ldu-d.The denotations of the PMSM parameters are shown in Table1. Suppose the direct and the quadrature-axis winding inductances are equal; that is, L=Ld=Lq. Meanwhile assume the time scale τ to be such that τ=L/R and define the normalized time t to be such that t=t-/τ and the scalar κ to be such that κ=B/(npτψr). Finally, the scaled state variables ω, id, and iq are defined as follows:
(2)ω=ω-τ,id=i-dκ,iq=i-qκ,
where ω, id, and iq are the normalized motor angular speed and the normalized quadrature-axis and direct-axis currents, respectively.Table 1
The denotation of the PMSM parameters.
Parameter
Denotation
i
-
d
The direct-axis currents (A)
ω
-
The velocity of the rotor (rad/s)
u
-
d
The direct-axis voltage (V)
T
-
L
The load torque (Nm)
L
d
The direct-axis winding inductance (H)
ψ
r
The permanent magnet flux (Wb)
J
The polar moment of inertia (kgm2)
i
-
q
The quadrature-axis currents (A)
t
-
The time (s)
u
-
q
The quadrature-axis voltage (V)
R
The stator winding resistance (Ω)
L
q
The quadrature-axis winding inductance (H)
B
The viscous damping coefficient (N/rad/s)
n
p
The number of pole pairsThen, the new normalized model for the PMSM is rewritten as(3)dωdt=σ(iq-ω)-TL,diqdt=-iq-idω+γω+uq,diddt=-id+iqω+ud,
where γ=-ψr/(κL), σ=Bτ/J, TL=τ2T-L/J, uq=u-q/(κR), ud=u-d/(κR), uq and ud denote the normalized quadrature-axis and direct-axis stator voltage, respectively, TL presents the normalized load torque, and σ and γ are previously defined system parameter.Figures1 and 2 reveal the chaotic behavior of the PMSM for the situation of σ=5.45, γ=20, uq=ud=0, TL=0, ω(0)=-5, iq(0)=0.01, and id(0)=20, in which it appears as an aperiodic, random, sudden, or intermittent morbid oscillation.Figure 1
Three-dimensional phase diagram with parametersσ=5.45 and γ=20.The chaotic time series of the PMSM.
(a)
The angular speed
(b)
The normalizedq-axis current
(c)
The normalizedd-axis currentIt is obvious that the model of the PMSM has high nonlinearity because of the coupling between the speed and the currents. In addition, the indeterminate system parametersσ and γ are enormously impacted by realistic conditions. So as to control efficiently the PMSM, uq and ud are used as the manipulated variables. Then, a nonlinear dynamics surface control approach based on RBF neural network is proposed to restrain the chaos, parameters variation, and external disturbance in the PMSM.For the sake of simplicity, the following symbols are introduced:(4)x1=ω,x2=iq,x3=id.Then, the mathematic model of the PMSM can be represented as follows:(5)x˙1=σx2-σx1-TL+Δ1,x˙2=-x2-x1x3+γx1+uq+Δ2,x˙3=-x3+x1x2+ud+Δ3,
where Δi, i=1–3, denote the external disturbance.Assumption 1.
The unknown disturbance termsΔi satisfy |Δi(x,t)|<di, i = 1–3, and the parameter σ is unknown and bounded. Moreover, it is assumed that |σ|≤G.Assumption 2.
The desired trajectoryyr is continuous, and its first-order derivative y˙r and second-order derivative y¨r are bounded and available.


## 3. Controller Design

### 3.1. RBF Neural Network

The type of RBF neural network is considered as a two-layer network, which contains a hidden layer and an output layer. In this paper, the RBF neural network will be used to approximate the unknown continuous functionf(z):Rn→R as follows:
(6)f^(z)=θ*Tξ(z),
where z∈Ω⊂Rn is the input vector with n being the neural network input dimension, θ*=[θ1*,θ2*,…,θl*]T∈Rl is the weight vector, l>1 is the node number of neuron, and ξ(z)=[ξ1(z),ξ2(z),…,ξn(z)]T∈Rl is a basic function vector with ξi(z) chosen as the commonly used Gaussian function in the following form:
(7)ξi(z)=exp⁡[-|z-μi|22σi2]=exp⁡[-(z-μi)T(z-μi)2σi2],i=1,2,…,l,
where μi=[μi1,μi2,…,μin]T is the center of the receptive field and σi is the width of ξi(z).For given scalarε>0, by choosing sufficiently large l, the RBF neural network can approximate any continuous function over a compact set Ω∈Rn to arbitrary accuracy as follows:
(8)f(z)=θTξ(z)+ε(z),∀z∈Ω∈Rn,
where ε(z) is the approximation error and θ is an unknown ideal constant weight vector, which is an artificial quantity required for analytical purpose. Typically, θ is chosen as the value of θ* that minimizes |ε(z)|, for all z∈Ω; that is,
(9)θ=argmin⁡θ*∈Rn{sup⁡z∈Ω|f(z)-θ*Tξ(z)|}.Assumption 3.
The approximation errorε is bounded, and it has positive constant εM which satisfies |εi|≤εM,i = 1–3.Assumption 4.
There exists a positive and bounded constantρi which satisfies |εM+di|≤ρi.


### 3.2. The Controller of Dynamics Surface Control Approach Based on RBF Neural Network

In this section, the controller of dynamics surface control approach will be developed based on the minimum weights of RBF neural network. The design procedure consists of three steps. Then, the detail process will be given.Step 1.
The first dynamic surface is defined asS1=x1-yr. Then, the time derivative of S1 can be obtained as follows:
(10)S˙1=σx2+f1+Δ1-y˙r,
where f1=-σx1-TL.The operating parameterσ is usually unknown due to the effect of the work environment. It is difficult for traditional methods to deal with the problem. In order to solve it, the adaptive technique is introduced to estimate the σ, and the adaptive RBF neural network is used to approximate the uncertain item f1 with little error. Therefore, for any given ε1, there exists a RBF neural network θ1Tξ1 such that
(11)f1=θ1Tξ1+ε1,
where ε1 is the approximation error and satisfies |ε1|≤εM.Substituting (11) into (10) yields the following:
(12)S˙1≤σx2+θ1Tξ1+ρ1-y˙r.The virtual control and related adaptive laws can be designed as follows:(13)α2=σ^σ^2+η(-k1S1-12a12λ^1S1ξ1Tξ1-12S1ρ12+y˙r),(14)λ^˙1=12a12γ1ξ1Tξ1S12-m1λ^1,(15)σ^˙=Γ1(S1α2-c1σ^),
where λ^1=∥θ^1∥2 belongs to a minimum weights of RBF neural network which greatly speeds up the solution speed, k1, m1, a1, γ1, c1, and Γ1 are the design constant, and η is a small positive constant.Introduce variablesλ~1 and σ~ as follows:
(16)λ~1=λ^1-λ1,σ~=σ^-σ.Letα2 be passed through a first-order filter with a time constant τ2 as follows:
(17)τ2α˙2f+α2f=α2,α2f(0)=α2(0).Define the filter error asy2=α2f-α2. With (17) and y2, it yields that
(18)α˙2f=-y2τ2.Then, the time derivative ofy2 can be obtained as follows:
(19)y˙2=-y2τ2-[σ^σ^2+η(-k1S˙1-θ^˙1Tξ1-θ^1T∂ξ1∂x1x˙1-12S˙1ρ12-S1ρ1∂ρ1∂x1x˙1+y¨r)]-[(η-σ^2)σ^˙(σ^2+η)2(-k1S1-θ^1Tξ1-12S1ρ12+y˙r)].It is obtained that(20)|y˙2+y2τ2|≤B2(S1,S2,y2,λ^1,σ^,yr,y˙r,y¨r).Using Young’s inequality, one has(21)y2y˙2≤-y22τ2+y22+14B22,
where B2(S1,S2,y2,λ^1,σ^,yr,y˙r,y¨r) is the continuous function.Substituting (13)–(21) into (12), (12) becomes
(22)S˙1≤σ(S2+y2)-σ~α2+σ^2σ^2+η×(-k1S1-12a12λ^1S1ξ1Tξ1-12S1ρ12+y˙r)+θ1Tξ1+ρ1-y˙r.One has(23)S1S˙1≤σ(S12+14S22)-ησ^2+η×(-k1S1-12a12λ^1S1ξ1Tξ1-12S1ρ12+y˙r)S1+σ(S12+14y22)-σ~α2S1-k1S12-12a12λ~1ξ1Tξ1S12+12+a122≤(2G-k1)S12+14GS22+14Gy22-σ~α2S1-12a12λ~1ξ1Tξ1S12+12+a122.Consider the Lyapunov function candidate as follows:(24)V1=12(S12+y22+1γ1λ~12+1Γ1σ~2).Then, the time derivative ofV1 is obtained as follows:
(25)V˙1≤(2G-k1)S12+14GS22+12+a122+(-1τ2+1+14G)y22+14B22-12m1γ1λ~12-12c1σ~2+12m1γ1λ12+12c1σ2.
For the terms -c1σ~σ^ and -(1/γ1)m1λ~1λ^1, one has -c1σ~σ^≤-(1/2)c1σ~2+(1/2)c1σ2 and
(26)-1γ1m1λ~1λ^1=-1γ1m1λ~1(λ~1+λ1)≤-12m1γ1λ~12+12m1γ1λ12.Step 2.
The second dynamic surface is given as(27)S2=x2-α2f.Then, differentiatingS2 gives
(28)S˙2=f2+Δ2+uq-α˙2f,
where f2=-x1x3-x2+γx1.To facilitate engineering application, a minimum-weights-based RBF neural network will be employed to approximate the nonlinear functionf2 again. Therefore, there exists a RBF neural network system such that
(29)f2=θ2Tξ2+ε2,
where ε2 is the approximation error and satisfies |ε2|≤εM.Substituting (29) into (28), one has
(30)S˙2≤θ2Tξ2+uq+ρ2-α˙2f.Similarly, the relevant control law and adaptive law are provided in the following forms:(31)uq=-k2S2-12a22λ^2S2ξ2Tξ2-12S2ρ22+α˙2f,(32)λ^˙2=12a22γ2ξ2Tξ2S22-m2λ^2,
where k2, m2, a2, and γ2 are the design constant and λ^2=∥θ^2∥2.With (31) and (32), (30) is written as follows:
(33)S˙2≤-k2S2+θ2Tξ2+ρ2-12a22λ^2S2ξ2Tξ2-12S2ρ22.One has(34)S2S˙2≤-k2S22-12a22λ~2S22ξ2Tξ2+a222+12.Choose the Lyapunov function candidate as follows:(35)V2=12(S22+1γ2λ~22).
Then, the time derivative of V2 is
(36)V˙2≤-k2S22+a222+12-12m2γ2λ~22+12m2γ2λ22.
For the term -(1/γ2)m2λ~2λ^2, one has -(1/γ2)m2λ~2λ^2≤-(1/2)(m2/γ2)λ~22+(1/2)(m2/γ2)λ22.Step 3.
Choose the last dynamic surface asS3=x3. Then, the time derivative of S3 is calculated as follows:
(37)S˙3=f3+ud+Δ3,
where f3=-x3+x1x2.In the same way, there is a minimum-weights-based RBF neural network such that(38)f3=θ3Tξ3+ε3,
where ε3 is the approximation error and satisfies |ε3|≤εM.Substituting (38) into (37) gives
(39)S˙3≤θ3Tξ3+ρ3+ud.At the present stage, the control input is designed as follows:(40)ud=-k3S3-12a32λ^3S3ξ3Tξ3-12S3ρ32,
where k3 is the positive constant and λ^3=∥θ^3∥2.According to the mention above, the adaptive law is chosen as follows:(41)λ^˙3=12a32γ3ξ3Tξ3S32-m3λ^3,
where m3, a3, and γ3 are the design constant.Similarly, (37) is given as follows:
(42)S˙3≤θ3Tξ3+ρ3-k3S3-12a32λ^3S3ξ3Tξ3-12S3ρ32.There exists(43)S3S˙3≤-12a32λ~3S32ξ3Tξ3-k3S32+a322+12.Choose the Lyapunov function candidate as follows:(44)V3=12(s32+1γ3λ~32).
Using the equality in (44), it can be verified easily that
(45)V˙3≤-k3S32+a322+12-12m3γ3λ~32+12m3γ3λ32.
For the term -(1/γ3)m3λ~3λ^3, one has -(1/γ3)m3λ~3λ^3≤-(1/2)(m3/γ3)λ~32+(1/2)(m3/γ3)λ32.Up to now, the design procedure of proposed controller of the PMSM is completed. The proposed controller significantly reduces the computation complexity compared with traditional backstepping control and dynamics surface control. Based on previous procedure, the configuration of the proposed control system is depicted in Figure3. The overall configuration consists of the PMSM with load, the space vector pulse width modulation (SVPWM), the voltage-source inverter, the power source rectifier, the automatic current regulator of the motor, the encoder used to detect speed and position, q-axis and d-axis controllers.Figure 3
Control schematic of the PMSM.


## 3.1. RBF Neural Network

The type of RBF neural network is considered as a two-layer network, which contains a hidden layer and an output layer. In this paper, the RBF neural network will be used to approximate the unknown continuous functionf(z):Rn→R as follows:
(6)f^(z)=θ*Tξ(z),
where z∈Ω⊂Rn is the input vector with n being the neural network input dimension, θ*=[θ1*,θ2*,…,θl*]T∈Rl is the weight vector, l>1 is the node number of neuron, and ξ(z)=[ξ1(z),ξ2(z),…,ξn(z)]T∈Rl is a basic function vector with ξi(z) chosen as the commonly used Gaussian function in the following form:
(7)ξi(z)=exp⁡[-|z-μi|22σi2]=exp⁡[-(z-μi)T(z-μi)2σi2],i=1,2,…,l,
where μi=[μi1,μi2,…,μin]T is the center of the receptive field and σi is the width of ξi(z).For given scalarε>0, by choosing sufficiently large l, the RBF neural network can approximate any continuous function over a compact set Ω∈Rn to arbitrary accuracy as follows:
(8)f(z)=θTξ(z)+ε(z),∀z∈Ω∈Rn,
where ε(z) is the approximation error and θ is an unknown ideal constant weight vector, which is an artificial quantity required for analytical purpose. Typically, θ is chosen as the value of θ* that minimizes |ε(z)|, for all z∈Ω; that is,
(9)θ=argmin⁡θ*∈Rn{sup⁡z∈Ω|f(z)-θ*Tξ(z)|}.Assumption 3.
The approximation errorε is bounded, and it has positive constant εM which satisfies |εi|≤εM,i = 1–3.Assumption 4.
There exists a positive and bounded constantρi which satisfies |εM+di|≤ρi.


## 3.2. The Controller of Dynamics Surface Control Approach Based on RBF Neural Network

In this section, the controller of dynamics surface control approach will be developed based on the minimum weights of RBF neural network. The design procedure consists of three steps. Then, the detail process will be given.Step 1.
The first dynamic surface is defined asS1=x1-yr. Then, the time derivative of S1 can be obtained as follows:
(10)S˙1=σx2+f1+Δ1-y˙r,
where f1=-σx1-TL.The operating parameterσ is usually unknown due to the effect of the work environment. It is difficult for traditional methods to deal with the problem. In order to solve it, the adaptive technique is introduced to estimate the σ, and the adaptive RBF neural network is used to approximate the uncertain item f1 with little error. Therefore, for any given ε1, there exists a RBF neural network θ1Tξ1 such that
(11)f1=θ1Tξ1+ε1,
where ε1 is the approximation error and satisfies |ε1|≤εM.Substituting (11) into (10) yields the following:
(12)S˙1≤σx2+θ1Tξ1+ρ1-y˙r.The virtual control and related adaptive laws can be designed as follows:(13)α2=σ^σ^2+η(-k1S1-12a12λ^1S1ξ1Tξ1-12S1ρ12+y˙r),(14)λ^˙1=12a12γ1ξ1Tξ1S12-m1λ^1,(15)σ^˙=Γ1(S1α2-c1σ^),
where λ^1=∥θ^1∥2 belongs to a minimum weights of RBF neural network which greatly speeds up the solution speed, k1, m1, a1, γ1, c1, and Γ1 are the design constant, and η is a small positive constant.Introduce variablesλ~1 and σ~ as follows:
(16)λ~1=λ^1-λ1,σ~=σ^-σ.Letα2 be passed through a first-order filter with a time constant τ2 as follows:
(17)τ2α˙2f+α2f=α2,α2f(0)=α2(0).Define the filter error asy2=α2f-α2. With (17) and y2, it yields that
(18)α˙2f=-y2τ2.Then, the time derivative ofy2 can be obtained as follows:
(19)y˙2=-y2τ2-[σ^σ^2+η(-k1S˙1-θ^˙1Tξ1-θ^1T∂ξ1∂x1x˙1-12S˙1ρ12-S1ρ1∂ρ1∂x1x˙1+y¨r)]-[(η-σ^2)σ^˙(σ^2+η)2(-k1S1-θ^1Tξ1-12S1ρ12+y˙r)].It is obtained that(20)|y˙2+y2τ2|≤B2(S1,S2,y2,λ^1,σ^,yr,y˙r,y¨r).Using Young’s inequality, one has(21)y2y˙2≤-y22τ2+y22+14B22,
where B2(S1,S2,y2,λ^1,σ^,yr,y˙r,y¨r) is the continuous function.Substituting (13)–(21) into (12), (12) becomes
(22)S˙1≤σ(S2+y2)-σ~α2+σ^2σ^2+η×(-k1S1-12a12λ^1S1ξ1Tξ1-12S1ρ12+y˙r)+θ1Tξ1+ρ1-y˙r.One has(23)S1S˙1≤σ(S12+14S22)-ησ^2+η×(-k1S1-12a12λ^1S1ξ1Tξ1-12S1ρ12+y˙r)S1+σ(S12+14y22)-σ~α2S1-k1S12-12a12λ~1ξ1Tξ1S12+12+a122≤(2G-k1)S12+14GS22+14Gy22-σ~α2S1-12a12λ~1ξ1Tξ1S12+12+a122.Consider the Lyapunov function candidate as follows:(24)V1=12(S12+y22+1γ1λ~12+1Γ1σ~2).Then, the time derivative ofV1 is obtained as follows:
(25)V˙1≤(2G-k1)S12+14GS22+12+a122+(-1τ2+1+14G)y22+14B22-12m1γ1λ~12-12c1σ~2+12m1γ1λ12+12c1σ2.
For the terms -c1σ~σ^ and -(1/γ1)m1λ~1λ^1, one has -c1σ~σ^≤-(1/2)c1σ~2+(1/2)c1σ2 and
(26)-1γ1m1λ~1λ^1=-1γ1m1λ~1(λ~1+λ1)≤-12m1γ1λ~12+12m1γ1λ12.Step 2.
The second dynamic surface is given as(27)S2=x2-α2f.Then, differentiatingS2 gives
(28)S˙2=f2+Δ2+uq-α˙2f,
where f2=-x1x3-x2+γx1.To facilitate engineering application, a minimum-weights-based RBF neural network will be employed to approximate the nonlinear functionf2 again. Therefore, there exists a RBF neural network system such that
(29)f2=θ2Tξ2+ε2,
where ε2 is the approximation error and satisfies |ε2|≤εM.Substituting (29) into (28), one has
(30)S˙2≤θ2Tξ2+uq+ρ2-α˙2f.Similarly, the relevant control law and adaptive law are provided in the following forms:(31)uq=-k2S2-12a22λ^2S2ξ2Tξ2-12S2ρ22+α˙2f,(32)λ^˙2=12a22γ2ξ2Tξ2S22-m2λ^2,
where k2, m2, a2, and γ2 are the design constant and λ^2=∥θ^2∥2.With (31) and (32), (30) is written as follows:
(33)S˙2≤-k2S2+θ2Tξ2+ρ2-12a22λ^2S2ξ2Tξ2-12S2ρ22.One has(34)S2S˙2≤-k2S22-12a22λ~2S22ξ2Tξ2+a222+12.Choose the Lyapunov function candidate as follows:(35)V2=12(S22+1γ2λ~22).
Then, the time derivative of V2 is
(36)V˙2≤-k2S22+a222+12-12m2γ2λ~22+12m2γ2λ22.
For the term -(1/γ2)m2λ~2λ^2, one has -(1/γ2)m2λ~2λ^2≤-(1/2)(m2/γ2)λ~22+(1/2)(m2/γ2)λ22.Step 3.
Choose the last dynamic surface asS3=x3. Then, the time derivative of S3 is calculated as follows:
(37)S˙3=f3+ud+Δ3,
where f3=-x3+x1x2.In the same way, there is a minimum-weights-based RBF neural network such that(38)f3=θ3Tξ3+ε3,
where ε3 is the approximation error and satisfies |ε3|≤εM.Substituting (38) into (37) gives
(39)S˙3≤θ3Tξ3+ρ3+ud.At the present stage, the control input is designed as follows:(40)ud=-k3S3-12a32λ^3S3ξ3Tξ3-12S3ρ32,
where k3 is the positive constant and λ^3=∥θ^3∥2.According to the mention above, the adaptive law is chosen as follows:(41)λ^˙3=12a32γ3ξ3Tξ3S32-m3λ^3,
where m3, a3, and γ3 are the design constant.Similarly, (37) is given as follows:
(42)S˙3≤θ3Tξ3+ρ3-k3S3-12a32λ^3S3ξ3Tξ3-12S3ρ32.There exists(43)S3S˙3≤-12a32λ~3S32ξ3Tξ3-k3S32+a322+12.Choose the Lyapunov function candidate as follows:(44)V3=12(s32+1γ3λ~32).
Using the equality in (44), it can be verified easily that
(45)V˙3≤-k3S32+a322+12-12m3γ3λ~32+12m3γ3λ32.
For the term -(1/γ3)m3λ~3λ^3, one has -(1/γ3)m3λ~3λ^3≤-(1/2)(m3/γ3)λ~32+(1/2)(m3/γ3)λ32.Up to now, the design procedure of proposed controller of the PMSM is completed. The proposed controller significantly reduces the computation complexity compared with traditional backstepping control and dynamics surface control. Based on previous procedure, the configuration of the proposed control system is depicted in Figure3. The overall configuration consists of the PMSM with load, the space vector pulse width modulation (SVPWM), the voltage-source inverter, the power source rectifier, the automatic current regulator of the motor, the encoder used to detect speed and position, q-axis and d-axis controllers.Figure 3
Control schematic of the PMSM.


## 4. Stability Analysis

For any givenp>0, the closed sets can be defined as follows:
(46)Π1={(S1,λ^1,σ^,y2):S12+y22+1γ1λ~12+1Γ1σ~2≤2p},Π2={∑i=12Si2(S1,S2,λ^1,λ^2,σ^,y2):∑i=12Si2+y22+∑i=121γiλ~i2+1Γ1σ~2≤2p},Π3={∑i=12Si2(S1,…,S3,λ^1,…,λ^3,σ^,y2):∑i=13Si2+y22+∑i=131γiλ~i2+1Γ1σ~2≤2p}.Theorem 5.
Suppose that the control laws in (31) and (40) with adaptive laws (14), (15), (32), and (41) are applied to the PMSM; Assumptions 1–4 stand, and if there exists a positive constant p that the initial condition satisfies V3≤p and the design constants ki, ai, γi, mi, Γ1, and c1 are chosen rationally, then all the signals of the closed-loop system are semiglobally uniformly ultimately bounded, and the output tracking error converges to a neighborhood of zero.Proof.
Define the Lyapunov function candidate as follows:(47)V=12(∑i=13Si2+y22+∑i=131γiλ~i2+1Γ1σ~2).
Consequently, one can obtain(48)V˙≤(2G-k1)S12-(k2-14G)S22-k3S32+(-1τ2+1+14G)y22-12c1σ~2-12∑i=13miγiλ~i2+12∑i=13miγiλi2+12c1σ2+∑i=13ai22+32+14B22≤-2a0V+b,
where b=(1/2)∑i=13(mi/γi)λi2+(1/2)c1σ2+∑i=13(ai2/2)+(3/2)+(1/4)B22.
Furthermore, (48) implies that
(49)0≤V(t)≤ba0+(V(0)-ba0)e-a0t≤ba0+V(0).


## 5. Performance Evaluation

The simulation is running to illustrate the effectiveness of the scheme presented in this paper under the assumption that the system parameters and nonlinear functions are uncertain. The initial conditionsx1(0)=x2(0)=1 and x3(0)=1.5 are used in this section. The controller parameters are chosen as follows: k1=35, k2=k3=15, γ1=γ2=γ3=2, m1=m2=m3=0.04, a1=a2=a3=30, τ1=0.01, η=0.01, Γ1=20, c1=0.02, λ~1(0)=λ~2(0)=λ~3(0)=0, and σ^(0)=0.7.To take into account the disturbance, the corresponding expressions are given as follows:(50)Δ1=Δ2=Δ3=0.04×x12cos⁡3(3t),TL={2.0Nm,0≤t≤43.5Nm,t>4.The simulation results are shown in Figures4(a)–4(f). Figures 4(a) and 4(b) explicitly illustrate that the state error of angular speed of the PMSM is gradually converged to zero within a short time. Meanwhile, these pictures show that the system successfully escapes from the chaotic behavior within 0.1 s and tracks the given trajectory with a great performance in spite of uncertainty, nonlinearity, and external disturbance. In Figures 4(c)–4(f), three kinds of curves basically overlap without disturbance on the whole time. Obviously, these pictures show the performance when the system parameters σ and γ have a perturbation. Furthermore, the indicators of the PMSM can still converge quickly when the model suffers from the disturbance. From the simulation results, it can clearly be seen that the proposed controller guarantees the boundedness of all the signals in the closed-loop system and also achieves the good tracking performance.The robustness analysis with 0.5sin(4t+1/2π) + 0.5sin(2t+1/2π).
(a)
The trajectory tracking
(b)
The rotor velocity tracking error
(c)
q-axis voltage
(d)
d-axis voltage
(e)
The periodic orbit
(f)
Thed-axis current


## 6. Conclusion

In this paper, an adaptive dynamic surface control method of chaos is applied to the permanent magnet synchronous motor based on the minimum weights of RBF neural network. The proposed controller guarantees the boundedness of all the signals in the closed-loop system, while the tracking error eventually converges to a small neighborhood of the origin. Moreover, the suggested controller contains minimum weights of RBF neural network. This makes the design scheme easier to be implemented in practical applications. Simulation results are given to show the effectiveness and robustness of the proposed controller. Finally, some potential future research works are pointed out, such as the recursive filtering for time-varying nonlinear systems and sliding mode design for time-invariant nonlinear systems.



---

*Source: 609340-2014-06-22.xml*