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

Authors: Jinpeng Yu; Junwei Gao; Yumei Ma; Haisheng Yu; Songfeng Pan
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## 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.


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## Body

## 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 d-q 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]:(2.1)dωdt=σ(iq-ω)-T̃L,diqdt=-iq-idω+γω+ũq,diddt=-id+iqω+ũd,
where ω, id and iq are state variables, which denote angle speed and the d-q axis currents, respectively. σ and γ are system operating parameters, which are positive. T̃L, ũd, and ũq stand for the d-q axis voltages and load torque,respectively.In system (2.1), the external inputs are set to zero, namely, T̃L=ũd=ũq=0 [1]. Then, the system (2.1) becomes an unforced system:(2.2)dωdt=σ(iq-ω),diqdt=-iq-idω+γω,diddt=-id+iqω.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 parametersσ and γ fall into a certain regime. For example, the PMSM displays chaos with σ = 5.45 and γ = 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 ud 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: x1=ω,x2=iq, and x3=id. By using these notations, the dynamic model of PMSM driver system can be described by the following differential equations:(2.3)ẋ1=σ(x2-x1),ẋ2=-x2-x1x3+γx1,ẋ3=-x3+x1x2+ud.Figure 1
Typical chaotic attractor in PMSM with system parametersσ = 5.45 and γ = 20.The control objective is to design an adaptive fuzzy controller such that the state variablex1 follows the given reference signal xd, 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:(2.4)Ri:IFx1isF1iand⋯andxnisFniTHENyisBi(i=1,2,…,N),
where x=[x1,…,xn]T∈Rn and y∈R are the input and output of the fuzzy system, respectively Fij and Bi are fuzzy sets in R. The fuzzy inference engine performs a mapping from fuzzy sets in Rn 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,…,xn]T∈Rn into a fuzzy set Ax 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(2.5)y(x)=∑j=1NWj∏i=1nμFij(xi)∑j=1N[∏i=1nμFij(xi)],
where Wj is the point at which fuzzy membership function μBj(Wj) achieves its maximum value, and it is further assumed that μBj(Wj)=1. Let pj(x)=∏i=1nμFij(xi)/∑j=1N[∏i=1nμFij(xi)], S(x)=[p1(x),p2(x),…,pN(x)]T, and W=[W1,…,WN]T, then the fuzzy logic system above can be rewritten as(2.6)y(x)=WTS(x).
If all memberships are taken as Gaussian functions, then the following lemma holds.Lemma 2.1 (see [19]).
Letf(x) be a continuous function defined on a compact set Ω. Then for any scalar ɛ>0, there exists a fuzzy logic system in the form of (2.6) such that
(2.7)supx∈Ω|f(x)-y(x)|≤ɛ.


## 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 contains3 steps. At each design step, a virtual control function αi(i=1,2) 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 signalxd, define the tracking error variable as z1=x1-xd. From the first differential equation of (2.3), the error dynamic system is given by ż1=σ(x2-x1)-ẋd.
Choose Lyapunov function candidate asV1=(1/2)z12, then the time derivative of V1 is computed by(3.1)V̇1=z1ż1=z1σ(x2-x1-ẋdσ).
Construct the virtual control law α1 as
(3.2)α1(x1,xd,ẋd)=-k¯1z1+x1+ẋdσ,
where k¯1 is a positive constant. By using (3.2), (3.1) can be rewritten in the following form:
(3.3)V̇1=-k¯1σz12+σz1z2=-k1z12+σz1z2,
with k1>0 being a design parameter and z2=x2-α1.Step 2.
Differentiatingz2 gives
(3.4)ż2=ẋ2-α̇1=-x2-x1x3+γx1-α̇1.
Now, choose the Lyapunov function candidate asV2=V1+(1/2)z22. Obviously, the time derivative of V2 is given by(3.5)V̇2=V̇1+z2ż2=-k1z12+z2(σz1-x2-x1x3+γx1-α̇1).
In the realistic model of PMSM, limited to the work conditions, the parameter γ is unknown. So it cannot be used to construct the control signal. Thus, let γ̂ be the estimation of γ. The corresponding adaptation laws will be specified later. The virtual control α2 is constructed as
(3.6)α2(Z2)=-1x1(-k2z2-σz1+x2-γ̂x1+α̇1),
where k2>0 is a positive design parameter and Z2=[x1,x2,xd,ẋd,ẍd,γ̂]T. Adding and subtracting α2 in the bracket in (3.5) shows that
(3.7)V̇2=-k1z12-k2z22-x1z2z3-z2(γ̂-γ)x1,
with z3=x3-α2.Step 3.
Differentiatingz3 results in the following differential equation:
(3.8)ż3=ẋ3-α̇2=-x3+x1x2+ud-α̇2.
Choose the Lyapunov function candidate as V3=V2+(1/2)z32. Furthermore, differentiating V3 yields
(3.9)V̇3=V̇2+z3ż3=V̇2+z3(-x3+x1x2+ud-α̇2)=-k1z12-k2z22-z2(γ̂-γ)x1+z3(f3+ud),
where
(3.10)α̇1=ẋ1-ẋd=σ(x2-x1)-ẋd,α̇2=∑i=12∂α2∂xiẋi+∑i=02∂α2∂xd(i)xd(i+1)+∂α2∂γ̂γ̂̇=∂α2∂x1σ(x2-x1)+∂α2∂x2(-x2-x1x3+γx1)+∑i=02∂α2∂xd(i)xd(i+1)+∂α2∂γ̂γ̂̇,f3(Z)=-x3+x1x2-x1z2-α̇2,Z=[x1,x2,x3,xd,ẋd,ẍd,γ̂]T.Notice thatf3 containing the derivative of α2, therewithal, the unknown parameter γ appears in the expression of f3. This will make the classical adaptive backstepping design become very complex and troubled, and the designed control law ud 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 f3. As shown later, the design procedure of ud becomes simple, and ud is of a simple structure.According to Lemma2.1, for any given ɛ3>0, there exists a fuzzy logic system W3TS(Z) such that(3.11)f3(Z)=W3TS(Z)+δ3(Z),
where δ3(Z) is the approximation error and satisfies |δ3|≤ɛ3. Consequently, a straightforward calculation produces the following inequality:(3.12)z3f3(Z)=z3(W3TS(Z)+δ3(Z))≤12l32z32∥W3∥2S2+12l32+12z32+12ɛ32.
Thus, it follows immediately from substituting (3.6) into (3.9) that(3.13)V̇3≤-k1z12-k2z22-z2(γ̂-γ)x1+12l32z32∥W3∥2S2+12l32+12z32+12ɛ32+z3ud.
At this present stage, the control law ud is designed as(3.14)ud=-k3z3-12z3-12l32z3θ̂S2,
where θ̂ is the estimation of the unknown constant θ which will be specified later. Define θ=∥W3∥2. Furthermore, using equality (3.14), it can be verified easily that(3.15)V̇3≤-∑i=13kizi2+12l32z32(∥W3∥2-θ̂)ST(Z)S(Z)+12l32+12ɛ32+z2(γ-γ̂)x1.Introduce variablesγ̃ and θ̃ as(3.16)γ̃=γ̂-γ,θ̃=θ̂-θ,
and choose the Lyapunov function candidate as(3.17)V=V3+12r1γ̃2+12r2θ̃2,
where ri, i=1,2 are positive constants. By differentiating V and taking (3.15)–(3.17) into account, one has(3.18)V̇≤-∑i=13kizi2+12l32z32θ̃ST(Z)S(Z)+z2(γ-γ̂)x1+12l32+12ɛ32+1r1γ̃γ̂̇+1r2θ̃θ̂̇=-∑i=13kizi2+12l32+12ɛ32+1r1γ̃(-r1z2x1+γ̂̇)+1r2θ̃[-12l32z32ST(Z)S(Z)+θ̂̇].
According to (3.18), the corresponding adaptive laws are chosen as follows:(3.19)γ̂̇=r1z2α̇1-m1γ̂,θ̂̇=12l32z32ST(Z)S(Z)-m2θ̂,
where mi, for i=1,2 and l3 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(4.1)V̇≤-∑i=13kizi2+12l32+12ɛ32-m1r1γ̃γ̂-m2r2θ̃θ̂.
For the term -γ̃γ̂, one has -γ̃γ̂≤-γ̃(γ̃+γ)≤-(1/2)γ̃2+(1/2)γ2. Similarly, -θ̃θ̂≤-(1/2)θ̃2+(1/2)θ2 holds. Consequently, by using these inequalities, (4.1) can be rewritten in the following form:(4.2)V̇≤-∑i=13kizi2-m12r1γ̃2-m22r2θ̃2+12l32+12ɛ32+m12r1γ2+m22r2θ2≤-a0V+b0,
where a0=min{2k1,2k2,2k3,m1,m2} and b0=(1/2)l32+(1/2)ɛ32+(m1/2r1)γ2+(m2/2r2)θ2. Furthermore, (4.2) implies that(4.3)V(t)≤(V(t0)-b0a0)e-a0(t-t0)+b0a0≤V(t0)+b0a0,∀t⩾t0.
As a result, all zi(i=1,2,3), γ̃, and θ̃ belong to the compact set(4.4)Ω={(zi,γ̃,θ̃)∣V≤V(t0)+b0a0,∀t⩾t0}.
Namely, all the signals in the closed-loop system are bounded. Especially, from (4.3) we have(4.5)limt→∞z12≤2b0a0.
From the definitions of a0 and b0, it is clear that to get a small tracking error by taking ri sufficiently large and li and ɛi small enough after giving the parameters ki and mi.


## 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 withud=0, 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: (5.1)k1=2,k2=20,k3=15,r1=r2=15,m1=m2=0.005,l3=0.2.Figure 2
Theω curve of the chaotic PMSM drive system without ud.Figure 3
Theid curve of the chaotic PMSM drive system without ud.Figure 4
Theiq curve of the chaotic PMSM drive system without ud.Figure 5
Theω curve of the chaotic PMSM drive system with ud.Figure 6
Theiq curve of the chaotic PMSM drive system with ud.Figure 7
Theid curve of the chaotic PMSM drive system with ud.Figure 8
The curve of the controllerud.And the fuzzy membership functions are:(5.2)μFi1=exp[-(x+5)22],μFi2=exp[-(x+4)22],μFi3=exp[-(x+3)22],μFi4=exp[-(x+2)22],μFi5=exp[-(x+1)22],μFi6=exp[-(x-0)22],μFi7=exp[-(x-1)22],μFi8=exp[-(x-2)22],μFi9=exp[-(x-3)22],μFi10=exp[-(x-4)22],μFi11=exp[-(x-5)22].
Give the reference signals xd=5 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.



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*Source: 269283-2010-08-16.xml*