# Adaptive Robust Backstepping Control of Permanent Magnet Synchronous Motor Chaotic System with Fully Unknown Parameters and External Disturbances

Authors: Yang Yu; Xudong Guo; Zengqiang Mi
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

The chaotic behavior of permanent magnet synchronous motor is directly related to the parameters of chaotic system. The parameters of permanent magnet synchronous motor chaotic system are frequently unknown. Hence, chaotic control of permanent magnet synchronous motor with unknown parameters is of great significance. In order to make the subject more general and feasible, an adaptive robust backstepping control algorithm is proposed to address the issues of fully unknown parameters estimation and external disturbances inhibition on the basis of associating backstepping control with adaptive control. Firstly, the mathematical model of permanent magnet synchronous motor chaotic system with fully unknown parameters is constructed, and the external disturbances are introduced into the model. Secondly, an adaptive robust backstepping control technology is employed to design controller. In contrast with traditional backstepping control, the proposed controller is more concise in structure and avoids many restricted problems. The stability of the control approach is proved by Lyapunov stability theory. Finally, the effectiveness and correctness of the presented algorithm are verified through multiple simulation experiments, and the results show that the proposed scheme enables making permanent magnet synchronous motor operate away from chaotic state rapidly and ensures the tracking errors to converge to a small neighborhood within the origin rapidly under the full parameters uncertainties and external disturbances.


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## Body

## 1. Introduction

In recent years, the permanent magnet synchronous motor (PMSM) is utilized widely in various industrial fields due to its constantly dropping production cost, simple structure, high torque, and high efficiency. However, Hemati found that PMSM would generate chaotic behavior with system parameters entering into a certain region [1]. Previous studies have shown that the chaotic movement of PMSM will produce irregular oscillations of torque and speed, exacerbate current noise, and worsen operation performance and may even damage the entire drive system. Therefore, research on PMSM chaos phenomenon has attracted extensive attention worldwide [2–5], and further studying on the control method of PMSM chaos is of extreme significance [6–8].The nonlinear characteristics of PMSM, such as multivariability, strong coupling, and high dimension, make it difficult to control for traditional linear control theory. Hence, a variety of modern and nonlinear control algorithms are introduced to suppress PMSM chaotic behavior. In terms of these control algorithms whether or not relying on the model parameters, the previous control methods can primarily be divided into two categories. The first type is on the basis of accurate model parameters, such as entrainment and migration control [9], exact feedback linearization control [10], and decoupling control [11]. However, the accuracy of these control methods directly depends on PMSM model parameters; if the system parameters deviate from the rated values, the control performance will go bad. The second type is based on unknown parameters, which have become the research focus of PMSM chaos suppression recently, mainly including sliding mode variable structure control [12, 13], fuzzy control [14], and H∞ control [15]. However, sliding mode variable structure control requires uncertain parameters to satisfy certain matching conditions, fuzzy control is dependent on the fuzzification of Takagi-Sugeno, and H∞ control is inclined to ignore the operating states under special conditions [16]. In essence, PMSM chaotic system is highly sensitive to initial states and parameters, and PMSM model parameters are susceptible to the temperature and humidity of the surrounding environment. Therefore, PMSM chaotic repression with unknown model parameters has applicability to a broader field and is more in line with reality [17]. Actually, the adaptive control (AC) provides a natural routine for PMSM chaotic control with unknown parameters, which has been presented in literatures [12, 13, 18].Backstepping control (BC) is one of the most popular nonlinear control methods newly proposed to address parameter uncertainty, specifically the uncertainty not satisfying matching condition, which has been successfully applied to many engineering fields such as motor drive, temperature control of boiler main steam, and rocket location tracking. The core idea of BC is that complex high-dimensional nonlinear systems are decomposed into many simple low-dimensional subsystems and virtual control variables are introduced to backstepping process to design concrete controllers. In addition, BC has been successfully applied to suppress Liu chaotic system [19] and Chen chaotic system [20]. Therefore, the idea of combining BC with AC provides a useful and feasible train of thought to control PMSM chaotic system with unknown parameters. Literatures [21, 22] have exactly practiced this idea.However, the conventional backstepping approach is confronted with two major problems of solving complicated “regression matrix” [23] and encountering “explosion of terms” [24]. In [25], the complexity of regression matrix is sufficiently manifested, which almost occupies one full page. Nevertheless, explosion of terms is an inherent shortcoming and is induced by repeated differentiations of virtual variables, particularly in design of adaptive backstepping controller [26]. Additionally, integration of BC with AC is frequently faced with the singularity arising from any estimation term emerging as a denominator of any control input. The overparameterization caused by the number of estimations larger than actual system parameters hinders the conventional adaptive backstepping control.In addition to the above problems, to the extent of our knowledge, mostly existing literatures on PMSM chaotic control only concentrate on the cases of single unknown parameter and partial unknown parameters [21, 22], and there is no way to address the issue of fully unknown parameters. Furthermore, the existing researches mainly aim at the situation of sudden power failure during PMSM operation [16]; the existing conclusions lack the generality. Hence, through combination of BC and AC, not only does this paper study the control issue of PMSM chaos suppression with fully unknown parameters, but also the external disturbances are taken into account in PMSM chaos model. Newly adaptive updating laws of unknown parameters are designed to totally estimate unknown parameters of PMSM chaotic model, and adaptive robust backstepping controllers on the basis of adaptive estimations and external disturbances are developed to drive PMSM to escape out of chaotic state quickly, inhibit the external disturbances, and accomplish the given signals tracking rapidly. The method proposed in this paper expands the applied range of backstepping control theory in PMSM chaotic system. Moreover, the study of chaos control problem with totally unknown parameters and external disturbances is more general and practical, and the results and conclusions obtained are more applicable.


## 2. PMSM Chaotic Model with Fully Unknown Parameters

For a PMSM, its mathematical model indq axis coordinate system can be described as follows [16]:(1)ω⌢˙=3pϕm2Ji⌢q-BJω⌢-τ⌢lJ,i⌢˙q=-RLqi⌢q-pω⌢·i⌢d-pϕmLq·ω⌢+1Lqu⌢q,i⌢˙d=-RLdi⌢d+pω⌢·i⌢q+1Ldu⌢d,where ω⌢ is the mechanical angular velocity of the rotating rotor, i⌢d and i⌢q are d axis and q axis currents of stator winding, respectively, u⌢d and u⌢q are d axis and q axis voltages of stator winding, p is the number of rotor pole pairs, ϕm is the flux generated by permanent magnets, J is the moment of inertia, B is the viscous damping coefficient, τ⌢l is the load torque, R is the phase resistance of the stator windings, and Ld and Lq are d axis and q axis inductances of stator winding, respectively. For a PMSM with uniform air gap, Ld=Lq. Hence, we use L to substitute Ld and Lq in the following paper.Selecting the affine transformationω⌢i⌢qi⌢d=1/pτ000k000kωiqid and time scale transformation t⌢=τt, the PMSM mathematical model described in (1) can be converted into dimensionless form as follows:(2)ω˙=1δiq-ω-τl,i˙q=-iq-ω·id+γ·ω+uq,i˙d=-id+ω·iq+ud,where τ=L/R, k=2B/3p2τϕm, ud=1/kRu⌢d, γ=-ϕm/kL, uq=1/kRu⌢q, δ=J/Bτ, and τl=pτ2τ⌢l/J.As presented in (2), the dynamic performance of PMSM depends on three parameters δ, γ, and τl. Considering the most general case, let δ=0.2, γ=50, τl=3.2, ud=-0.6, and uq=0.8. If the initial state is selected as ω,iq,id=0,0,0, PMSM system will run on a chaotic state and display the chaotic behavior. A typical chaotic attractor of PMSM is manifested in Figure 1.Figure 1
Chaotic attractor of PMSM system.In reality, the three parametersδ, γ, and τl in (2) tend to be unknown or to have uncertainties resulting from operating conditions. In other words, when all the parameters δ, γ, and τl cannot be determined, (2) actually represents PMSM chaotic system model with fully unknown parameters.


## 3. Design of Adaptive Robust Controller with Backstepping Approach

Taking a more general situation into account, the PMSM chaotic model described in (2) is immersed by external disturbances. The model can be rewritten as follows:(3)ω˙=1δiq-ω-τl,i˙q=-iq-ω·id+γ·ω+uq+Δ1x,t,i˙d=-id+ω·iq+ud+Δ2x,t,where Δ1(x,t) and Δ2(x,t) represent the external disturbances, x indicates the system states, and x=x1,x2,x3=ω,iq,id.


### 3.1. Control Objective and Assumptions

Control problem in the paper can be described as follows: for PMSM chaotic system (3) with fully unknown parameters δ, γ, and τl and external disturbances Δ1 and Δ2, adaptive laws of unknown parameters δ, γ, and τl are designed and adaptive robust controllers ud and uq are constructed to ensure PMSM breaks away from chaos rapidly and runs into an expected orbit. Simultaneously, the fully unknown parameters δ, γ, and τl can be estimated accurately and the external disturbances can be inhibited effectively.For convenience of controller design, the control system is supposed to hold some reasonable assumptions as follows.Assumption 1.
The state variables for PMSM chaotic systemω,iq,id are observable.Assumption 2.
The external disturbancesΔi(x,t) satisfy the condition Δix,t≤dixfit,i=1,2, where dix is a known function, fit is an unknown but bounded time-varying function, and fit≤fimax, where fimax is a constant.Assumption 3.
The desired speed andd axis current reference signals ω∗ and id∗ and their derivatives are known and bounded.The estimated values of unknown system parameters are described asδ^, γ^, and τ^l; then, the estimation errors δ~, γ~, and τ~l can be expressed as follows:(4)δ~=δ^-δ,γ~=γ^-γ,τ~l=τ^l-τl.


### 3.2. Controller Design

The essence of adaptive robust backstepping controller is to design controller through combination of backstepping method and adaptive approach; then, a reasonably stable function is built in accordance with Lyapunov stability theory to guarantee error variables to be effectively stabilized and meanwhile ensure the output of closed loop system tracks reference signals quickly. On the basis of this, the adaptive robust backstepping controller is designed as follows.Step 1.
For the speed reference signalω∗, define the tracking error eω as follows:(5)eω=ω-ω∗.Taking PMSM chaotic system model (3) into account, the derivative of (5) can be written as(6)e˙ω=ω˙-ω˙∗=1δiq-ω-τl-ω˙∗.
Define the tracking erroreq of q axis stator current iq as follows:(7)eq=iq-iq∗,where iq∗ is the expected output value of iq.
For thed axis current reference signal id∗, its tracking error ed is defined as follows:(8)ed=id-id∗.
By substitution of (7) into (6), we can obtain(9)e˙ω=1δeq+iq∗-ω-τl-ω˙∗=1δiq∗-ω-τ^l+τ~l-ω˙∗.Let(10)iq∗=ω+δ^τ^l+ω˙∗-k1eω,(11)id∗=0,where k1 represents the positive control gain.
Through substitution of (10) into (9), (12) can be obtained:(12)e˙ω=1δeq+δ^δτ^l+ω˙∗-k1eω-τ^l+τ~l-ω˙∗=1δeq+δ+δ~δτ^l+ω˙∗-k1eω-τ^l+τ~l-ω˙∗=1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω.Lyapunov function V1 is selected as follows:(13)V1=12eω2.Then, the derivative of V1 can be described as(14)V˙1=eωe˙ω=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω.Step 2.
To stabilize the outputq axis current of PMSM, the derivative of eq is conducted as follows:(15)e˙q=i˙q-i˙q∗=-iq-ω·id+γ·ω+uq+Δ1-ω˙+δ^˙τ^l+ω˙∗-k1·eω+δ^τ^˙l+ω¨∗-k1·e˙ω=-iq-ω·id+γ·ω+uq+Δ1-ω˙+δ^˙τ^l+ω˙∗-k1·eω+δ^τ^˙l+ω¨∗-k1·e˙ω=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-ω˙-δ^τ^˙l+ω¨∗-k1·e˙ω=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-e˙ω-ω˙∗-δ^τ^˙l+ω¨∗+δ^·k1ω-ω˙∗.By substitution of (12) into (15), we can get(16)e˙q=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^·k1ω-ω˙∗.Combined with the mathematical model of PMSM chaotic system, (16) can be further calculated as follows:(17)e˙q=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+kr·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^·k1·iq-ω-τlδ-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^δ·k1·iq-ω-δ^·k1·τl-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ~l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^δ·k1·iq-ω-δ^·k1·τ^l-τ~l-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗.By substitution of δ^=δ~+δ into (17), the following equation can be obtained:(18)e˙q=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1eω-1δeq-δ~δτ^l+ω˙∗-k1eω-τ~l+k1eω-ω˙∗-δ^τ^˙l+ω¨∗+δ+δ~δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1eω-1δeq-δ~δτ^l+ω˙∗-k1eω-τ~l+k1eω-ω˙∗-δ^τ^˙l+ω¨∗+k1·iq-ω+δ~δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗.Lyapunov function V2 is chosen as follows:(19)V2=V1+12eq2.Then, the derivative of V2 can be described as(20)V˙2=V˙1+eqe˙q=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1eω-1δeq-δ~δτ^l+ω˙∗-k1eω-τ~l+k1eω-ω˙∗-δ^τ^˙l+ω¨∗+k1·iq-ω+δ~δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗.The first control variable is selected as(21)uq=uqs+uqr,where uqs and uqr are the model compensation and robust control inputs, respectively.
Then,uqs and uqr can be, respectively, chosen as(22)uqs=-k2·eq+iq+ω·id-γ^·ω+δ^˙τ^l+ω˙∗-k1eω+ω˙∗+δ^τ^˙l+ω¨∗+δ^·k1·τ^l+δ^·kr·ω˙∗-k1iq-ω,(23)uqr=-d12x4ε1eq,where k2 is another positive control gain and ε1 is a positive number chosen arbitrarily.
By substitution of (21) and (22) into (20), we can acquire(24)V˙2=V˙1+eqe˙q=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-k2eq-1δ·eq+k1eω-γ~·ω+uqr+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-k2eq-1δ·eq+k1eω-γ~·ω-d12x4ε1eq+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1.Additionally, (25)eq-d12x4ε1eq+Δ1=-d12x4ε1eq2+Δ1eq≤-d12x4ε1eq2+d1xf1maxeq=-d1xeq2ε1-ε1f1max2+ε1f1max2.Step 3.
Differentiating the tracking errored of d axis current id, we can get(26)e˙d=i˙d-i˙d∗=-id+ω·iq+ud+Δ2-i˙d∗.Lyapunov function V3 is chosen as(27)V3=V2+12ed2.Then, the derivative of V3 can be represented as(28)V˙3=V˙2+ede˙d=V˙2+ed-id+ω·iq+ud+Δ2-i˙d∗.In terms of (28), d axis output stator voltage ud can be calculated:(29)ud=uds+udr,where(30)uds=-k3·ed+id-ω·iq+id∗,(31)udr=-d22x4ε2ed,where k3 is the positive control gain and ε2 is a positive number chosen arbitrarily.
By substitution of (30) and (31) into (26) and (28), respectively, the following equations can be acquired: (32)e˙d=-k3·ed+udr+Δ2=-k3·ed-d22x4ε2ed+Δ2,(33)V˙3=V˙2+ede˙d=V˙2+ed-k3·ed-d22x4ε2ed+Δ2,(34)ed-d22x4ε2ed+Δ2=-d22x4ε2ed2+Δ2ed≤-d22x4ε2ed2+d2xf2maxed=-d2xed2ε2-ε2f2max2+ε2f2max2.Step 4.
Lyapunov functionV of PMSM chaotic system with fully unknown parameters and external disturbances is selected as follows:(35)V=V3+12θ1τ~l2+12θ2γ~2+12δ·θ3δ~2,where θ1, θ2, and θ3 represent positive adaptive gains.
Combined with equationsδ~˙=δ^˙, γ~˙=γ^˙, and τ~˙l=τ^˙l, derivative of selected Lyapunov function V can be calculated as follows: (36)V˙=V˙3+τ~lθ1τ^˙l+γ~θ2γ^˙+δ~δ·θ3δ^˙=V˙2+ed-k3·ed+udr+Δ2=V˙1+eq-k2eq-1δ·eq+k1eω-γ~·ω+uqr+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1+ed-k3·ed+udr+Δ2=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-k2eq-1δ·eq+k1eω-γ~·ω+uqr+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1+ed-k3·ed+udr+Δ2=1δeqeω+δ~δτ^l+ω˙∗-k1eω·eω+τ~leω-k1eω2-k3ed2-k2eq2-γ~ωeq-1δeq2+equqr+Δ1+edudr+Δ2-δ~δτ^l+ω˙∗-k1eω-k1iq-ω·eq+τ~lδ^·k1-1eq+k1eωeq+τ~lθ1τ^˙l+γ~θ2γ^˙+δ~δθ3δ^˙=1δeqeω+k1eωeq+δ~δτ^l+ω˙∗-k1eωeω-τ^l+ω˙∗-k1eωeq+k1iq-ωeq+δ^˙θ3-k1eω2-k2ed2-k3eq2+τ~leω-eq+δ^k1eq+τ^˙lθ1+γ~-ωeq+γ^˙θ2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2.In terms of (36), the adaptive laws of unknown parameters δ, γ, and τl can be selected, respectively, as follows:(37)δ^˙=-θ3τ^l+ω˙∗-k1eω·eω-eq+k1iq-ωeq,(38)γ^˙=θ2ωeq,(39)τ^˙l=-θ1eω-eq+δ^k1eq.By substitution of (37), (38), and (39) into (36), (36) can be simplified as follows:(40)V˙=1δeqeω+k1eqeω-k1eω2-k3ed2-k2eq2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2.


### 3.3. Stability Analysis

Theorem 4.
For PMSM chaotic system model (3) with fully unknown parameters and external disturbances, design of adaptive control laws (37), (38), and (40) and selection of suitable controller gains k1, k2, k3, ε1, and ε2 and adaptive gains θ1, θ2, and θ3, the proposed adaptive robust backstepping controllers (21) and (29) can ensure the tracking error signals (5), (7), and (8) of PMSM chaotic systems are asymptotically stable. That is to say, PMSM chaotic system can run out of chaos quickly through the proposed controllers (21) and (29) and track the given reference signals.Through stability analysis, we want to verify the correctness of the theorem.According to (40), new expression can be obtained as follows through some mathematical computations:(41)V˙=1δeqeω+k1eqeω-k1eω2-k3ed2-k2eq2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2=-12δeω-eq2+12δeω2+12δeq2-k12eω-eq2+k12eω2+k12eq2-k1eω2-k3ed2-k2eq2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2=-12δeω-eq2-k12eω-eq2-k3ed2+12δ+k12-k1eω2+12δ+k12-k2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2.Appropriate controller gains k1 and k2 are selected as follows:(42)12δ+k12-k1<0,12δ+k12-k2-1δ<0.Equation (42) can be replaced by the following:(43)k1>1δ,k1-2k2<1δ.Then, by substitution of (43) into (41), we can obtain(44)V˙=-12δeω-eq2-k12eω-eq2-k3ed2+12δ+k12-k1eω2+12δ+k12-k2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2≤-12δeω-eq2-k12eω-eq2-k3ed2-k4eω2-k5eq2-d1xeq2ε1-ε1f1max2+ε1f1max2-d2xed2ε2-ε2f2max2+ε2f2max2≤-12δeω-eq2-k12eω-eq2-k3ed2-k4eω2-k5eq2-d1xeq2ε1-ε1f1max2-d2xed2ε2-ε2f2max2+ε0,where k4=-1/2δ+k1/2-k1≥0, k5=-1/2δ+k1/2-k2-1/δ≥0, and ε0=ε1f1max2+ε2f2max2.Let(45)Wit=-12δeω-eq2-k12eω-eq2-k3ed2-k4eω2-k5eq2-d1xeq2ε1-ε1f1max2-d2xed2ε2-ε2f2max2+ε0,where i(t)=eω,eq,ed.By integration of (45), we can get(46)∫t0tWitdt=-∫t0tV˙itdt⟹∫t0tWitdt=Vt0-Vt.Since V(t0) is bounded and V(t) is bounded and nonincreasing, hence(47)limt→∞⁡∫t0tWitdt<∞.Moreover, Wi(t) is uniform continuous and W˙i(t) is bounded. In accordance with Barbalat’s Lemma, the following equation can be obtained:(48)limt→∞⁡Wit=0.Apparently, through selection of suitable controller gains k1, k2, k3, ε1, and ε2, V˙ can be ensured to be negative definite.The above derivation has proved that the selected suitable controller gainsk1, k2, k3, ε1, and ε2 and adaptive gains θ1, θ2, and θ3 can make the inequalities of V≥0 and V˙<0 hold. In addition, equation of V=0 is not satisfied until eω=eq=ed=τ~l=γ~=δ~=0. In summary, PMSM chaotic system is globally asymptotically stable at the equilibrium point of eω,eq,ed=0,0,0.


## 3.1. Control Objective and Assumptions

Control problem in the paper can be described as follows: for PMSM chaotic system (3) with fully unknown parameters δ, γ, and τl and external disturbances Δ1 and Δ2, adaptive laws of unknown parameters δ, γ, and τl are designed and adaptive robust controllers ud and uq are constructed to ensure PMSM breaks away from chaos rapidly and runs into an expected orbit. Simultaneously, the fully unknown parameters δ, γ, and τl can be estimated accurately and the external disturbances can be inhibited effectively.For convenience of controller design, the control system is supposed to hold some reasonable assumptions as follows.Assumption 1.
The state variables for PMSM chaotic systemω,iq,id are observable.Assumption 2.
The external disturbancesΔi(x,t) satisfy the condition Δix,t≤dixfit,i=1,2, where dix is a known function, fit is an unknown but bounded time-varying function, and fit≤fimax, where fimax is a constant.Assumption 3.
The desired speed andd axis current reference signals ω∗ and id∗ and their derivatives are known and bounded.The estimated values of unknown system parameters are described asδ^, γ^, and τ^l; then, the estimation errors δ~, γ~, and τ~l can be expressed as follows:(4)δ~=δ^-δ,γ~=γ^-γ,τ~l=τ^l-τl.


## 3.2. Controller Design

The essence of adaptive robust backstepping controller is to design controller through combination of backstepping method and adaptive approach; then, a reasonably stable function is built in accordance with Lyapunov stability theory to guarantee error variables to be effectively stabilized and meanwhile ensure the output of closed loop system tracks reference signals quickly. On the basis of this, the adaptive robust backstepping controller is designed as follows.Step 1.
For the speed reference signalω∗, define the tracking error eω as follows:(5)eω=ω-ω∗.Taking PMSM chaotic system model (3) into account, the derivative of (5) can be written as(6)e˙ω=ω˙-ω˙∗=1δiq-ω-τl-ω˙∗.
Define the tracking erroreq of q axis stator current iq as follows:(7)eq=iq-iq∗,where iq∗ is the expected output value of iq.
For thed axis current reference signal id∗, its tracking error ed is defined as follows:(8)ed=id-id∗.
By substitution of (7) into (6), we can obtain(9)e˙ω=1δeq+iq∗-ω-τl-ω˙∗=1δiq∗-ω-τ^l+τ~l-ω˙∗.Let(10)iq∗=ω+δ^τ^l+ω˙∗-k1eω,(11)id∗=0,where k1 represents the positive control gain.
Through substitution of (10) into (9), (12) can be obtained:(12)e˙ω=1δeq+δ^δτ^l+ω˙∗-k1eω-τ^l+τ~l-ω˙∗=1δeq+δ+δ~δτ^l+ω˙∗-k1eω-τ^l+τ~l-ω˙∗=1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω.Lyapunov function V1 is selected as follows:(13)V1=12eω2.Then, the derivative of V1 can be described as(14)V˙1=eωe˙ω=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω.Step 2.
To stabilize the outputq axis current of PMSM, the derivative of eq is conducted as follows:(15)e˙q=i˙q-i˙q∗=-iq-ω·id+γ·ω+uq+Δ1-ω˙+δ^˙τ^l+ω˙∗-k1·eω+δ^τ^˙l+ω¨∗-k1·e˙ω=-iq-ω·id+γ·ω+uq+Δ1-ω˙+δ^˙τ^l+ω˙∗-k1·eω+δ^τ^˙l+ω¨∗-k1·e˙ω=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-ω˙-δ^τ^˙l+ω¨∗-k1·e˙ω=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-e˙ω-ω˙∗-δ^τ^˙l+ω¨∗+δ^·k1ω-ω˙∗.By substitution of (12) into (15), we can get(16)e˙q=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^·k1ω-ω˙∗.Combined with the mathematical model of PMSM chaotic system, (16) can be further calculated as follows:(17)e˙q=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+kr·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^·k1·iq-ω-τlδ-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^δ·k1·iq-ω-δ^·k1·τl-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ~l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^δ·k1·iq-ω-δ^·k1·τ^l-τ~l-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1·eω-1δeq-δ~δτ^l+ω˙∗-k1·eω-τ^l+k1·eω-ω˙∗-δ^τ^˙l+ω¨∗+δ^δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗.By substitution of δ^=δ~+δ into (17), the following equation can be obtained:(18)e˙q=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1eω-1δeq-δ~δτ^l+ω˙∗-k1eω-τ~l+k1eω-ω˙∗-δ^τ^˙l+ω¨∗+δ+δ~δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗=-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1eω-1δeq-δ~δτ^l+ω˙∗-k1eω-τ~l+k1eω-ω˙∗-δ^τ^˙l+ω¨∗+k1·iq-ω+δ~δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗.Lyapunov function V2 is chosen as follows:(19)V2=V1+12eq2.Then, the derivative of V2 can be described as(20)V˙2=V˙1+eqe˙q=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-iq-ω·id+γ^·ω-γ~·ω+uq+Δ1-δ^˙τ^l+ω˙∗-k1eω-1δeq-δ~δτ^l+ω˙∗-k1eω-τ~l+k1eω-ω˙∗-δ^τ^˙l+ω¨∗+k1·iq-ω+δ~δ·k1·iq-ω-δ^·k1·τ^l+δ^·k1·τ~l-δ^·k1·ω˙∗.The first control variable is selected as(21)uq=uqs+uqr,where uqs and uqr are the model compensation and robust control inputs, respectively.
Then,uqs and uqr can be, respectively, chosen as(22)uqs=-k2·eq+iq+ω·id-γ^·ω+δ^˙τ^l+ω˙∗-k1eω+ω˙∗+δ^τ^˙l+ω¨∗+δ^·k1·τ^l+δ^·kr·ω˙∗-k1iq-ω,(23)uqr=-d12x4ε1eq,where k2 is another positive control gain and ε1 is a positive number chosen arbitrarily.
By substitution of (21) and (22) into (20), we can acquire(24)V˙2=V˙1+eqe˙q=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-k2eq-1δ·eq+k1eω-γ~·ω+uqr+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-k2eq-1δ·eq+k1eω-γ~·ω-d12x4ε1eq+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1.Additionally, (25)eq-d12x4ε1eq+Δ1=-d12x4ε1eq2+Δ1eq≤-d12x4ε1eq2+d1xf1maxeq=-d1xeq2ε1-ε1f1max2+ε1f1max2.Step 3.
Differentiating the tracking errored of d axis current id, we can get(26)e˙d=i˙d-i˙d∗=-id+ω·iq+ud+Δ2-i˙d∗.Lyapunov function V3 is chosen as(27)V3=V2+12ed2.Then, the derivative of V3 can be represented as(28)V˙3=V˙2+ede˙d=V˙2+ed-id+ω·iq+ud+Δ2-i˙d∗.In terms of (28), d axis output stator voltage ud can be calculated:(29)ud=uds+udr,where(30)uds=-k3·ed+id-ω·iq+id∗,(31)udr=-d22x4ε2ed,where k3 is the positive control gain and ε2 is a positive number chosen arbitrarily.
By substitution of (30) and (31) into (26) and (28), respectively, the following equations can be acquired: (32)e˙d=-k3·ed+udr+Δ2=-k3·ed-d22x4ε2ed+Δ2,(33)V˙3=V˙2+ede˙d=V˙2+ed-k3·ed-d22x4ε2ed+Δ2,(34)ed-d22x4ε2ed+Δ2=-d22x4ε2ed2+Δ2ed≤-d22x4ε2ed2+d2xf2maxed=-d2xed2ε2-ε2f2max2+ε2f2max2.Step 4.
Lyapunov functionV of PMSM chaotic system with fully unknown parameters and external disturbances is selected as follows:(35)V=V3+12θ1τ~l2+12θ2γ~2+12δ·θ3δ~2,where θ1, θ2, and θ3 represent positive adaptive gains.
Combined with equationsδ~˙=δ^˙, γ~˙=γ^˙, and τ~˙l=τ^˙l, derivative of selected Lyapunov function V can be calculated as follows: (36)V˙=V˙3+τ~lθ1τ^˙l+γ~θ2γ^˙+δ~δ·θ3δ^˙=V˙2+ed-k3·ed+udr+Δ2=V˙1+eq-k2eq-1δ·eq+k1eω-γ~·ω+uqr+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1+ed-k3·ed+udr+Δ2=eω1δeq+δ^δτ^l+ω˙∗-k1eω+τ~l-k1eω+eq-k2eq-1δ·eq+k1eω-γ~·ω+uqr+Δ1-δ~δτ^l+ω˙∗-k1eω-k1·iq-ω+τ~lδ^·k1-1+ed-k3·ed+udr+Δ2=1δeqeω+δ~δτ^l+ω˙∗-k1eω·eω+τ~leω-k1eω2-k3ed2-k2eq2-γ~ωeq-1δeq2+equqr+Δ1+edudr+Δ2-δ~δτ^l+ω˙∗-k1eω-k1iq-ω·eq+τ~lδ^·k1-1eq+k1eωeq+τ~lθ1τ^˙l+γ~θ2γ^˙+δ~δθ3δ^˙=1δeqeω+k1eωeq+δ~δτ^l+ω˙∗-k1eωeω-τ^l+ω˙∗-k1eωeq+k1iq-ωeq+δ^˙θ3-k1eω2-k2ed2-k3eq2+τ~leω-eq+δ^k1eq+τ^˙lθ1+γ~-ωeq+γ^˙θ2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2.In terms of (36), the adaptive laws of unknown parameters δ, γ, and τl can be selected, respectively, as follows:(37)δ^˙=-θ3τ^l+ω˙∗-k1eω·eω-eq+k1iq-ωeq,(38)γ^˙=θ2ωeq,(39)τ^˙l=-θ1eω-eq+δ^k1eq.By substitution of (37), (38), and (39) into (36), (36) can be simplified as follows:(40)V˙=1δeqeω+k1eqeω-k1eω2-k3ed2-k2eq2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2.


## 3.3. Stability Analysis

Theorem 4.
For PMSM chaotic system model (3) with fully unknown parameters and external disturbances, design of adaptive control laws (37), (38), and (40) and selection of suitable controller gains k1, k2, k3, ε1, and ε2 and adaptive gains θ1, θ2, and θ3, the proposed adaptive robust backstepping controllers (21) and (29) can ensure the tracking error signals (5), (7), and (8) of PMSM chaotic systems are asymptotically stable. That is to say, PMSM chaotic system can run out of chaos quickly through the proposed controllers (21) and (29) and track the given reference signals.Through stability analysis, we want to verify the correctness of the theorem.According to (40), new expression can be obtained as follows through some mathematical computations:(41)V˙=1δeqeω+k1eqeω-k1eω2-k3ed2-k2eq2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2=-12δeω-eq2+12δeω2+12δeq2-k12eω-eq2+k12eω2+k12eq2-k1eω2-k3ed2-k2eq2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2=-12δeω-eq2-k12eω-eq2-k3ed2+12δ+k12-k1eω2+12δ+k12-k2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2.Appropriate controller gains k1 and k2 are selected as follows:(42)12δ+k12-k1<0,12δ+k12-k2-1δ<0.Equation (42) can be replaced by the following:(43)k1>1δ,k1-2k2<1δ.Then, by substitution of (43) into (41), we can obtain(44)V˙=-12δeω-eq2-k12eω-eq2-k3ed2+12δ+k12-k1eω2+12δ+k12-k2-1δeq2+eq-d12x4ε1eq+Δ1+ed-d22x4ε2ed+Δ2≤-12δeω-eq2-k12eω-eq2-k3ed2-k4eω2-k5eq2-d1xeq2ε1-ε1f1max2+ε1f1max2-d2xed2ε2-ε2f2max2+ε2f2max2≤-12δeω-eq2-k12eω-eq2-k3ed2-k4eω2-k5eq2-d1xeq2ε1-ε1f1max2-d2xed2ε2-ε2f2max2+ε0,where k4=-1/2δ+k1/2-k1≥0, k5=-1/2δ+k1/2-k2-1/δ≥0, and ε0=ε1f1max2+ε2f2max2.Let(45)Wit=-12δeω-eq2-k12eω-eq2-k3ed2-k4eω2-k5eq2-d1xeq2ε1-ε1f1max2-d2xed2ε2-ε2f2max2+ε0,where i(t)=eω,eq,ed.By integration of (45), we can get(46)∫t0tWitdt=-∫t0tV˙itdt⟹∫t0tWitdt=Vt0-Vt.Since V(t0) is bounded and V(t) is bounded and nonincreasing, hence(47)limt→∞⁡∫t0tWitdt<∞.Moreover, Wi(t) is uniform continuous and W˙i(t) is bounded. In accordance with Barbalat’s Lemma, the following equation can be obtained:(48)limt→∞⁡Wit=0.Apparently, through selection of suitable controller gains k1, k2, k3, ε1, and ε2, V˙ can be ensured to be negative definite.The above derivation has proved that the selected suitable controller gainsk1, k2, k3, ε1, and ε2 and adaptive gains θ1, θ2, and θ3 can make the inequalities of V≥0 and V˙<0 hold. In addition, equation of V=0 is not satisfied until eω=eq=ed=τ~l=γ~=δ~=0. In summary, PMSM chaotic system is globally asymptotically stable at the equilibrium point of eω,eq,ed=0,0,0.


## 4. Numerical Simulation and Discussions

In order to illustrate the superiority of the proposed approach adequately, the simulation is carried out in MATLAB environment for three cases under the initial condition ofω,iq,id=0,0,0. Let ω,iq,id=x1,x2,x3 and the control parameters are selected as k1=10, k2=30000, k3=5, and ε1=ε2=0.01; the adaptive gains are chosen as θ1=6.2, θ2=100, and θ3=0.06. The simulation time is chosen as 100 s and the designed controller is put into effect at the time of 20 s.


### 4.1. Test-I

The PMSM chaotic system is tested with the parametersδ=0.2, γ=50, and τl=3.2. In order to be consistent with the reality better, we assume that the three parameters of PMSM chaotic system are all unknown with the initial condition of δ,γ,τl=0,0,0, and the expected reference signals are set as ω∗=10 and id∗=1. Furthermore, the external disturbances Δ1x,t=20x3sin⁡(5t) and Δ2x,t=10sin⁡(5t) are injected into the PMSM chaotic system. The simulation results given in Figures 2–4 apparently show PMSM runs in a chaotic state with no control inputs. Therefore, introduction of the presented control approach to suppress chaos in PMSM system will be of great importance and necessity. Figures 5–12 show that the proposed controller is utilized to control the PMSM chaotic system, where Figures 5–7 display the curves of state variables changing over time for PMSM chaotic system, which demonstrate the PMSM system stays away from the previous chaotic state when the designed controller is added to PMSM chaotic system, and track the desired signals accurately and rapidly. Furthermore, Figures 5–7 indicate the proposed controller shown in Figures 8-9 can inhibit the external disturbances.Figure 2
Theω curve of PMSM chaotic system with no control inputs ud and uq.Figure 3
Theiq curve of PMSM chaotic system with no control inputs ud and uq.Figure 4
Theid curve of PMSM chaotic system with no control inputs ud and uq.Figure 5
Theω curve of PMSM chaotic system added the controller inputs ud and uq.Figure 6
Theiq curve of PMSM chaotic system added the controller inputs ud and uq.Figure 7
Theid curve of PMSM chaotic system added the controller inputs ud and uq.Figure 8
The controller inputud.Figure 9
The controller inputuq.Figure 10
Theδ~ curve of estimation error of δ.Figure 11
Theγ~ curve of estimation error of γ.Figure 12
Theτ~l curve of estimation error of τl.Figures10–12 show the estimated errors δ~, γ~, and τ~l of unknown parameters δ, γ, and τl for PMSM chaotic system, which testify the effectiveness of constructed adaptive laws and demonstrate the proposed approach has a good robustness against the uncertainties in system parameters.


### 4.2. Test-II

In reality, the motor parameters are frequently varying with the design values. As a result, the parametersδ, γ, and τl in Test-I are changed into δ=0.1, γ=25, and τl=1.6 in Test-II, respectively. Simultaneously, the expected reference signals are also changed and set as ω∗=20 and id∗=0. In a word, the unknown motor parameters and expected reference signals all differ from Test-I, which is able to validate the proposed control algorithm better. The simulation results are shown in Figures 13–20. Figures 13–15 indicate the motor’s output states ω, iq, and id added the controller inputs ud and uq shown in Figures 16-17, which demonstrate that the designed controller can guarantee the outputs track references well and suppress the external disturbances effectively. Figures 18–20 indicate that the designed adaptive law can estimate the fully unknown parameters precisely even if the fully unknown parameters are changed.Figure 13
Theω curve of PMSM chaotic system added the controller inputs ud and uq.Figure 14
Theiq curve of PMSM chaotic system added the controller inputs ud and uq.Figure 15
Theid curve of PMSM chaotic system added the controller inputs ud and uq.Figure 16
The controller inputud.Figure 17
The controller inputuq.Figure 18
Theδ~ curve of estimation error of δ.Figure 19
Theγ~ curve of estimation error of γ.Figure 20
Theτ~l curve of estimation error of τl.


### 4.3. Test-III

For Test-III, the external disturbances are enlarged in addition to changing the unknown motor parameters and expected reference signals on the basis of Test-II, which are described asΔ1x,t=40x3sin⁡(5t) and Δ2x,t=20sin⁡(5t). The control difficulty in Test-III is larger than the previous two experiments and Test-III is a more general instance to verify the controller’s performance. The simulation results are shown in Figures 21–28. Figures 21–23 give the curves of the state variables ω, iq, and id, which manifest these variables are controlled to their references and chaos is eliminated when adding the proposed controllers ud and uq shown in Figures 24-25. Figures 21–23 also illustrate the enlarged external disturbances are restrained by the controllers. The estimation errors of the fully unknown parameters are provided in Figures 26–28, which proves the effectiveness of the adaptive laws again.Figure 21
Theω curve of PMSM chaotic system added the controller inputs ud and uq.Figure 22
Theiq curve of PMSM chaotic system added the controller inputs ud and uq.Figure 23
Theid curve of PMSM chaotic system added the controller inputs ud and uq.Figure 24
The controller inputud.Figure 25
The controller inputuq.Figure 26
Theδ~ curve of estimation error of δ.Figure 27
Theγ~ curve of estimation error of γ.Figure 28
Theτ~l curve of estimation error of τl.Remark 5.
Previous researches on parameter estimation of PMSM chaotic system mostly assumed that only partial parameters of the system are unknown. The paper takes fully nondeterministic parametersδ, γ, and τl into account; it undoubtedly extends the theory of parameter estimation for PMSM chaotic system.Remark 6.
The action time of control inputs is 20 s in the simulation. The aim of doing this is to observe the effect of the control approach better. In reality, as long as the chaos occurs, the controller will be put into effect.Remark 7.
On the basis of considering fully unknown parameters, the external disturbances are introduced into the PMSM chaotic model. Hence, the designed control consists of two parts. One is to guarantee the state variables to track the reference signals; another is to suppress the external disturbances. In general, the simultaneous consideration of fully unknown parameters and external disturbances makes the research results more general and practical.


## 4.1. Test-I

The PMSM chaotic system is tested with the parametersδ=0.2, γ=50, and τl=3.2. In order to be consistent with the reality better, we assume that the three parameters of PMSM chaotic system are all unknown with the initial condition of δ,γ,τl=0,0,0, and the expected reference signals are set as ω∗=10 and id∗=1. Furthermore, the external disturbances Δ1x,t=20x3sin⁡(5t) and Δ2x,t=10sin⁡(5t) are injected into the PMSM chaotic system. The simulation results given in Figures 2–4 apparently show PMSM runs in a chaotic state with no control inputs. Therefore, introduction of the presented control approach to suppress chaos in PMSM system will be of great importance and necessity. Figures 5–12 show that the proposed controller is utilized to control the PMSM chaotic system, where Figures 5–7 display the curves of state variables changing over time for PMSM chaotic system, which demonstrate the PMSM system stays away from the previous chaotic state when the designed controller is added to PMSM chaotic system, and track the desired signals accurately and rapidly. Furthermore, Figures 5–7 indicate the proposed controller shown in Figures 8-9 can inhibit the external disturbances.Figure 2
Theω curve of PMSM chaotic system with no control inputs ud and uq.Figure 3
Theiq curve of PMSM chaotic system with no control inputs ud and uq.Figure 4
Theid curve of PMSM chaotic system with no control inputs ud and uq.Figure 5
Theω curve of PMSM chaotic system added the controller inputs ud and uq.Figure 6
Theiq curve of PMSM chaotic system added the controller inputs ud and uq.Figure 7
Theid curve of PMSM chaotic system added the controller inputs ud and uq.Figure 8
The controller inputud.Figure 9
The controller inputuq.Figure 10
Theδ~ curve of estimation error of δ.Figure 11
Theγ~ curve of estimation error of γ.Figure 12
Theτ~l curve of estimation error of τl.Figures10–12 show the estimated errors δ~, γ~, and τ~l of unknown parameters δ, γ, and τl for PMSM chaotic system, which testify the effectiveness of constructed adaptive laws and demonstrate the proposed approach has a good robustness against the uncertainties in system parameters.


## 4.2. Test-II

In reality, the motor parameters are frequently varying with the design values. As a result, the parametersδ, γ, and τl in Test-I are changed into δ=0.1, γ=25, and τl=1.6 in Test-II, respectively. Simultaneously, the expected reference signals are also changed and set as ω∗=20 and id∗=0. In a word, the unknown motor parameters and expected reference signals all differ from Test-I, which is able to validate the proposed control algorithm better. The simulation results are shown in Figures 13–20. Figures 13–15 indicate the motor’s output states ω, iq, and id added the controller inputs ud and uq shown in Figures 16-17, which demonstrate that the designed controller can guarantee the outputs track references well and suppress the external disturbances effectively. Figures 18–20 indicate that the designed adaptive law can estimate the fully unknown parameters precisely even if the fully unknown parameters are changed.Figure 13
Theω curve of PMSM chaotic system added the controller inputs ud and uq.Figure 14
Theiq curve of PMSM chaotic system added the controller inputs ud and uq.Figure 15
Theid curve of PMSM chaotic system added the controller inputs ud and uq.Figure 16
The controller inputud.Figure 17
The controller inputuq.Figure 18
Theδ~ curve of estimation error of δ.Figure 19
Theγ~ curve of estimation error of γ.Figure 20
Theτ~l curve of estimation error of τl.


## 4.3. Test-III

For Test-III, the external disturbances are enlarged in addition to changing the unknown motor parameters and expected reference signals on the basis of Test-II, which are described asΔ1x,t=40x3sin⁡(5t) and Δ2x,t=20sin⁡(5t). The control difficulty in Test-III is larger than the previous two experiments and Test-III is a more general instance to verify the controller’s performance. The simulation results are shown in Figures 21–28. Figures 21–23 give the curves of the state variables ω, iq, and id, which manifest these variables are controlled to their references and chaos is eliminated when adding the proposed controllers ud and uq shown in Figures 24-25. Figures 21–23 also illustrate the enlarged external disturbances are restrained by the controllers. The estimation errors of the fully unknown parameters are provided in Figures 26–28, which proves the effectiveness of the adaptive laws again.Figure 21
Theω curve of PMSM chaotic system added the controller inputs ud and uq.Figure 22
Theiq curve of PMSM chaotic system added the controller inputs ud and uq.Figure 23
Theid curve of PMSM chaotic system added the controller inputs ud and uq.Figure 24
The controller inputud.Figure 25
The controller inputuq.Figure 26
Theδ~ curve of estimation error of δ.Figure 27
Theγ~ curve of estimation error of γ.Figure 28
Theτ~l curve of estimation error of τl.Remark 5.
Previous researches on parameter estimation of PMSM chaotic system mostly assumed that only partial parameters of the system are unknown. The paper takes fully nondeterministic parametersδ, γ, and τl into account; it undoubtedly extends the theory of parameter estimation for PMSM chaotic system.Remark 6.
The action time of control inputs is 20 s in the simulation. The aim of doing this is to observe the effect of the control approach better. In reality, as long as the chaos occurs, the controller will be put into effect.Remark 7.
On the basis of considering fully unknown parameters, the external disturbances are introduced into the PMSM chaotic model. Hence, the designed control consists of two parts. One is to guarantee the state variables to track the reference signals; another is to suppress the external disturbances. In general, the simultaneous consideration of fully unknown parameters and external disturbances makes the research results more general and practical.


## 5. Conclusions

In this paper, a control approach is proposed to address the control issue of chaos in PMSM system with fully unknown parameters and external disturbances. Main conclusions are acquired as the following:(1)
Through combination of adaptive control with backstepping control, the presented adaptive robust backstepping control scheme resolves the main problems of the conventional backstepping algorithm encountered. And the stability of the designed controller is proved by Lyapunov theory.(2)
The simulation results show that the designed controller is able to make the PMSM operate out of chaotic state quickly, and the adaptive laws are established to estimate the unknown parameters accurately. Furthermore, the proposed algorithm can ensure the unknown parameters converge to the actual values fast and restrain the external disturbance effectively.(3)
The design method in this paper is simple and effective. For PMSM chaotic system with fully unknown parameters, the control variables in proposed approach can be self-adjusted with the changing of system parameters. Therefore, our findings are more practical and more convenient for engineering applications. Future research will discuss the application of the proposed control approach into practical implementation.



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