Yang Yu, Xudong Guo, and Zengqiang Mi. 1. Introduction

Transcription

1 Mathematical Problems in Engineering Volume 6, Article ID 3694, 4 pages Research Article Adaptive Robust Backstepping Control of Permanent Magnet Synchronous Motor Chaotic System with Fully Unknown Parameters and External Disturbances Yang Yu, Xudong Guo, and Zengqiang Mi State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Baoding 73, China Correspondence should be addressed to Yang Yu; ncepu yy@63.com Received 5 December 5; Revised 6 March 6; Accepted 3 March 6 Academic Editor: Shengjun Wen Copyright 6 Yang Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 andavoidsmanyrestrictedproblems.thestabilityofthecontrolapproachisprovedbylyapunovstabilitytheory.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.. 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 []. 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 [ 5], and further studying on the control method ofpmsmchaosisofextremesignificance[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 [], and decoupling control []. 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 [, 3], fuzzy control [4], and H control [5]. However, sliding mode variable structure control requires uncertain parameters to satisfy certain matching conditions, fuzzy control is

2 Mathematical Problems in Engineering dependent on the fuzzification of Takagi-Sugeno, and H control is inclined to ignore the operating states under special conditions [6]. 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 toabroaderfieldandismoreinlinewithreality[7].actually, the adaptive control (AC) provides a natural routine for PMSM chaotic control with unknown parameters, which has been presented in literatures [, 3, 8]. 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 highdimensional 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 [9] and Chen chaotic system []. 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 [, ] have exactly practiced this idea. However, the conventional backstepping approach is confronted with two major problems of solving complicated regression matrix [3] and encountering explosion of terms [4]. In [5], 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 [6]. 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 [, ], 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 [6]; 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 takenintoaccountinpmsmchaosmodel.newlyadaptive 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.. PMSM Chaotic Model with Fully Unknown Parameters For a PMSM, its mathematical model in dq axis coordinate system can be described as follows [6]: ω = 3pφ m J i q B J ω τ l J, i q = R i L q p ω i d pφ m ω+ u q L q L q, q i d = R i L d +p ω i q + u d L d, 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, φ m 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 L d and L q are d axis and q axis inductances of stator winding, respectively. For a PMSM with uniform air gap, L d =L q.hence,weusel to substitute L d and L q in the following paper. Selecting the affine transformation [ i q ] = i d [ /pτ k k ][ ω i q i d ] and time scale transformation t = τt,the PMSM mathematical model described in () can be converted into dimensionless form as follows: ω= δ (i q ω) τ l, i q = i q ω i d +γ ω+u q, i d = i d +ω i q +u d, where τ = L/R, k=b/3p τφ m, u d =(/kr) u d, γ= φ m /kl, u q =(/kr) u q, δ = J/Bτ,andτ l =pτ τ l /J. As presented in (), the dynamic performance of PMSM depends on three parameters δ, γ, andτ l. Considering the most general case, let δ =., γ=5, τ l = 3., u d =.6, and u q =.8. Iftheinitialstateisselectedas(ω, i q,i d ) = (,, ), PMSM system will run on a chaotic state and display the chaotic behavior. A typical chaotic attractor of PMSM is manifested in Figure. In reality, the three parameters δ, γ, andτ l in () tend to beunknownortohaveuncertaintiesresultingfromoperating conditions. In other words, when all the parameters δ, γ, and τ l cannot be determined, () actually represents PMSM chaotic system model with fully unknown parameters. ω () ()

3 Mathematical Problems in Engineering 3 i d i q ω Figure : Chaotic attractor of PMSM system. 3. Design of Adaptive Robust Controller with Backstepping Approach Taking a more general situation into account, the PMSM chaotic model described in () is immersed by external disturbances. The model can be rewritten as follows: ω= δ (i q ω) τ l, i q = i q ω i d +γ ω+u q +Δ (x,t), i d = i d +ω i q +u d +Δ (x,t), where Δ (x,t) and Δ (x,t) represent the external disturbances, x indicates the system states, and x =(x,x,x 3 )= (ω, i q,i d ). 3.. 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, andτ l and external disturbances Δ and Δ,adaptivelawsofunknown parameters δ, γ, andτ l are designed and adaptive robust controllers u d and u q are constructed to ensure PMSM breaks away from chaos rapidly and runs into an expected orbit. Simultaneously, the fully unknown parameters δ, γ,andτ l 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. The state variables for PMSM chaotic system (ω, i q,i d ) are observable. Assumption. The external disturbances Δ i (x,t) satisfy the condition Δ i (x,t) d i (x)f i (t), i =,, whered i (x) is a known function, f i (t) is an unknown but bounded timevarying function, and f i (t) f imax,wheref imax is a constant. Assumption 3. The desired speed and d axis current reference signals and i d and their derivatives are known and bounded. 5 5 (3) The estimated values of unknown system parameters are described as δ, γ,and τ l ; then, the estimation errors δ, γ,and τ l canbeexpressedasfollows: δ = δ δ, γ = γ γ, τ l = τ l τ l. 3.. 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. For the speed reference signal, define the tracking error e ω as follows: e ω =ω. (5) Taking PMSM chaotic system model (3) into account, the derivative of (5) can be written as e ω = (4) ω = δ (i q ω) τ l. (6) Define the tracking error e q of q axis stator current i q as follows: e q =i q i q, (7) where i q is the expected output value of i q. For the d axis current reference signal i d, its tracking error e d is defined as follows: e d =i d i d. (8) By substitution of (7) into (6), we can obtain Let e ω = δ (e q +i q ω) τ l = δ (i q ω) τ l + τ l ω. (9) i q =ω+ δ( τ l + k e ω ), () i d =, () where k represents the positive control gain. Through substitution of () into (9), () can be obtained: e ω = δ e q + δ δ ( τ l + k e ω ) τ l + τ l = δ e q + δ+ δ δ ( τ l + k e ω ) τ l + τ l = δ e q + δ δ ( τ l + k e ω )+ τ l k e ω. ()

4 4 Mathematical Problems in Engineering Lyapunov function V is selected as follows: V = e ω. (3) Then, the derivative of V canbedescribedas V =e ω e ω =e ω ( δ e q + δ δ ( τ l + k e ω )+ τ l k e ω ). (4) Step. To stabilize the output q axis current of PMSM, the derivative of e q is conducted as follows: e q = i q i q = i q ω i d +γ ω+u q +Δ [ ω + δ( τ l + k e ω )+ δ( τ l + k e ω )] = i q ω i d +γ ω+u q +Δ [ ω + δ( τ l + k e ω )+ δ( τ l + k e ω )] = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + ω k e ω ) ω δ( τ l + k e ω )= i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) e ω δ( τ l + )+ δ k (ω ). By substitution of () into (5), we can get e q = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k e ω δ( τ l + )+ δ k (ω ). (5) (6) Combined with the mathematical model of PMSM chaotic system, (6) can be further calculated as follows: e q = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k r e ω δ( τ l + )+ δ k i q ω τ l δ δ k = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k e ω δ( τ l + )+ δ δ k (i q ω) δ k τ l δ k = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k e ω δ( τ l + )+ δ δ k (i q ω) δ k ( τ l τ l ) δ k = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k e ω δ( τ l + )+ δ δ k (i q ω) δ k τ l + δ k τ l δ k. (7) By substitution of δ = δ+δinto (7), the following equation can be obtained: e q = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k e ω δ( τ l + )+ δ+ δ k δ (i q ω) δ k τ l + δ k τ l δ k ω = i q ω i d + γ ω γ ω+u q +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k e ω δ( τ l + )+k (i q ω)+ δ δ k (i q ω) δ k τ l + δ k τ l δ k. Lyapunov function V is chosen as follows: (8) V =V + e q. (9)

5 Mathematical Problems in Engineering 5 Then, the derivative of V canbedescribedas V = V +e q e q =e ω ( δ e q + δ δ ( τ l + k e ω )+ τ l δ (( τ l + k e ω ) k (i q ω)) + τ l ( δ k )). k e ω )+e q ( i q ω i d + γ ω γ ω+u q (4) +Δ δ( τ l + k e ω ) δ e q δ ( τ l + k e ω ) τ l +k e ω δ( τ l + )+k (i q ω)+ δ δ k (i q ω) δ k τ l + δ k τ l δ k ). () Additionally, e q ( d (x) 4ε e q +Δ )= d (x) e q 4ε +Δ e q d (x) 4ε e q +d (x) f max e q = ( d (x) e q ε f max ) +ε f max. (5) The first control variable is selected as u q =u qs +u qr, () where u qs and u qr are the model compensation and robust control inputs, respectively. Then, u qs and u qr can be, respectively, chosen as u qs = k e q +i q +ω i d γ ω + δ( τ l + k e ω )+ + δ( τ l + )+ δ k τ l + δ k r k (i q ω), () u qr = d (x) 4ε e q, (3) where k is another positive control gain and ε is a positive number chosen arbitrarily. By substitution of () and () into (), we can acquire V = V +e q e q =e ω ( δ e q + δ δ ( τ l + k e ω )+ τ l k e ω )+e q ( k e q δ e q +k e ω γ ω+u qr +Δ δ (( τ l + k e ω ) k (i q ω)) + τ l ( δ k ))=e ω ( δ e q + δ δ ( τ l + k e ω )+ τ l k e ω )+e q ( k e q δ e q +k e ω γ ω d (x) 4ε e q +Δ Step 3. Differentiating the tracking error e d of d axis current i d,wecanget e d = i d i d = i d +ω i q +u d +Δ i d. (6) Lyapunov function V 3 is chosen as V 3 =V + e d. (7) Then, the derivative of V 3 can be represented as V 3 = V +e d e d = V +e d ( i d +ω i q +u d +Δ i d ). (8) In terms of (8), d axis output stator voltage u d can be calculated: where u d =u ds +u dr, (9) u ds = k 3 e d +i d ω i q +i d, (3) u dr = d (x) 4ε e d, (3) where k 3 is the positive control gain and ε is a positive number chosen arbitrarily. By substitution of (3) and (3) into (6) and (8), respectively, the following equations can be acquired: e d = k 3 e d +u dr +Δ = k 3 e d d (x) e 4ε d +Δ, (3) V 3 = V +e d e d = V +e d ( k 3 e d d (x) 4ε e d +Δ ), (33)

6 6 Mathematical Problems in Engineering e d ( d (x) 4ε e d +Δ )= d (x) e d 4ε +Δ e d d (x) 4ε e d +d (x) f max e d = ( d (x) e d ε f max ) +ε f max. (34) Step 4. Lyapunov function V of PMSM chaotic system with fully unknown parameters and external disturbances is selected as follows: V=V 3 + τ l + γ + δ, (35) θ θ δ θ 3 where θ, θ,andθ 3 represent positive adaptive gains. Combined with equations δ = δ, γ = γ, and τ l = τ l, derivative of selected Lyapunov function V canbecalculated as follows: V= V 3 + τ l τ θ l + γ γ+ δ θ δ θ 3 +u dr +Δ )= ω+u qr +Δ δ = V +e d ( k 3 e d V +e q ( k e q δ e q +k e ω γ δ (( τ l + k e ω ) k (i q ω)) + τ l ( δ k )) + e d ( k 3 e d +u dr +Δ ) + δ δ [( τ l + k e ω )e ω ( τ l + k e ω )e q δ +k (i q ω)e q + ] k θ e ω k e d k 3e q 3 + τ l [e ω e q + δk τ e q + l γ ]+ γ[ ωe θ q + ] θ δ e q +e q ( d (x) 4ε +Δ ). e q +Δ )+e d ( d (x) e 4ε d (36) In terms of (36), the adaptive laws of unknown parameters δ, γ,andτ l canbeselected,respectively,asfollows: δ = θ 3 [( τ l + k e ω ) (e ω e q ) +k (i q ω)e q ], (37) γ =θ ωe q, (38) τ l = θ [e ω e q + δk e q ]. (39) By substitution of (37), (38), and (39) into (36), (36) can be simplified as follows: =e ω ( δ e q + δ δ ( τ l + k e ω )+ τ l k e ω ) +e q ( k e q δ e q +k e ω γ ω+u qr +Δ δ (( τ l + k e ω ) k (i q ω)) + τ l ( δ k )) + e d ( k 3 e d +u dr +Δ )= δ V= δ e qe ω +k e q e ω k e ω k 3e d k e q δ e q +e q ( d (x) 4ε e q +Δ ) +e d ( d (x) 4ε e d +Δ ) Stability Analysis (4) e q e ω + δ δ ( τ l + k e ω ) e ω + τ l e ω k e ω k 3 e d k e q γωe q δ e q +e q (u qr +Δ ) +e d (u dr +Δ ) δ (( τ l + k e ω ) k (i q ω)) e q + τ l ( δ k )e q +k e ω e q + τ l τ θ l + γ γ+ δ δ = θ δθ 3 δ e qe ω +k e ω e q Theorem 4. For PMSM chaotic system model (3) with fully unknown parameters and external disturbances, design of adaptive control laws (37), (38), and (4) and selection of suitable controller gains k, k, k 3, ε,andε and adaptive gains θ, θ,andθ 3, the proposed adaptive robust backstepping controllers () and (9) 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 () and (9) and track the given reference signals. Through stability analysis, we want to verify the correctness of the theorem.

7 Mathematical Problems in Engineering 7 According to (4), new expression can be obtained as follows through some mathematical computations: V= δ e qe ω +k e q e ω k e ω k 3e d k e q δ e q +e q ( d (x) e 4ε q +Δ )+e d ( d (x) e 4ε d +Δ ) = δ (e ω e q ) + δ e ω + δ e q k (e ω e q ) + k e ω + k e q k e ω k 3e d k e q δ e q +e q ( d (x) e 4ε q +Δ )+e d ( d (x) e 4ε d +Δ ) = δ (e ω e q ) k (e ω e q ) k 3 e d +( δ + k k )e ω +( δ + k k δ )e q +e q ( d (x) 4ε e q +Δ ) +e d ( d (x) 4ε e d +Δ ). (4) Appropriate controller gains k and k are selected as follows: δ (e ω e q ) k (e ω e q ) k 3 e d k 4e ω k 5 e q (d (x) e q ε f max ) +ε f max ( d (x) e d ε f max ) +ε f max δ (e ω e q ) k (e ω e q ) k 3 e d k 4e ω k 5 e q (d (x) e q ε f max ) ( d (x) e d ε f max ) +ε, (44) where k 4 = (/δ + k / k ), k 5 = (/δ + k / k /δ),andε =ε f max +ε f max. Let W (i (t)) = δ (e ω e q ) k (e ω e q ) k 3 e d k 4 e ω k 5e q ( d (x) e q ε f max ) (45) ( δ + k k )<, ( δ + k k δ )<. (4) ( d (x) e d ε f max ) +ε, where i(t) = (e ω,e q,e d ). By integration of (45), we can get Equation(4)canbereplacedbythefollowing: k > δ, k k < δ. (43) Then, by substitution of (43) into (4), we can obtain V= δ (e ω e q ) k (e ω e q ) k 3 e d +( δ + k k )e ω +( δ + k k δ )e q +e q ( d (x) 4ε e q +Δ ) +e d ( d (x) 4ε e d +Δ ) t t W (i (t)) dt = t t V (i (t)) dt t W (i (t)) dt = V (t ) V(t). t (46) Since V(t ) is bounded and V(t) is bounded and nonincreasing, hence t lim W (i (t)) dt <. (47) t t Moreover, W(i(t)) is uniform continuous and W(i(t)) is bounded. In accordance with Barbalat s Lemma, the following equation can be obtained: lim W (i (t)) =. (48) t Apparently, through selection of suitable controller gains k, k, k 3, ε,andε, V canbeensuredtobenegativedefinite. The above derivation has proved that the selected suitable controller gains k, k, k 3, ε,andε and adaptive gains θ, θ,

8 8 Mathematical Problems in Engineering ω 5 i q Figure : The ω curve of PMSM chaotic system with no control Figure 3: The i q curve of PMSM chaotic system with no control and θ 3 can make the inequalities of V and V<hold. In addition, equation of V=is not satisfied until e ω =e q = e d = τ l = γ = δ =.Insummary,PMSMchaoticsystem is globally asymptotically stable at the equilibrium point of (e ω,e q,e d )=(,,). 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 of (ω, i q,i d ) = (,, ). Let(ω, i q,i d )=(x,x,x 3 ) and the control parameters are selected as k =, k = 3, k 3 =5, and ε =ε =.; the adaptive gains are chosen as θ = 6., θ =, andθ 3 =.6. The simulation time is chosen as s and the designed controller is put into effect at the time of s. 4.. Test-I. The PMSM chaotic system is tested with the parameters δ =., γ = 5,andτ l = 3.. 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, τ l ) = (,,), and the expected reference signals are set as = and i d =.Furthermore, the external disturbances Δ (x,t) = x 3 sin(5t) and Δ (x,t)=sin(5t) are injected into the PMSM chaotic system. The simulation results given in Figures 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 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, i d Figure 4: The i d curve of PMSM chaotic system with no control ω Figure 5: The ω curve of PMSM chaotic system added the controller

9 Mathematical Problems in Engineering 9 i q Figure 6: The i q curve of PMSM chaotic system added the controller u q Figure 9: The controller input u q. i d Figure 7: The i d curve of PMSM chaotic system added the controller Estimation error of δ Figure : The δ curve of estimation error of δ u d Estimation error of γ Figure 8: The controller input u d. Figure : The γ curve of estimation error of γ.

10 Mathematical Problems in Engineering Estimation error of τ l i q Figure : The τ l curve of estimation error of τ l. Figure 4: The i q curve of PMSM chaotic system added the controller ω 5 i d Figure 3: The ω curve of PMSM chaotic system added the controller Figure 5: The i d curve of PMSM chaotic system added the controller Figures 5 7 indicate the proposed controller shown in Figures 8-9 can inhibit the external disturbances. Figures show the estimated errors δ, γ, and τ l of unknown parameters δ, γ, and τ l 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 Test-II. In reality, the motor parameters are frequently varying with the design values. As a result, the parameters δ, γ, andτ l in Test-I are changed into δ =., γ=5,and τ l =.6 in Test-II, respectively. Simultaneously, the expected reference signals are also changed and set as = and i d =. 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 3. Figures 3 5 indicate the motor s output states ω, i q,andi d added the u d Figure 6: The controller input u d.

11 Mathematical Problems in Engineering u q Estimation error of τ l Figure 7: The controller input u q. Figure : The τ l curve of estimation error of τ l. 5 Estimation error of δ Figure 8: The δ curve of estimation error of δ. ω Figure : The ω curve of PMSM chaotic system added the controller Estimation error of γ Figure 9: The γ curve of estimation error of γ. controller inputs u d and u q shown in Figures 6-7, which demonstrate that the designed controller can guarantee the outputs track references well and suppress the external disturbances effectively. Figures 8 indicate that the designed adaptive law can estimate the fully unknown parameters precisely even if the fully unknown parameters are changed 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 as Δ (x,t)=4x 3 sin(5t) and Δ (x,t)= sin(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 8. Figures 3 give the curves of the state variables ω, i q,andi d,which manifest these variables are controlled to their references and chaos is eliminated when adding the proposed controllers

12 Mathematical Problems in Engineering i q u q Figure : The i q curve of PMSM chaotic system added the controller Figure 5: The controller input u q. 5. i d Figure 3: The i d curve of PMSM chaotic system added the controller Estimation error of δ Figure 6: The δ curve of estimation error of δ. u d Estimation error of γ Figure 4: The controller input u d Figure 7: The γ curve of estimation error of γ.

13 Mathematical Problems in Engineering 3 Estimation error of τ l Figure 8: The τ l curve of estimation error of τ l. u d and u q shown in Figures 4-5. Figures 3 also illustrate the enlarged external disturbances are restrained by thecontrollers.theestimationerrorsofthefullyunknown parameters are provided in Figures 6 8, which proves the effectiveness of the adaptive laws again. 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 parameters δ, γ, andτ l into account; it undoubtedly extends the theory of parameter estimation for PMSM chaotic system. Remark 6. The action time of control inputs is 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. Onthebasisofconsideringfullyunknownparameters, 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: () 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. () The simulation results show that the designed controller is able to make the PMSM operate out of chaoticstatequickly,andtheadaptivelawsareestablished 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. Competing Interests The author declares that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work is partially supported by the National Natural Science Foundation of China (Grant no ) and the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant no. 4MS93). References [] N. Hemati, Strange attractors in brushless DC motors, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,vol.4,no.,pp.4 45,994. [] Z.Li,J.B.Park,Y.H.Joo,B.Zhang,andG.Chen, Bifurcations and chaos in a permanent-magnet synchronous motor, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications,vol.49,no.3,pp ,. [3] Z. Jing, C. Yu, and G. Chen, Complex dynamics in a permanent-magnet synchronous motor model, Chaos, Solitons and Fractals,vol.,no.4,pp ,4. [4] C.-L. Li, S.-M. Yu, and X.-S. Luo, Fractional-order permanent magnet synchronous motor and its adaptive chaotic control, Chinese Physics B,vol.,no.,ArticleID56,. [5] Y.-Y. Hou, Finite-time chaos suppression of permanent magnet synchronous motor systems, Entropy, vol. 6, no. 4, pp , 4. [6] H.P.RenandD.Liu, Nonlinearfeedbackcontrolofchaosin permanent magnet synchronous motor, IEEE Transactions on Circuits and Systems II: Express Briefs, vol.53,no.,pp.45 5, 6. [7] A. Loría, Robust linear control of (chaotic) permanent-magnet synchronous motors with uncertainties, IEEE Transactions on CircuitsandSystems.I.RegularPapers,vol.56,no.9,pp.9, 9. [8] M. Messadi, A. Mellit, K. Kemih, and M. Ghanes, Predictive control of a chaotic permanent magnet synchronous generator in a wind turbine system, Chinese Physics B, vol.4,no., Article ID 5, 5.

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