This paper proposes an adaptive chaos synchronization control scheme for nonlinear permanent magnet synchronous motor (PMSM) systems by using extended state observer (ESO). Frist of all, a chaotic PMSM system is built through an affine transformation and a time scale transformation of the mathematical PMSM model. Then, an adaptive sliding mode controller is developed based on the extended state observer to achieve the synchronization performance of two chaotic PMSM systems. Moreover, an adaptive parameter law of the control gain is designed to reduce the chattering problem existing in the traditional sliding mode control. Finally, the effectiveness of the proposed method is verified by simulation results.
National Natural Science Foundation of China9143811791538202614033431. Introduction
The research on chaos control and synchronization has been widely studied since the chaos phenomenon was discovered [1–4]. As far as we know, chaos problem has been proved to exist in various practical systems. In the engineering field, chaos may cause irregular operation and affect the stability of the motors. Consequently, how to deal with the chaos problem in motors is still a significant problem.
Lots of control and synchronization schemes have been represented since the discovery of chaotic impact on the motor control performance, such as finite-time control [5, 6], unidirectional correlation control [7], sliding mode control [8–11], linear control [12], dynamic surface control [13, 14], optimal control [15], and neural network control [16]. In [5], an adaptive finite-time control method is proposed for PMSM system to suppress the chaos behavior with parameter uncertainties. The uncertain parameters which are caused by external factors can be solved and the chaos in motor can be effectively stabilized. Reference [14] presents a dynamic surface control method based on neural network (NN) for PMSM system. The NN is adopted to approximate the system nonlinearities like disturbance and unknown parameters, and the control performance is guaranteed by using the designed control method.
Most of the methods mentioned above can effectively eliminate the influence of chaos in PMSM, but the robustness of the system may be not guaranteed when the system has different initial conditions or disturbances. Due to the strong robustness and antidisturbance ability, sliding mode control (SMC) has been widely applied to the chaotic control in PMSM systems. In [8], a sliding mode control based on fuzzy neural network is investigated for the chaotic PMSM to suppress the chaos and improve the tracking control performance. Reference [9] proposes a high robust controller based on the traditional sliding mode control for PMSM with interference and uncertainties. The system has a good control performance and robustness by using the proposed method. But unfortunately, the robustness of SMC usually has a strong dependence on the control gain, and the control performance may become worse when the disturbance or uncertainties of the system are beyond of the control range. Consequently, in this paper, an adaptive law of the control gain is designed to solve this problem, and the extended state observer is employed to estimate the system uncertainties and unknown parameters. For the dual-motor cooperative control system with chaos, the adaptive sliding mode control is investigated to guarantee the synchronization control performance, and the chattering problem in traditional sliding mode control is also improved.
The rest of this paper is organized as follows. The chaotic PMSM model and the extended state observer are derived in Sections 2 and 3, respectively. In Section 4, an adaptive sliding mode control scheme is developed for chaotic PMSM system and the system analysis is provided. Section 5 provides the simulation results and the conclusion is given by Section 6.
2. System Description
The mathematical model of a nonlinear PMSM system is expressed as(1)diddt=-Rsid+npLqωmiq+udLddiqdt=-Rsiq-npLdωmid-npψfωm+uqLqdωmdt=npψfiq+Ld-Lqidiq-TL-BωmJ,where id, iq are the components of stator current in d-axis and q-axis; ud, uq are the components of stator voltage in d-axis and q-axis; Ld, Lq are the equivalent inductances of stator windings in d-axis and q-axis; ψf is the flux which is generated by the permanent magnet; Rs is the stator resistance; TL denotes the load torque; J is the rotary inertia; B is the coefficient of friction; np represents the pole pairs of motor; ωm is the mechanical angular velocity of rotor.
Define (2)x=idiqωm,x~=i~di~qω~m,b=LqLd,k=βnpτψf,τ=LdRs,λ=λd000λq000λω=bk000k0001τ,and choose the affine transformation x=λx~ and time scale transformation t=τt~ for system (1); then (1) can be transformed into(3)di~ddt=-i~d+ω~mi~q+u~ddi~qdt=-i~q-ω~mi~d+γω~m+u~qdω~mdt=σi~q-ω~m+εi~di~q-T~L,where γ=-ψf/kLq, σ=βτ/J, ε=npbτ2k2Ld-Lq/J, u~d=1/kRud, u~q=1/kRuq, and T~L=τ2/JTL. Since the conversion is linear transformation, it does not change the characteristics of the system. The purpose of the transformation is to simplify system (1) and to obtain the corresponding chaotic model. For the uniform air gap, we have Ld=Lq=L, and then (3) can be rewritten as(4)di~ddt=-i~d+ω~mi~q+u~ddi~qdt=-i~q-ω~mi~d+γω~m+u~qdω~mdt=σi~q-ω~m-T~L,where i~d, i~q, and ω~m are the state variables, which represent the stator currents of direct axis and quadrature axis and the angular frequency of rotor, respectively; u~d and u~q denote the stator voltages of the direct axis and quadrature axis, respectively; T~L is the external torque; σ, γ are the constant parameters. The external torque is set as T~L=0 when u~d=0 and u~q=0. Then, we can obtain the following chaotic PMSM model:(5)di~ddt=-i~d+ω~mi~qdi~qdt=-i~q-ω~mi~d+γω~mdω~mdt=σi~q-ω~m.
Define x1=ω~m, x2=i~q, and x3=i~d, and from (5) we have(6)x˙1=σx2-x1x˙2=γx1-x1x3-x2x˙3=x1x2-x3,where x1, x2, and x3 are the states. The PMSM system (6) is regarded as the active system, and the corresponding passive PMSM system is given by(7)y˙1=σy2-y1y˙2=γy1-y1y3-y2+uy˙3=y1y2-y3,where y1, y2, and y3 are the states and u is the system controller.
The objective of this paper is to design the controller u such that the synchronization performance of the states xi and yi, i=1,2,3 between chaotic systems (6) and (7) are achieved.
3. Extend State Observer Design
Define the synchronization errors as e1=y1-x1, e2=y2-x2, and e3=y3-x3, and we can obtain the following error system:(8)e˙1=σe2-e1e˙2=γe1-y1y3+x1x3-e2+ue˙3=y1y2-x1x2-e3.
In (8), we have the following fact:(9)y1y3-x1x3=-e1e3+e1y3+e3y1y1y2-x1x2=-e1e2+e1y2+e2y1.
Substituting (9) into (8) yields(10)e˙1=σe2-e1e˙2=γe1+e1e3-e1y3-e3y1-e2+ue˙3=-e1e2+e1y2+e2y1-e3.
Then, error system (10) can be divided into the following two subsystems:(11)e˙1=σe2-e1e˙2=γe1+e1e3-e1y3-e3y1-e2+u,(12)e˙3=-e1e2+e1y2+e2y1-e3.
Remark 1.
From (12), it can be easily concluded that e˙3=-e3 when e1 and e2 converge to zero, which leads e3 to converge to zero finally. That means the error e3 acts as the interior dynamics of whole system (10). Therefore, the control task is transferred to design the controller u for subsystem (11) and guarantee the convergence of e1 and e2.
Define g1=e1, g2=σe2-e1, and then subsystem (11) can be transformed into the following Brunovsky form:(13)g˙1=g2g˙2=ae+bu,where ae=σγe1+e1e3-e3y1-e1y3-e2-σe2-e1, b=σ.
In order to facilitate the design of controller u, the system uncertainty ae and unknown parameter b in (13) should be measured by designing an observer. Define a0=ae+Δbu, Δb=b-b0, where b0 is the estimation of b and can be given by the prior experience directly. Then, designing an extended state g3=a0, system (13) can be transformed into(14)g˙1=g2g˙2=g3+b0ug˙3=a˙0.
Define zi, i=1,2,3, as the observation values of the states gi in (14), and the corresponding observer errors are given as e0i=zi-gi; then the nonlinear extended state observer is expressed as(15)z˙1=z2-β1e01z˙2=z3-β2fale01,α1,δ+b0uz˙3=-β3fale01,α2,δ,where β1,β2,β3>0 are the observer tuning gains; fal· is a nonlinear continuous function with the following form:(16)fale01,αi,δ=e01δ1-αie01≤δe01αisigne01e01>δ,where δ>0 denotes the interval length of the linear segment; 0<αi<1 is a constant.
4. Controller Design and Stability Analysis4.1. Controller Design
In order to stabilize the system tracking errors e1 and e2 to the zero, an adaptive controller u is designed in this subsection based on the sliding mode control technique.
The sliding mode surface is designed as(17)s=g2+λ1g1.
Differentiate s, and we can obtain(18)s˙=g˙2+λ1g˙1=g3+b0u+λ1g2,where λ1>0 is the control parameter.
According to (15) and (18), the traditional sliding mode controller using extended state observer (SMC + ESO) depicted in [17] is given by(19)u∗=1b0-z3-λ1z2-k∗signs,where k∗>0 is a constant satisfying the condition that k∗≥d3+λ1d2, in which d2 and d3 are the upper bounds of estimation error.
Unfortunately, the control gain k∗ cannot be obtained accurately since the upper bounds d2 and d3 are difficult to be measured. This may lead to a negative influence on the system control performance. To solve the problem, an adaptive sliding mode controller using extended state observer (ASMC + ESO) is developed with the following expression:(20)u=1b0-z3-λ1z2-ksigns,where k=kt is the adaptive control parameter designed as(21)k˙=kmssigns-ϵk>μμk≤μwith km, ϵ and μ being small positive constants and used to guarantee k>0.
4.2. Stability Analysis
Before the system stability analysis, the following two lemmas are introduced.
Lemma 2 (see [18]).
The parameter kt has an upper bound in the nonlinear uncertain system (13) with the sliding mode surface (17); namely, there exists a desired value k∗>0 which can guarantee that kt≤k∗, ∀t>0.
Lemma 3 (see [19]).
Suppose there is a continuous positive definite function Vt which satisfies the following differential function:(22)V˙t≤-αVηt∀t>0,Vt0>0,where α>0 and 0<η<1 are constants. Then, there exists a finite-time t1 for the given time t0, and we have the following inequality and equality relationships:(23)V1-ηt≤V1-ηt0-α1-ηt-t0t0≤t≤t1,Vt≡0,∀t≥t1,where t1=t0+V1-ηt0/α1-η.
Theorem 4.
Considering the uncertain PMSM subsystem (13), the sliding mode surface (17), the controller (20), and the parameter adaptive law (21), the sliding surface s can converge to zero within a finite time.
Proof.
Define a Lyapunov function for system (13):(24)V=12s2+12βk~2,where k~=k-k∗.
Differentiating V, we have(25)V˙=ss˙+1βk~k~˙=sg3+b0u+λ1g2+1βk-k∗k˙.
Substitute (20) into (25), and we can obtain(26)V˙=sg3-z3+λ1g2-z2-ksigns+1βk-k∗k˙≤sg3-z3+λ1g2-z2-ks+k∗s-k∗s+1βk-k∗k˙≤-k∗-d3+λ1d2s-k-k∗s+1βk-k∗k˙=-k∗-d3+λ1d2s+k-k∗1βk˙-s.
Introducing a new parameter αk>0, (26) can be written as(27)V˙≤-k∗-d3+λ1d2s-k-k∗s-1βk˙+αkk-k∗-αkk-k∗.
According to Lemma 2 and (27), it can be concluded that(28)V˙≤-αds-ξ-αkk-k∗=-2αds2-2βαkk-k∗2β-ξ≤-αmV1/2-ξ,where αd=k∗-d3+λ1d2>0, ξ=1/βk˙-s-αkk-k∗, and αm=min2αd,2βαk.
Since μ and ε are both small constants, without loss of generality, we only discuss the situation for k>μ. When k>μ, two different cases are discussed according to relationship between s and ϵ.
(a) When s>ϵ, (21) can be rewritten as k˙=kms,ξ=1/βkms-s-αkk-k∗. Choosing β<kmϵ/αk+ϵ, we can conclude ξ>0.
(b) When s≤ϵ, we have k˙=-kms, it can be concluded that ξ<0,k˙<0, and k is gradually decreasing. Thus, we can obtain s˙>0 when k is reduced to 0≤k≤d3+λ1d2. Consequently, sliding mode s will increase and achieve to the range of s>ϵ.
Similarly, we can guarantee ξ>0 by choosing the appropriate parameter β when k>μ. Then, from (28) and the above discussion, we have V˙≤-αmV1/2. According to Lemma 3, there exists a finite-time t1 satisfying Vt≡0 as t≥t1. This can ensure the convergence of the sliding mode s within a finite time.
Theorem 5.
The state variables e1,e2,e3 in error system (8) will converge to zero when the states g1,g2 in system (13) achieve the sliding surface s=0.
Proof.
System (13) have invariant characteristics when the states g1,g2 achieve s=0, and from (14) and (17) we have g˙1+λ1g1=0. Then, solving the first-order differential function, we can obtain g1=e-λ1t, which means that the state g1 can converge to zero when time tends to infinity. Also, according to (17), we have the similar result that g2 will converge to zero when time tends to infinity.
Consequently, the state variables e1,e2,e3 in error system (8) have the following relationship according to (11) and (12): e1,e2 converge to zero and e˙3=-e3 when time tends to infinity. Thus, we can conclude that e1,e2,e3 will asymptotically stabilized to zero. This completes the proof.
5. Simulation
In order to verify the effectiveness of the proposed method, a traditional sliding mode control based on extended state observer (SMC + ESO) is adopted to compare with the proposed adaptive control method (ASMC + ESO). The initial conditions and parameters in the simulations are set the same for a fair comparison; that is, the sampling time is set as Ts=0.01; the initial conditions are given as x10,x20,x30=-5,1,-3, y10,y20,y30=-1,0.01,20; the parameters of sliding mode control and extended state observer are chosen as λ1=10, b0=5, β1=60, β2=200, β3=0.01, α1=0.5, α2=0.25, α3=0.125, δ=0.01, and σ=5.46; the control parameter in SMC + ESO is given by k∗=12; the control parameters in ASMC + ESO are set as km=0.15, ϵ=0.01, and μ=0.0001.
Case 1 (the controller u works at initial time t=0).
The simulation results are shown in Figures 1–5. Figure 1 provides the synchronization performance of the system states. The observer errors of ESO and synchronization errors are shown in Figures 2 and 3, respectively. The control signal is given by Figure 4. As shown in Figures 1–3, the compared two control methods, that is, SMC + ESO and ASMC + ESO, can both achieve satisfactory chaos synchronization control performance; the observer errors of ESO and system synchronization errors can rabidly converge to zero. However, from Figure 4 we can see that the amplitude of ASMC + ESO is smaller than SMC + ESO, and the chattering phenomenon of control signal in ASMC + ESO is also smaller when the system is stable. The adaptation curve of the parameter kt is shown in Figure 5. As can be seen from Figure 5, the parameter k(t) converges to 8.2, which is slightly less than the parameter k∗ in SMC + ESO.
Synchronization performance of the system states in Case 1.
Synchronization of x1,y1 for SMC + ESO and ASMC + ESO
Synchronization of x2,y2 for SMC + ESO and ASMC + ESO
Synchronization of x3,y3 for SMC + ESO and ASMC + ESO
Observer errors of ESO in Case 1.
Synchronization errors in Case 1.
Control signals in Case 1.
Adaptive parameter kt in Case 1.
Case 2 (the controller u works at t=2s).
The parameters and initial conditions are all the same as those in Case 1 for fair comparison. The synchronization performance of the system states, observer errors of ESO, synchronization errors, control signals, and the adaptive parameter kt are shown in Figures 6–10, respectively. As can be seen from Figures 6–8, the control input is delayed to be working by 2 seconds, and the error system is not well controlled by using SMC + ESO; however, the ASMC + ESO can still have a good chaos synchronization control performance after a slight chattering. From Figure 9, we can clearly see that the chattering phenomenon of the control signal in SMC + ESO is significantly larger than that of ASMC + ESO. The reason is that, for the fixed parameter k in SMC + ESO, the condition that k∗≥d3+λ1d2 may not be always satisfied any more, but the control parameter kt in ASMC + ESO is an adaptive parameter, which can always satisfy the condition. From Figure 10, it can be seen that the parameter kt will converge to 42, which is much larger than the setting value k∗=12 in SMC + ESO.
Synchronization performance of the system states in Case 2.
Synchronization of x1,y1 for SMC + ESO and ASMC + ESO
Synchronization of x2,y2 for SMC + ESO and ASMC + ESO
Synchronization of x3,y3 for SMC + ESO and ASMC + ESO
Observer errors of ESO in Case 2.
Synchronization errors in Case 2.
Control signals in Case 2.
Adaptive parameter k(t) in Case 2.
6. Conclusion
In this paper, an adaptive sliding mode control method using extended state observer is presented to guarantee the synchronization control performance for two chaotic PMSM systems. An adaptive parameter is designed for the control gain to improve the suitability for different control situation and reduce the chattering in the control signal, and the extended state observer is adopted to estimate the system uncertainties. The simulation results indicate that the system can achieve a good synchronization control performance for different initial conditions.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was supported by the Major Research Plan of the National Natural Science Foundation of China (Grants nos. 91438117 and 91538202) and National Natural Science Foundation of China (Grant no. 61403343).
AhmadI.Bin SaabanA.IbrahimA. B.ShahzadM.Global chaos synchronization of new chaotic system using linear active control201521137938610.1002/cplx.21573MR34078772-s2.0-84942293405ChenQ.RenX. M.NaJ.Robust finite-time chaos synchronization of uncertain permanent magnet synchronous motors201558126226910.1016/j.isatra.2015.07.0052-s2.0-84943581226DittoW. L.SinhaS.Exploiting chaos for applications201525909761510.1063/1.49229762-s2.0-84934300851GuanJ.Adaptive modified generalized function projection synchronization between integer-order and fractional-order chaotic systems2016127104211421610.1016/j.ijleo.2016.01.030WangJ. K.ChenX. Q.FuJ. K.Adaptive finite-time control of chaos in permanent magnet synchronous motor with uncertain parameters20147821321132810.1007/s11071-014-1518-72-s2.0-84911002697HouY.-Y.Finite-time chaos suppression of permanent magnet synchronous motor systems20141642234224310.3390/e16042234MR31955782-s2.0-84899152422SuK. L.LiC. L.Chaos control of permanent magnet synchronous motors via unidirectional correlation2014125143693369610.1016/j.ijleo.2014.01.1312-s2.0-84902215242NguyenT.-B.-T.LiaoT.-L.YanJ.-J.Adaptive sliding mode control of chaos in permanent magnet synchronous motor via fuzzy neural networks201420141186841510.1155/2014/868415MR31704922-s2.0-84896132148YangX. H.LiuX. P.HuL. L.Robust sliding mode variable structure synchronization control of chaos in permanent magnet synchronous motor2012189398ChenQ.NanY.-R.ZhengH.-H.RenX.-M.Full-order sliding mode control of uncertain chaos in a permanent magnet synchronous motor based on a fuzzy extended state observer2015241111050410.1088/1674-1056/24/11/1105042-s2.0-84947294630YangX. H.LiuX. P.LiuH. S.Fuzzy sliding-mode control in permanent magnet synchronous motor20137152653510.4156/jdcta.vol7.issue1.61LoriaA.Robust linear control of (chaotic) permanent-magnet synchronous motors with uncertainties20095692109212210.1109/tcsi.2008.2011587MR26498282-s2.0-70349251408LuoS. H.Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain20142435880588503313510.1063/1.4895810MR34044332-s2.0-84907414315LuoS.Nonlinear dynamic surface control of chaos in permanent magnet synchronous motor based on the minimum weights of RBF neural network20142014960934010.1155/2014/6093402-s2.0-84904198711WeiQ.WangX.-Y.HuX.-P.Optimal control for permanent magnet synchronous motor20142081176118410.1177/10775463124646802-s2.0-84902155417NaJ.ChenQ.RenX. M.GuoY.Adaptive prescribed performance motion control of servo mechanisms with friction compensation201461148649410.1109/tie.2013.22406352-s2.0-84880883039LiS.ZhouM.YuX.Design and implementation of terminal sliding mode control method for PMSM speed regulation system2013941879189110.1109/TII.2012.22268962-s2.0-84886670034YuS.YuX. H.ShirinzadehB.ManZ.Continuous finite-time control for robotic manipulators with terminal sliding mode200541111957196410.1016/j.automatica.2005.07.001ChenQ.YuL.NanY.Finite-time tracking control for motor servo systems with unknown dead-zones201326694095610.1007/s11424-013-2153-yMR31514932-s2.0-84890364654