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Speed Synchronization of Multiple Induction Motors with Total Sliding Mode Control
Systems Engineering — Theory & Practice Volume 29, Issue 10, October 2009 Online English edition of the Chinese language journal Cite this article as: SETP, 2009, 29(10): 110–117
Speed Synchronization of Multiple Induction Motors with Total Sliding Mode Control ZHAO De-zong1,∗ , LI Chun-wen1,2 , REN Jun1 1. Department of Automation, Tsinghua University, Beijing 100084, China 2. Key Laboratory of Information Electric Apparatus in Henan Province, Zhengzhou 450002, China
Abstract: A speed synchronization control strategy for multiple induction motors, based on adjacent cross-coupling control structure, is developed by employing total sliding mode control method. The proposed control strategy is to stabilize speed tracking of each induction motor while synchronizing its speed with the speed of the other motors so as to make speed synchronization error amongst induction motors converge to zero. The global stability and the convergence of the designed controller are proved by using Lyapunov method. Simulation results demonstrate the effectiveness of the proposed method. Key words: multiple induction motors; speed synchronization; total sliding mode control; adjacent cross-coupling control
1 Introduction With the rapid development of modern manufacturing, a challenging problem in modern induction motor control field is that the motion of multiple motors must be controlled in a synchronous manner, for instance of distributed papermaking machines, continuous rolling mills, print works, and spinning works. The performance of synchronization control affects the reliability and precision of product seriously[1] . In multiple-motor applications, the synchronization performance of the system may be degraded by many factors such as disturbances in the load and system parameters variation due to environmental changes[2−3] , thus high performance synchronization control of multiple induction motors is important in practice. Many control schemes have been proposed in implementing speed synchronization research. In master slave structure[4−5] , the input of the slave motor is the output of the master motor, but the disturbances on the load torque may result in large speed-tracking errors. Parallel crosscoupling control is used to remedy this problem by sharing the feedback information of both control loops, which is proposed by Koren[6] . But when the number of synchronization motors n > 2, parallel cross-coupling control is not applicable because the control structure is difficult to determine. Perez[7−8] proposed the relative cross-coupling control scheme, which is applicable to the number of synchronization motors n > 2. Unfortunately, online computation of an n-dimensional square matrix is needed when designing the above controllers, which increases the computation work heavily. If the scale of the system n is large, the control structure is complicated and the computation pressure is hard. In order to simplify the control structure, Sun[9] proposed adjacent cross-coupling control, which is applied in
robotic synchronization combined with adaptive control, so that simple control structure and high synchronization performance are achieved. In this article, total sliding mode control is adopted in adjacent cross-coupling control structure to implement speed synchronization of multiple induction motors. The speed synchronization strategy is to stabilize synchronization errors between each motor and the other two motors to zero. The tracking error controllers and the synchronization error controllers both include an equivalent control law and a robust control law, which makes the stable tracking performances and synchronization performances are ensured against the system uncertainties. Simulations on a fourmotor system are given, and the performance of the proposed control strategy in several typical working conditions have been studied. Lyapunov stability analysis and simulation results demonstrate the effectiveness of the proposed control strategy.
2 Induction motor model The four-order nonlinear state-space model of the induction motor in rotor field-oriented coordinate is shown as ⎤ ω ⎥ d ⎢ ⎢ ψdr ⎥ dt ⎣ ids ⎦ iqs ⎡ ⎡
⎤ μψdr iqs − (Bω) /J − TL /J ⎢ ⎥ −αψdr + αM ids ⎥ =⎢ 2 ⎣ ⎦ −γids + αβψdr + np ωiqs + αM iqs /ψdr −γiqs + βnp ωψdr − np ωiqs − αM iqs ids /ψdr T +1/(σLs ) 0 0 uds uqs (1)
Received date: September 3, 2008 ∗ Corresponding author: Tel: +86-10-62797962; E-mail: [email protected] Foundation item: Supported by the National Nature Science Foundation(No.69774011, 60433050) c 2009, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
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Figure 2. Diagram of the speed controller of motor i
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Define the synchronization errors of a subset of all possible pairs of two motors from the total of n motors as:
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Figure 1. Diagram of adjacent cross-coupling control
Denote the rotor mechanical speed by ω, and the rotor flux in the direct axis by ψdr . ids , iqs are the stator currents, respectively; uds and uqs are the corresponding applied stator voltages, respectively. Rs , Rr are the stator and rotor resistances, respectively; Ls , Lr are the stator and rotor self-inductances, respectively; M is the statorrotor mutual inductance. B is the friction coefficient, J is the motor-load moment of inertia, TL is the load torque, and np is the number of the pole pairs. The inverse of the rotor time constant is α = Rr /Lr , and the leakage factor is σ = 1 − M 2 / (Ls L r ). Definitions of
the other parameters in (1) are γ = M 2 Rr / σLs L2r + (Rs / (σLs )), β = M/ (σLs Lr ), and μ = np M/ (JLr ). The induction motor can be divided into two subsystems: the mechanical system and the electromagnetic system. The first equation of the induction model is the mechanical system, while the other equations construct the electromagnetic system. Using singular perturbation theory, the mechanical system can be considered as the slow subsystem, and the electromagnetic system is considered as the fast subsystem. The variables of the electromagnetic system ψdr , ids , and iqs converge to stable value fast, then we aim to control the output speed converge to the reference speed primarily. The mechanical equation of the induction motor can be rewritten as J ω(t) ˙ + Bω(t) + Tl = Te
(2)
where the electromagnetic torque is Te = KT i∗qs with the coefficient is defined as the torque constant: KT = (3np /2)(L2m /Lr i∗ds ), where i∗ds and i∗qs denote the flux and torque current commands, respectively.
3 Adjacent cross-coupling control structure Considering a multiple-motor system including n induction motors, the tracking error of motor i is denoted as ei (t) = ωi∗ (t) − ωi (t), where ωi∗ (t) and ωi (t) denote the reference speed and the output speed of motor i, respectively. In the synchronization motion, besides ei (t) → 0, it is also aimed to regulate motion relationships among multiple motors in the tracking so that e1 (t) = · · · = ei (t) = · · · = en (t)
(3)
ε1 (t) ε2 (t)
= = .. .
e1 (t) − e2 (t) e2 (t) − e3 (t)
εn (t)
=
en (t) − e1 (t)
(4)
If εi (t) = 0, ∀i = 1 · · · n are held, then (3) is achieved. The relevant synchronization errors of motor i are εi−1 (t) and εi (t), namely we only need to synchronize motion of motor i with motor (i − 1) and motor (i + 1), respectively. Accordingly, a concept named coupled speed error, denoted by e∗i (t), is introduced here[10] : e∗1 (t) e∗2 (t)
= = .. .
ε1 (t) − εn (t) ε2 (t) − ε1 (t)
e∗n (t)
=
εn (t) − εn−1 (t)
(5)
When e∗1 (t) = · · · = e∗i (t) = · · · = e∗n (t) = 0 is achieved, then ε1 (t) = · · · = εi (t) = εn is held, therefore e1 (t) = · · · = ei (t) = · · · = en (t) = 0 is held. The aim of designing synchronization error controllers is driving e∗i (t) converge to zero fast for i = 1 · · · n. The diagram of adjacent cross-coupling control is illustrated as Figure 1. Synchronous speed controller includes two subcontrollers: one tracking error controller, which is used to track the reference speed value accurately, and one synchronization error controller, which is used to drive the synchronization error between the controlled motor and its adjacent motors to zero. The inner structure of controller i is shown as Figure 2, where Ui1 and Ui2 are the output of the tracking error controller and the synchronization error controller, respectively. Ui is the complete control law of motor i. In our control scheme, each control loop only considers their own motion response and those of the other two motors, but not all other motors, for synchronization. For a multiple-motor system with n induction motors, adjacent cross-coupling control requires 2n controllers while relative cross-coupling control requires n2 controllers, so adjacent cross-coupling control significantly simplifies the implementation especially when the number of motors is large.
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
4 Speed controller design The mechanical equation of the induction motor model can be represented as ˙ X(t) = Am X(t) + Bm (t) + Cm TL
(6)
where X(t) = ω(t), Am = −B/J, Bm = −KT /J, Cm = −1/J, and U (t) = i∗qs . Assuming the parameters of the controlled system are well known and the external load disturbance is absent, rewrite (6) as the nominal model: ˙ X(t) = AmN X(t) + BmN U (t)
(7)
¯ J¯ and BmN = −K ¯ T /J¯ are the nominal where AmN = −B/ values of Am and Bm , respectively. Considering the parameters deviation and the disturbances of the load torques, the dynamics of the controlled mechanical subsystem is rewritten as ˙ X(t)
=
(AmN + ΔA)X(t) + (BmN + ΔB)U (t) + (CmN + ΔC)TL
=
AmN X(t) + BmN U (t) + W (t)
(8)
where CmN = −1/J¯ is the nominal value of Cm ; ΔA, ΔB, and ΔC denote the mechanical uncertainties; W (t) is the lumped uncertainty and is defined as W (t) = ΔAX(t) + ΔBU (t) + (CmN + ΔCTL )
Thus, an equivalent control law is designed as
t −β 0 e(τ )dτ + (AR + β)e(0)+ −1 (15) UE (t) = BmE (AR − AmN )X(t) + BR UR (t) Substituting (15) into (14), then the derivative of the sliding mode surface is ˙ S(t) = (AR + β)S(t)
In conventional sliding mode control[11] , if (AR + β) < 0 is held, then lim S → 0 is achieved. In the presented scheme, t→∞
β is chosen such that (AR + β) = 0, namely the selection ˙ of β = −AR will result in S(t) = 0. According to (13), ˙ S(0) = 0 is guaranteed. S(t) = 0 and S(0) = 0 assure that the system is in the sliding mode surface at the beginning, and make the whole system be free of reaching mode that exists in conventional sliding mode control. The presented control scheme is called total sliding mode control because of its free sliding mode surface[12−13] . The equivalent controller under ideal condition has been designed completely. However, if the system uncer˙ tainties occur, then controller (15) can not ensure S(t) =0 be maintained. Considering the extra disturbances and inner perturbations, we should design a robust controller to stabilize the system. Considering the speed error (11), the tracking error dynamic can be rewritten as the speed error between the reference model (10) and the model (8): e(t) ˙
=
X˙ R (t) = AR XR (t) + BR UR (t)
(10)
where UR is the speed command; XR is the output the reference speed; AR and BR are given coefficients according to the desired system specifications. The speed error between the reference model (10) and the nominal model (7) is e(t) = XR (t) − X(t)
(11)
The differential of the speed error is represented as =
˙ X˙ R (t) − X(t)
=
AR (t)e(t) + (AR − AmN )X(t) + BR UR (t) − BmN U (t)
−1 (ka S(t) + kb sign(S(t))) UF (t) = BmE
(12)
t 0
UT (t) = UE (t) + UF (t)
Theorem 1 For the mechanical subsystem (8), which is controlled by controller (19) with ka > Am + β and kb > |W (t)|, the speed-tracking error trajectory can always be kept on the tracking error-sliding mode surface and asymptotical convergence of the controlled system can be guaranteed. Proof. Differentiating (13) with respect to time and using (17), we can obtain t ˙ S(t) = (AR + β)S(t) − β(AR + β) e(τ )dτ + 0
BR UR (t) − BmN UT (t) − W (t) (13)
where β is is a positive constant. Differentiating (13) with respect to time, we obtain: ˙ S(t)
=
(AR + β)S(t) − β(AR + β)
(19)
(AR + β)e(0) + (AR − AmN )X(t)) +
e(τ )dτ − e(0)
(18)
where ka and kb are positive constants. The complete speed tracking error controller is shown as
Define a tracking error-sliding mode surface as S(t) = e(t) + β
(17)
The robust controller is designed as
Define the reference model of the controlled system as
AR e(t) + (AR − AmN )X(t) + BR UR (t) − BmN U (t) − W (t)
(9)
4.1 Tracking error controller
e(t) ˙
(16)
t
e(τ )dτ
(20)
Substituting controller (22) into (23), we obtain ˙ S(t)
=
(AR + β)S(t) − ka S(t) − kb sgn(S(t)) − W (t)
(21)
Define a positive definite function as VT = 12 S 2 (t), with the derivative respect to time: V˙ T
+(AR + β)e(0) + (AR − AmN )X(t) +
˙ = S(t)S(t) = (AR + β − ka )S 2 (t) − kb |S(t)| − W (t)S(t)
BR UR (t) − BmN U (t)
≤
0
(14)
(AR + β − ka )S 2 (t) + (|W (t)| − kb )|S(t)|
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
If ka and kb are chosen to satisfy ka > Am + β and kb > |W (t)|, respectively, then V˙ T ≤ 0 is achieved. From Lyapunov theorem, the controlled system is asymptotically stable. 4.2 Synchronization error controller Define the synchronization error-sliding mode surface of motor i as t Si∗ (t) = e∗i (t) + β e(τ )dτ − e∗i (0) (22) 0
namely Si∗ (t) = 2Si (t) − Si+1 (t) − Si−1 (t)
(23)
where Si (t) denotes the tracking error-sliding mode surface of motor i. The speed synchronization error controller of motor i is designed as Ui∗ (t)
=
2UT i (t) −
BmN (i+1) UT (i+1) (t) − BmN
BmN (i−1) UT (i−1) (t) BmN
(24)
where BmN i is the coefficient BmN of motor i, and UT i is the tracking error control law of motor i. Theorem 2 For the mechanical subsystem (8), which is controlled by controller (24) with ka > Am + β and kb > |W (t)|, then the speed synchronization error trajectory can always be kept on the synchronization errorsliding mode surface and asymptotical convergence of the controlled system can be guaranteed. Proof. Differentiating (22) with respect to time, we get S˙ i∗ (t)
=
AR e∗i (t) + 2(AR − AmN i )Xi (t)) − (AR − AmN (i+1) )Xi+1 (t)) − (AR − AmN (i−1) Xi−1 (t)) +
βe∗i (t) − Wi∗ (t) − BmN i Ui∗ (t)
(25)
where AmN i is the coefficient AmN of motor i; Wi∗ (t) is the lumped uncertainties in synchronization of motor i and has the expression of Wi∗ (t) = 2Wi (t) − Wi+1 (t) − Wi−1 (t)
=
Lemma 1 Suppose f (x) is defined on interval I. If f (x) is differentiable and f˙(x) is bounded, then f (x) is uniformly continuous. Lemma 2 (Barbalat Lemma) If lim f (t) is bounded, x→∞ and f˙(t) is uniformly continuous, then lim f˙(t) = 0. t→∞
Lemma 3 (LaSalle’s invariance principle) Suppose there is a neighborhood D of 0 and a continuously differentiable positive definite function V : D → R whose orbital derivative V˙ is negative semidefinite. Let H be the union of all complete orbits contained in {x ∈ D|V˙ (x) = 0}. Then, there is a neighborhood U of 0 such that for any x0 ∈ U , ω(x0 ) ⊆ H . Thoerem 3 The proposed sliding mode cross-coupling controller (29) guarantees asymptotic convergence to 0 of both tracking errors and synchronization errors, i.e., lim ei (t) = 0 and lim ei (t) = 0 for i = 1, · · · , n. t→∞ n
t→∞
Proof. A positive definite function is defined as V (t) =
( 12 Si2 (t)), with its derivative =
n i=1 n
(Si (t)S˙ i (t))
(AR + β − ka )Si2 (t) + (|Wi (t)| − kb )|Si (t)|
i=1
If ka = Am +β and kb > |W (t)| are satisfied, then V˙ (t) ≤ 0 is guaranteed. Considering (21), the derivatives of the sliding mode surfaces are shown as: (27)
Considering (20) and (21), Eq. (27) can be represented as (28)
If the tracking error controllers (19) of motor i , i + 1 , and i − 1 are properly designed, then lim Si∗ (t) = lim Si (t) = t→∞
5 Stability analysis
=
βe∗i (t) − Wi∗ (t) − BmN i UT i (t) − BmN (i+1) UT (i+1) (t) −
t→∞
When we design the speed controller using slidingmode control theory, the chattering phenomenon is introduced around the sliding-mode surface. In order to reduce the chattering, the sign function sign(·) is replaced by the saturation function sat(·). The saturation function is designed as: ⎧ ⎨ 1 Si > 1 −1 Si < −1 sat(Si ) = i = 1, · · · , n (30) ⎩ Si |Si | ≤ 1
V˙ (t)
(AR − AmN (i−1) Xi−1 (t)) +
S˙ i∗ (t) = 2S˙ i (t) − S˙ i+1 (t) − S˙ i−1 (t)
(29)
4.3 Chattering reduction technique
(26)
AR e∗i (t) + 2(AR − AmN i )Xi (t)) − (AR − AmN (i+1) )Xi+1 (t)) −
BmN (i−1) UT (i−1) (t)
Ui (t) = UT i (t) + Ui∗ (t)
i=1
Substituting (24) into (25), we have S˙ i∗ (t)
lim Si+1 (t) = lim Si−1 (t) = 0 is held. It can conclude t→∞ that the asymptotic stability of the synchronization errorsliding mode surface can be guaranteed whether the system uncertainties exist or not, and the sliding mode is assured throughout the whole control period. The tracking error controller and the synchronization error controller construct the complete controller of motor i: t→∞
S˙ i (t) = kb sign(Si (t) − Wi (t))
i = 1, · · · , n.
(31)
Because Wi (t) is bounded in terms of L2 norm, S˙ i (t) is also bounded. Using Lemma 1, Si (t) is uniformly continuous. Using lemma 2, lim Si (t) = 0 is also held. t→∞ The invariant set of the controlled system in the set Ω = (xi , x˙ i ) : V˙ (t) = 0 contains the equilibrium points of the
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
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Figure 3. Structure diagram of the control subsystem of motor i
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Figure 4. System performance with high reference speed
Figure 5. System performance with load torques change
sliding mode surfaces. Using lemma 3, Si (t) = 0 is globally asymptotically stable. Considering (13) and (22), the analytical expressions of ei (t) and e∗i (t) are as follows: ei (t) = ei (0)e−βt e∗i (t) = e∗i (0)e−βt
(32)
From (32), we can derive that lim ei (t) = 0 are held. When
t→∞
e∗i (t)
lim e∗ (t) t→∞ i
=
= 0, from (5), ε1 (t) =
· · · = εi (t) = · · · = εn (t) is achieved. Considering (4) and ei (t) = 0, then ε1 (t) = · · · = εi (t) = · · · = εn (t) = 0 is achieved, namely the synchronization errors of the controlled system converge to 0 simultaneously. Table 1. Parameters of the four motors
Parameters PN /Kw ωN /(rad/m) ψr∗ /Wb TLN /Nm Rr /Ω Rs /Ω Lr /H Ls /H Lm /H 2 ¯ J/(kgm ) B/(Nms/rad) np
Motor 1 1.1 1410 0.86 6.0 5.5 6.7 0.47 0.475 0.45 0.015 0.01 2
Motor 2 1.1 1410 0.86 6.0 5.5 6.7 0.47 0.475 0.45 0.015 0.01 2
Motor 3 1.1 1475 0.9 7.0 4.45 5.46 0.492 0.492 0.475 0.008 0.005 2
Motor 4 1.1 1475 0.9 7.0 4.45 5.46 0.492 0.492 0.475 0.008 0.005 2
Motor Motor Motor Motor
6 Simulation In this section, computer simulations are carried out on a multiple-motor system including 4 induction motors to verify the effectiveness of the proposed control approach. The schematic structure diagram of the control subsystem of motor i within the multiple-motor system is shown as Figure 3. The parameters of the 4 motors are listed in Table 1. Figure 4 shows the controlled system performance under high reference speed. The parameters of the four motors are given as nominal value. The speed command is ω ∗ = 1000r/m. The load torques of the four motors are TL1 = 2Nm, TL2 = 3Nm, TL3 = 1Nm, and TL4 = 4Nm. We can see that the tracking errors and the synchronization errors of the four motors are all converge to zero in 1s. The response of the controlled multiple-motor system is fast with high precision. In order to testify the robustness of the controlled system, the load torques changed at 3s simultaneously, while the command speed is set as ω ∗ = 1000r/m. The values of the load torques are listed in Table 2. Figure 5 illustrates the system performance under load torques change. The response curves deteriorated slightly at 3s and Table 2. Load torques of the four motors
Load Torque TL /Nm (0s < t ≤ 3s) TL /Nm (3s < t ≤ 5s)
Motor 1 2 2.5
Motor 2 3 4
Motor 3 1 2
Motor 4 4 3
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Figure 6. System performance with reference speed change
converge to the desired value fast. Figure 6 illustrates the response curves under reference speed changed. The command speed changed from ω ∗ = 1000r/m to ω ∗ = −1000r/m at 3s, whereas the load torques and the other parameters of the motors are maintained at the normal values. The tracking errors and the synchronization errors deviated from zero dramatically, but be driven back to zero fast. Figure 7 shows the response curves under system parameter perturbations. The values of the moment of inertias are changed as: J1 = 0.02kgm2 , J2 = 0.017kgm2 , J1 = 0.017kgm2 , and J1 = 0.012kgm2 . From the simulation results, we can see that the performance of the controlled system varied slightly without tuning the parameters of the controllers. The tracking errors and the synchronization errors converge to zero within 1s and maintain on it.
7 Conclusions A new synchronization control strategy of multiple induction motors is proposed in this article. Sliding-mode control is employed into adjacent cross-coupling control. The presented control scheme comprises two parts: tracking error controller and synchronization error controller. Each controller includes an equivalent controller and a robust controller. Global stability of the proposed control scheme has been proven by Lyapunov method, Barbalat’s lemma, and LaSalle’s invariance principle. The proposed sliding mode control scheme has a total sliding motion without a reaching phase as in the conventional sliding mode control, such that the response and precision is better than the conventional sliding mode control. The proposed control method is simpler than traditional relative cross-coupling control. Simulation results demonstrated the fast response and high robustness of the proposed synchronization control approach.
References [1] Chiu T C, Tomizuka M. Coordinate position control of multi-axis mechanical system. Dynamic Systems and Con-
Figure 7. System performance with inner parameters change
trol, 1994, 55(1): 159–166. [2] Lee H S, Tomizuka M. Robust motion controller design for high-accuracy positioning. IEEE Transactions on Industrial Electronics, 1996, 43(1): 48–55. [3] Xiao Y, Zhu K Y. Optimal synchronization control of highprecision motion systems. IEEE Transactions on Industrial Electronics, 2006, 53(4): 1160–1169. [4] Tomizuka M, Hu J S, Chiu T C, et al. Synchronization of two motion control axes under adaptive feedforward control. ASME Journal of Dynamic Systems, Measurement, and Control, 1992, 114(2): 196–203. [5] Perez-Pinal F J, Nunez C, Alvarez R, et al. Comparison of multi-motor synchronization techniques. Proceedings of the 30th annual conference of the IEEE industrial electronics society, 2004: 1670–1675. [6] Koren Y. Cross-coupled biaxial computer for manufacturing systems. ASME Journal of Dynamic Systems, Measurement, and Control, 1980, 102(4): 265–272. [7] Perez-Pinal F J, Calderon G, Araujo-Vargas I. Relative coupling strategy. Proceeding of the IEEE International Electric Machines and Drives Conference, 2003: 1162–1166. [8] Yeh S S, Hsu P L. Analysis and design of integrated control for multi-axis motion systems. IEEE Transactions on Control System Technology, 2003, 11(3): 375–382. [9] Sun D, Mills J K. Adaptive synchronized control for coordination of two robot manipulators. Proceedings of the IEEE international conference on robotics and automation, 2002: 976–981. [10] Sun D. Position synchronization of multiple motion axes with adaptive coupling control. Automatica, 2003, 39(6): 997– 1005. [11] Chen F, Dunnigan M W. Sliding-mode torque and flux control for an induction machine. IEEE Proceedings of Electric Power Applications, 2003, 150(2): 227–236. [12] Wai R J, Lin K M, Lin C Y. Total sliding-mode speed control of field-oriented induction motor servo drive. Proceedings of the 5th Asian Control Conference, 2004: 1354–1361. [13] Wai R J, Lin K M. Robust decoupled control of direct fieldoriented induction motor drive. IEEE Transactions on Industrial Electronics, 2005, 52(3): 837–854.
Speed Synchronization of Multiple Induction Motors with Total Sliding Mode Control ZHAO De-zong1,∗ , LI Chun-wen1,2 , REN Jun1 1. Department of Automation, Tsinghua University, Beijing 100084, China 2. Key Laboratory of Information Electric Apparatus in Henan Province, Zhengzhou 450002, China
Abstract: A speed synchronization control strategy for multiple induction motors, based on adjacent cross-coupling control structure, is developed by employing total sliding mode control method. The proposed control strategy is to stabilize speed tracking of each induction motor while synchronizing its speed with the speed of the other motors so as to make speed synchronization error amongst induction motors converge to zero. The global stability and the convergence of the designed controller are proved by using Lyapunov method. Simulation results demonstrate the effectiveness of the proposed method. Key words: multiple induction motors; speed synchronization; total sliding mode control; adjacent cross-coupling control
1 Introduction With the rapid development of modern manufacturing, a challenging problem in modern induction motor control field is that the motion of multiple motors must be controlled in a synchronous manner, for instance of distributed papermaking machines, continuous rolling mills, print works, and spinning works. The performance of synchronization control affects the reliability and precision of product seriously[1] . In multiple-motor applications, the synchronization performance of the system may be degraded by many factors such as disturbances in the load and system parameters variation due to environmental changes[2−3] , thus high performance synchronization control of multiple induction motors is important in practice. Many control schemes have been proposed in implementing speed synchronization research. In master slave structure[4−5] , the input of the slave motor is the output of the master motor, but the disturbances on the load torque may result in large speed-tracking errors. Parallel crosscoupling control is used to remedy this problem by sharing the feedback information of both control loops, which is proposed by Koren[6] . But when the number of synchronization motors n > 2, parallel cross-coupling control is not applicable because the control structure is difficult to determine. Perez[7−8] proposed the relative cross-coupling control scheme, which is applicable to the number of synchronization motors n > 2. Unfortunately, online computation of an n-dimensional square matrix is needed when designing the above controllers, which increases the computation work heavily. If the scale of the system n is large, the control structure is complicated and the computation pressure is hard. In order to simplify the control structure, Sun[9] proposed adjacent cross-coupling control, which is applied in
robotic synchronization combined with adaptive control, so that simple control structure and high synchronization performance are achieved. In this article, total sliding mode control is adopted in adjacent cross-coupling control structure to implement speed synchronization of multiple induction motors. The speed synchronization strategy is to stabilize synchronization errors between each motor and the other two motors to zero. The tracking error controllers and the synchronization error controllers both include an equivalent control law and a robust control law, which makes the stable tracking performances and synchronization performances are ensured against the system uncertainties. Simulations on a fourmotor system are given, and the performance of the proposed control strategy in several typical working conditions have been studied. Lyapunov stability analysis and simulation results demonstrate the effectiveness of the proposed control strategy.
2 Induction motor model The four-order nonlinear state-space model of the induction motor in rotor field-oriented coordinate is shown as ⎤ ω ⎥ d ⎢ ⎢ ψdr ⎥ dt ⎣ ids ⎦ iqs ⎡ ⎡
⎤ μψdr iqs − (Bω) /J − TL /J ⎢ ⎥ −αψdr + αM ids ⎥ =⎢ 2 ⎣ ⎦ −γids + αβψdr + np ωiqs + αM iqs /ψdr −γiqs + βnp ωψdr − np ωiqs − αM iqs ids /ψdr T +1/(σLs ) 0 0 uds uqs (1)
Received date: September 3, 2008 ∗ Corresponding author: Tel: +86-10-62797962; E-mail: [email protected] Foundation item: Supported by the National Nature Science Foundation(No.69774011, 60433050) c 2009, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
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Figure 2. Diagram of the speed controller of motor i
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Define the synchronization errors of a subset of all possible pairs of two motors from the total of n motors as:
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Figure 1. Diagram of adjacent cross-coupling control
Denote the rotor mechanical speed by ω, and the rotor flux in the direct axis by ψdr . ids , iqs are the stator currents, respectively; uds and uqs are the corresponding applied stator voltages, respectively. Rs , Rr are the stator and rotor resistances, respectively; Ls , Lr are the stator and rotor self-inductances, respectively; M is the statorrotor mutual inductance. B is the friction coefficient, J is the motor-load moment of inertia, TL is the load torque, and np is the number of the pole pairs. The inverse of the rotor time constant is α = Rr /Lr , and the leakage factor is σ = 1 − M 2 / (Ls L r ). Definitions of
the other parameters in (1) are γ = M 2 Rr / σLs L2r + (Rs / (σLs )), β = M/ (σLs Lr ), and μ = np M/ (JLr ). The induction motor can be divided into two subsystems: the mechanical system and the electromagnetic system. The first equation of the induction model is the mechanical system, while the other equations construct the electromagnetic system. Using singular perturbation theory, the mechanical system can be considered as the slow subsystem, and the electromagnetic system is considered as the fast subsystem. The variables of the electromagnetic system ψdr , ids , and iqs converge to stable value fast, then we aim to control the output speed converge to the reference speed primarily. The mechanical equation of the induction motor can be rewritten as J ω(t) ˙ + Bω(t) + Tl = Te
(2)
where the electromagnetic torque is Te = KT i∗qs with the coefficient is defined as the torque constant: KT = (3np /2)(L2m /Lr i∗ds ), where i∗ds and i∗qs denote the flux and torque current commands, respectively.
3 Adjacent cross-coupling control structure Considering a multiple-motor system including n induction motors, the tracking error of motor i is denoted as ei (t) = ωi∗ (t) − ωi (t), where ωi∗ (t) and ωi (t) denote the reference speed and the output speed of motor i, respectively. In the synchronization motion, besides ei (t) → 0, it is also aimed to regulate motion relationships among multiple motors in the tracking so that e1 (t) = · · · = ei (t) = · · · = en (t)
(3)
ε1 (t) ε2 (t)
= = .. .
e1 (t) − e2 (t) e2 (t) − e3 (t)
εn (t)
=
en (t) − e1 (t)
(4)
If εi (t) = 0, ∀i = 1 · · · n are held, then (3) is achieved. The relevant synchronization errors of motor i are εi−1 (t) and εi (t), namely we only need to synchronize motion of motor i with motor (i − 1) and motor (i + 1), respectively. Accordingly, a concept named coupled speed error, denoted by e∗i (t), is introduced here[10] : e∗1 (t) e∗2 (t)
= = .. .
ε1 (t) − εn (t) ε2 (t) − ε1 (t)
e∗n (t)
=
εn (t) − εn−1 (t)
(5)
When e∗1 (t) = · · · = e∗i (t) = · · · = e∗n (t) = 0 is achieved, then ε1 (t) = · · · = εi (t) = εn is held, therefore e1 (t) = · · · = ei (t) = · · · = en (t) = 0 is held. The aim of designing synchronization error controllers is driving e∗i (t) converge to zero fast for i = 1 · · · n. The diagram of adjacent cross-coupling control is illustrated as Figure 1. Synchronous speed controller includes two subcontrollers: one tracking error controller, which is used to track the reference speed value accurately, and one synchronization error controller, which is used to drive the synchronization error between the controlled motor and its adjacent motors to zero. The inner structure of controller i is shown as Figure 2, where Ui1 and Ui2 are the output of the tracking error controller and the synchronization error controller, respectively. Ui is the complete control law of motor i. In our control scheme, each control loop only considers their own motion response and those of the other two motors, but not all other motors, for synchronization. For a multiple-motor system with n induction motors, adjacent cross-coupling control requires 2n controllers while relative cross-coupling control requires n2 controllers, so adjacent cross-coupling control significantly simplifies the implementation especially when the number of motors is large.
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
4 Speed controller design The mechanical equation of the induction motor model can be represented as ˙ X(t) = Am X(t) + Bm (t) + Cm TL
(6)
where X(t) = ω(t), Am = −B/J, Bm = −KT /J, Cm = −1/J, and U (t) = i∗qs . Assuming the parameters of the controlled system are well known and the external load disturbance is absent, rewrite (6) as the nominal model: ˙ X(t) = AmN X(t) + BmN U (t)
(7)
¯ J¯ and BmN = −K ¯ T /J¯ are the nominal where AmN = −B/ values of Am and Bm , respectively. Considering the parameters deviation and the disturbances of the load torques, the dynamics of the controlled mechanical subsystem is rewritten as ˙ X(t)
=
(AmN + ΔA)X(t) + (BmN + ΔB)U (t) + (CmN + ΔC)TL
=
AmN X(t) + BmN U (t) + W (t)
(8)
where CmN = −1/J¯ is the nominal value of Cm ; ΔA, ΔB, and ΔC denote the mechanical uncertainties; W (t) is the lumped uncertainty and is defined as W (t) = ΔAX(t) + ΔBU (t) + (CmN + ΔCTL )
Thus, an equivalent control law is designed as
t −β 0 e(τ )dτ + (AR + β)e(0)+ −1 (15) UE (t) = BmE (AR − AmN )X(t) + BR UR (t) Substituting (15) into (14), then the derivative of the sliding mode surface is ˙ S(t) = (AR + β)S(t)
In conventional sliding mode control[11] , if (AR + β) < 0 is held, then lim S → 0 is achieved. In the presented scheme, t→∞
β is chosen such that (AR + β) = 0, namely the selection ˙ of β = −AR will result in S(t) = 0. According to (13), ˙ S(0) = 0 is guaranteed. S(t) = 0 and S(0) = 0 assure that the system is in the sliding mode surface at the beginning, and make the whole system be free of reaching mode that exists in conventional sliding mode control. The presented control scheme is called total sliding mode control because of its free sliding mode surface[12−13] . The equivalent controller under ideal condition has been designed completely. However, if the system uncer˙ tainties occur, then controller (15) can not ensure S(t) =0 be maintained. Considering the extra disturbances and inner perturbations, we should design a robust controller to stabilize the system. Considering the speed error (11), the tracking error dynamic can be rewritten as the speed error between the reference model (10) and the model (8): e(t) ˙
=
X˙ R (t) = AR XR (t) + BR UR (t)
(10)
where UR is the speed command; XR is the output the reference speed; AR and BR are given coefficients according to the desired system specifications. The speed error between the reference model (10) and the nominal model (7) is e(t) = XR (t) − X(t)
(11)
The differential of the speed error is represented as =
˙ X˙ R (t) − X(t)
=
AR (t)e(t) + (AR − AmN )X(t) + BR UR (t) − BmN U (t)
−1 (ka S(t) + kb sign(S(t))) UF (t) = BmE
(12)
t 0
UT (t) = UE (t) + UF (t)
Theorem 1 For the mechanical subsystem (8), which is controlled by controller (19) with ka > Am + β and kb > |W (t)|, the speed-tracking error trajectory can always be kept on the tracking error-sliding mode surface and asymptotical convergence of the controlled system can be guaranteed. Proof. Differentiating (13) with respect to time and using (17), we can obtain t ˙ S(t) = (AR + β)S(t) − β(AR + β) e(τ )dτ + 0
BR UR (t) − BmN UT (t) − W (t) (13)
where β is is a positive constant. Differentiating (13) with respect to time, we obtain: ˙ S(t)
=
(AR + β)S(t) − β(AR + β)
(19)
(AR + β)e(0) + (AR − AmN )X(t)) +
e(τ )dτ − e(0)
(18)
where ka and kb are positive constants. The complete speed tracking error controller is shown as
Define a tracking error-sliding mode surface as S(t) = e(t) + β
(17)
The robust controller is designed as
Define the reference model of the controlled system as
AR e(t) + (AR − AmN )X(t) + BR UR (t) − BmN U (t) − W (t)
(9)
4.1 Tracking error controller
e(t) ˙
(16)
t
e(τ )dτ
(20)
Substituting controller (22) into (23), we obtain ˙ S(t)
=
(AR + β)S(t) − ka S(t) − kb sgn(S(t)) − W (t)
(21)
Define a positive definite function as VT = 12 S 2 (t), with the derivative respect to time: V˙ T
+(AR + β)e(0) + (AR − AmN )X(t) +
˙ = S(t)S(t) = (AR + β − ka )S 2 (t) − kb |S(t)| − W (t)S(t)
BR UR (t) − BmN U (t)
≤
0
(14)
(AR + β − ka )S 2 (t) + (|W (t)| − kb )|S(t)|
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
If ka and kb are chosen to satisfy ka > Am + β and kb > |W (t)|, respectively, then V˙ T ≤ 0 is achieved. From Lyapunov theorem, the controlled system is asymptotically stable. 4.2 Synchronization error controller Define the synchronization error-sliding mode surface of motor i as t Si∗ (t) = e∗i (t) + β e(τ )dτ − e∗i (0) (22) 0
namely Si∗ (t) = 2Si (t) − Si+1 (t) − Si−1 (t)
(23)
where Si (t) denotes the tracking error-sliding mode surface of motor i. The speed synchronization error controller of motor i is designed as Ui∗ (t)
=
2UT i (t) −
BmN (i+1) UT (i+1) (t) − BmN
BmN (i−1) UT (i−1) (t) BmN
(24)
where BmN i is the coefficient BmN of motor i, and UT i is the tracking error control law of motor i. Theorem 2 For the mechanical subsystem (8), which is controlled by controller (24) with ka > Am + β and kb > |W (t)|, then the speed synchronization error trajectory can always be kept on the synchronization errorsliding mode surface and asymptotical convergence of the controlled system can be guaranteed. Proof. Differentiating (22) with respect to time, we get S˙ i∗ (t)
=
AR e∗i (t) + 2(AR − AmN i )Xi (t)) − (AR − AmN (i+1) )Xi+1 (t)) − (AR − AmN (i−1) Xi−1 (t)) +
βe∗i (t) − Wi∗ (t) − BmN i Ui∗ (t)
(25)
where AmN i is the coefficient AmN of motor i; Wi∗ (t) is the lumped uncertainties in synchronization of motor i and has the expression of Wi∗ (t) = 2Wi (t) − Wi+1 (t) − Wi−1 (t)
=
Lemma 1 Suppose f (x) is defined on interval I. If f (x) is differentiable and f˙(x) is bounded, then f (x) is uniformly continuous. Lemma 2 (Barbalat Lemma) If lim f (t) is bounded, x→∞ and f˙(t) is uniformly continuous, then lim f˙(t) = 0. t→∞
Lemma 3 (LaSalle’s invariance principle) Suppose there is a neighborhood D of 0 and a continuously differentiable positive definite function V : D → R whose orbital derivative V˙ is negative semidefinite. Let H be the union of all complete orbits contained in {x ∈ D|V˙ (x) = 0}. Then, there is a neighborhood U of 0 such that for any x0 ∈ U , ω(x0 ) ⊆ H . Thoerem 3 The proposed sliding mode cross-coupling controller (29) guarantees asymptotic convergence to 0 of both tracking errors and synchronization errors, i.e., lim ei (t) = 0 and lim ei (t) = 0 for i = 1, · · · , n. t→∞ n
t→∞
Proof. A positive definite function is defined as V (t) =
( 12 Si2 (t)), with its derivative =
n i=1 n
(Si (t)S˙ i (t))
(AR + β − ka )Si2 (t) + (|Wi (t)| − kb )|Si (t)|
i=1
If ka = Am +β and kb > |W (t)| are satisfied, then V˙ (t) ≤ 0 is guaranteed. Considering (21), the derivatives of the sliding mode surfaces are shown as: (27)
Considering (20) and (21), Eq. (27) can be represented as (28)
If the tracking error controllers (19) of motor i , i + 1 , and i − 1 are properly designed, then lim Si∗ (t) = lim Si (t) = t→∞
5 Stability analysis
=
βe∗i (t) − Wi∗ (t) − BmN i UT i (t) − BmN (i+1) UT (i+1) (t) −
t→∞
When we design the speed controller using slidingmode control theory, the chattering phenomenon is introduced around the sliding-mode surface. In order to reduce the chattering, the sign function sign(·) is replaced by the saturation function sat(·). The saturation function is designed as: ⎧ ⎨ 1 Si > 1 −1 Si < −1 sat(Si ) = i = 1, · · · , n (30) ⎩ Si |Si | ≤ 1
V˙ (t)
(AR − AmN (i−1) Xi−1 (t)) +
S˙ i∗ (t) = 2S˙ i (t) − S˙ i+1 (t) − S˙ i−1 (t)
(29)
4.3 Chattering reduction technique
(26)
AR e∗i (t) + 2(AR − AmN i )Xi (t)) − (AR − AmN (i+1) )Xi+1 (t)) −
BmN (i−1) UT (i−1) (t)
Ui (t) = UT i (t) + Ui∗ (t)
i=1
Substituting (24) into (25), we have S˙ i∗ (t)
lim Si+1 (t) = lim Si−1 (t) = 0 is held. It can conclude t→∞ that the asymptotic stability of the synchronization errorsliding mode surface can be guaranteed whether the system uncertainties exist or not, and the sliding mode is assured throughout the whole control period. The tracking error controller and the synchronization error controller construct the complete controller of motor i: t→∞
S˙ i (t) = kb sign(Si (t) − Wi (t))
i = 1, · · · , n.
(31)
Because Wi (t) is bounded in terms of L2 norm, S˙ i (t) is also bounded. Using Lemma 1, Si (t) is uniformly continuous. Using lemma 2, lim Si (t) = 0 is also held. t→∞ The invariant set of the controlled system in the set Ω = (xi , x˙ i ) : V˙ (t) = 0 contains the equilibrium points of the
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
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Figure 3. Structure diagram of the control subsystem of motor i
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Figure 4. System performance with high reference speed
Figure 5. System performance with load torques change
sliding mode surfaces. Using lemma 3, Si (t) = 0 is globally asymptotically stable. Considering (13) and (22), the analytical expressions of ei (t) and e∗i (t) are as follows: ei (t) = ei (0)e−βt e∗i (t) = e∗i (0)e−βt
(32)
From (32), we can derive that lim ei (t) = 0 are held. When
t→∞
e∗i (t)
lim e∗ (t) t→∞ i
=
= 0, from (5), ε1 (t) =
· · · = εi (t) = · · · = εn (t) is achieved. Considering (4) and ei (t) = 0, then ε1 (t) = · · · = εi (t) = · · · = εn (t) = 0 is achieved, namely the synchronization errors of the controlled system converge to 0 simultaneously. Table 1. Parameters of the four motors
Parameters PN /Kw ωN /(rad/m) ψr∗ /Wb TLN /Nm Rr /Ω Rs /Ω Lr /H Ls /H Lm /H 2 ¯ J/(kgm ) B/(Nms/rad) np
Motor 1 1.1 1410 0.86 6.0 5.5 6.7 0.47 0.475 0.45 0.015 0.01 2
Motor 2 1.1 1410 0.86 6.0 5.5 6.7 0.47 0.475 0.45 0.015 0.01 2
Motor 3 1.1 1475 0.9 7.0 4.45 5.46 0.492 0.492 0.475 0.008 0.005 2
Motor 4 1.1 1475 0.9 7.0 4.45 5.46 0.492 0.492 0.475 0.008 0.005 2
Motor Motor Motor Motor
6 Simulation In this section, computer simulations are carried out on a multiple-motor system including 4 induction motors to verify the effectiveness of the proposed control approach. The schematic structure diagram of the control subsystem of motor i within the multiple-motor system is shown as Figure 3. The parameters of the 4 motors are listed in Table 1. Figure 4 shows the controlled system performance under high reference speed. The parameters of the four motors are given as nominal value. The speed command is ω ∗ = 1000r/m. The load torques of the four motors are TL1 = 2Nm, TL2 = 3Nm, TL3 = 1Nm, and TL4 = 4Nm. We can see that the tracking errors and the synchronization errors of the four motors are all converge to zero in 1s. The response of the controlled multiple-motor system is fast with high precision. In order to testify the robustness of the controlled system, the load torques changed at 3s simultaneously, while the command speed is set as ω ∗ = 1000r/m. The values of the load torques are listed in Table 2. Figure 5 illustrates the system performance under load torques change. The response curves deteriorated slightly at 3s and Table 2. Load torques of the four motors
Load Torque TL /Nm (0s < t ≤ 3s) TL /Nm (3s < t ≤ 5s)
Motor 1 2 2.5
Motor 2 3 4
Motor 3 1 2
Motor 4 4 3
ZHAO De-zong et al./Systems Engineering — Theory & Practice, 2009, 29(10): 110–117
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Motor Motor Motor Motor
Figure 6. System performance with reference speed change
converge to the desired value fast. Figure 6 illustrates the response curves under reference speed changed. The command speed changed from ω ∗ = 1000r/m to ω ∗ = −1000r/m at 3s, whereas the load torques and the other parameters of the motors are maintained at the normal values. The tracking errors and the synchronization errors deviated from zero dramatically, but be driven back to zero fast. Figure 7 shows the response curves under system parameter perturbations. The values of the moment of inertias are changed as: J1 = 0.02kgm2 , J2 = 0.017kgm2 , J1 = 0.017kgm2 , and J1 = 0.012kgm2 . From the simulation results, we can see that the performance of the controlled system varied slightly without tuning the parameters of the controllers. The tracking errors and the synchronization errors converge to zero within 1s and maintain on it.
7 Conclusions A new synchronization control strategy of multiple induction motors is proposed in this article. Sliding-mode control is employed into adjacent cross-coupling control. The presented control scheme comprises two parts: tracking error controller and synchronization error controller. Each controller includes an equivalent controller and a robust controller. Global stability of the proposed control scheme has been proven by Lyapunov method, Barbalat’s lemma, and LaSalle’s invariance principle. The proposed sliding mode control scheme has a total sliding motion without a reaching phase as in the conventional sliding mode control, such that the response and precision is better than the conventional sliding mode control. The proposed control method is simpler than traditional relative cross-coupling control. Simulation results demonstrated the fast response and high robustness of the proposed synchronization control approach.
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Figure 7. System performance with inner parameters change
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