Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
FrA15.2
Speed Synchronization of Multiple Induction Motors with Adjacent Cross Coupling Control Dezong Zhao, Chunwen Li, and Jun Ren
Abstract—In this paper, a new control approach for real time speed synchronization of multiple induction motors during speed acceleration and load changes is developed. The control strategy is to stabilize speed tracking of each motor while synchronizing its motion with other motors’ motions so that differential speed errors amongst multiple motors converge to zero. An adjacent cross coupling control architecture incorporating sliding mode control method is proposed, and the asymptotic convergence to zero of both speed tracking errors and synchronization errors have been realized via Lyapunov stability analysis. A sliding mode flux observer is adopted to implement direct field orientation of an induction motor with high robustness. Simulations conducted on a multiple motors system demonstrate the effectiveness of the proposed control strategy.
I
I. INTRODUCTION
NDUCTION motor is the most widely used motor in motion control field for its merits of simple structure, low price, and high reliability [1]. With the rapid development of modern manufacturing, a challenging problem in modern induction motor applications is that the motion of multiple motors must be controlled in a synchronous manner [2], for instance of distributed paper-making machines, continuous rolling mills, computer number control and manufacturing assembly. The reliability and precision of the products are affected by the performance of synchronization seriously. It is recognized that poor synchronization of relevant motors results in depraved accuracy of the workpiece or even in unusable products [3,4]. The synchronization performance of the multiple motors system may be degraded by many factors such as mismatched dynamics, disturbances, and parameters variation [5]. In the past, controllers for the multiple motors were designed individually. Decoupling design may be preferable if the disturbance in one motor does not affect the other motors. But for coordinated motion, decoupling design may cause damage to the overall performance. Parallel cross coupling control is used to remedy this defect by sharing the feedback information in bi-axis motion system [6]. Some control strategies have been applied on bi-axis motion
synchronization based on parallel cross coupling control [7]-[9]. Assuming there are n motors in the multiple motors system, on condition that n 2 , relative cross coupling control is applicable. Several control schemes have been applied based on relative cross coupling control [10]-[13]. Unfortunately, online computation of n order square matrixes is needed in implementing relative cross coupling control, which increases the computation work heavily. If n is large, the computation pressure is hard. In robots control field, an adaptive cross coupling control strategy for position synchronization of multi-axis is attractive. It is based on the assumption that motions of all the axes are synchronized if every pair of the axes is synchronized [14]-[16]. Considering the defects of relative cross coupling control and the advantages of adaptive cross coupling control, we proposed the adjacent cross coupling control structure of multi induction motors. Synchronization errors are defined to be differential position errors amongst multiple motors. Note that the consideration of all synchronization errors in each motor may result in intensive online computational work, especially when the number of motors is large. The synchronization strategy proposed in this literature is to stabilize synchronization errors between each motor and its two adjacent motors to zero. In this paper, sliding mode control is applied with adjacent cross coupling control to implement speed tracking and speed synchronization. PI controllers are adopted in the inner loop to regulate the stator currents. II. THEORETICAL ASPECTS In this section, we review the induction motor model and the sliding mode theory which we are concerned. A. Induction Motor Model The dynamic model of a three-phase squirrel induction motor under stator fixed reference frame can be described as the following differential equations [17]: i s i s r n p r u s Ls , (1)
This work was supported by National Natural Science Foundation of China (69774011, 60433050), and Excellent Scientist Foundation of He’nan Scientific Committee (084200510009). Dezong Zhao, and Jun Ren are with Department of Automation & Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing, 100084, P. R. China. (e-mail:
[email protected],
[email protected]). Chunwen Li is with Department of Automation, Tsinghua University & Key Laboratory of Information Electric Apparatus in Henan Province of China. (e-mail:
[email protected]).
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
6805
i s i s r n p r u s Ls ,
(2)
r Mi s r n p r ,
(3)
r Mi s r n p r ,
(4)
r i s r i s Tl B J .
(5)
FrA15.2
where the two-dimensional vectors is i s
r r r
T
,
and
u u s
u s
T
T
T
T Sob i s i s iˆ s i s iˆ s i s . (10) The definitions of the other parameters are n p n p A , B , E0 0 , n p n p K 0. From (1), (2), and (7), the error dynamics of the stator currents is obtained as is is A r I s , (11),
i s ,
are stator
currents, rotor fluxes, and stator voltages, respectively. is the rotor mechanical speed. Rs , Rr are the stator and rotor resistances, respectively; Ls , Lr are the stator and rotor self-inductances, respectively; M is the stator-rotor mutual inductance. Tl is the load torque, B is the friction coefficient, J is the motor-load moment of inertia, and n p
where
is the number of the pole pairs. The leakage factor is 1 M 2 Ls Lr , and the rotor time constant is
T
T Sob 2 , where We select Lyapunov candidate function V1 Sob its time derivative on the state trajectories of current error system is T V1 Sob Sob
Tr Lr Rr . The other parameters are 1 Tr ,
M Ls Lr , M 2 Rr Ls L2r Rs Ls , 3n p M
2 JLr
and
. Equation (5) is the mechanical
i s r n p r i s E0 sign i s
equation and can be expressed in terms of the electromagnetic torque Te : Te J B Tl . (6)
B. Brief Review of Sliding Mode We consider a nonlinear system described by the following equations [18] X f X , U and Y H X , Where X t R n is the state vector, U t R m is the control signal, and Y t R p is the output. Assuming that the system is controllable and observable, the sliding mode control objective consists of designing an equilibrium surface S X , t R m such that the state trajectories of the plant restricted to the equilibrium surface have a desired behavior, such as tracking, regulation, and stability. Secondly, determine a switching control law U X , t that is able to drive the state trajectory to the equilibrium and maintain it on the surface. Similarly, the basic sliding mode observer design procedure consists of performing the two steps. First, design the manifold S Y , t R p such that the estimation error trajectories restricted to S Y , t have desired stable dynamics. Second, determine the observer gain to derive the estimation error trajectories to S Y , t and maintain it on the set, once intercepted, for all subsequent time. III. FLUX OBSERVER DESIGN The sliding mode flux observer is designed as [19]: iˆs iˆs Aˆ r u Ls I s ,
ˆ r Mis Bˆ r KI s ,
(7)
E i
i s r n p r i s E0 sign i s 0
s
,
i s i 2s i 2s i s f s i s f s
where f s r n p r , f s r n p r . If
E0 is sufficiently large to satisfy E0 max f s , f s , then S Sob until Sob 0 , which means the sliding surface T ob
Sob 0 is attractive in finite time. When the sliding mode happens, S S 0 is achieved. ob
ob
After the sliding mode occurs, substituting is is 0 into (11) we can obtain the equivalent control component I seq : I seq A r .
(12)
Note that the equivalent control component I seq in (12) is only for analysis of the proposed observer convergence. It is not necessary to obtain it in the observer implementation. We can obtain the dynamics of the observed fluxes in stator currents sliding mode by substituting the equivalent control component I seq into (8): ˆ Mi Bˆ KI . (13) r
s
r
seq
The error dynamics equation for flux estimation is obtained by (3), (4), and (13): r 1 K B r . (14) We select a Lyapunov candidate function V2 rT r 2 , its derivative is V2 rT r
(8)
where iˆs and ˆ r are the estimates of is and r , respectively; I s are discontinuous functions of the current errors: I s E0 sign Sob ,
T
r r r ˆ r r ˆ r r .
(9) 6806
r 1 K r n p 1 K r
r 1 K r n p 1 K r 1 K 2r 2 r
.
FrA15.2
By choosing appropriate parameter K , then rT r is
e1 t e2 t , , ei t ei 1 t , , en t e1 t .
guaranteed until r 0 , which means the estimated rotor fluxes converge to the real ones in finite time. Because of lim rT r 0 , the sliding surface r 0 is invariant. So the
Define speed synchronization errors of a subset of all possible pairs of two motors from the total of n motors in the following way: 1 t e1 t e2 t (20) i t ei t ei 1 t . n t en t e1 t
r 0
robustness of the flux observer is guaranteed. The angle of the rotor flux can be calculated employing the estimated flux: ˆe arctan ˆ r ˆ r . (15) The direct rotor field orientation can be established according to Park’s transformation. The stator currents and the rotor fluxes under synchronous rotating d q reference frame are given by ˆ sin ˆe i ˆ ids ˆ dr cos e s r , (16) i ˆ qr sin ˆe cos ˆe i s ˆ r qs
where ˆ qr 0 and ˆ dr ˆ2r ˆ 2 r are satisfied after the
If i t 0 for i 1 n , the goal of multi-motor synchronization of (19) is achieved. The relevant synchronization errors of motor i are i 1 t and i t , namely we only need to synchronize motions of motor i with motor (i-1) and motor (i+1). A concept named speed coupled error, denoted by ei* t , is introduced here. ei* t contains the speed synchronization errors i 1 t and i t , i.e., e1* t 1 t n t
direct rotor field orientation is implemented. The mechanical equation (6) can be represented as following under synchronous rotating d q reference frame: a f biqs , (17)
ei* t i t i 1 t .
where a B J , f Tl J , and b dr* . dr* is the reference rotor flux on d axis.
(21)
e t n t n 1 t * n
Note that i t appears in ei* t and ei*1 t with opposite sign. When
IV. SPEED CONTROLLER DESIGN In the following sections, the symbols with subscript i denote corresponding variables and parameters of motor i .
A. Adjacent Cross Coupling Control Considering a multiple motors system including n induction motors, the dynamic model of every motor is given by (1)-(5), with different parameters. Define the speed tracking error of motor i as (18) ei t i t * t ,
e1* t ei* t en* t 0
is achieved,
considering (20) and (21), e1 (t ) ei (t ) en (t ) is held. The aim of designing the speed synchronization controller is driving ei* t converges to zero.
where * t and i t denote the desired speed and the
In our scheme, the speed controller is composed by two sub controllers: one speed tracking controller which is used to track the reference speed value and one speed synchronization controller which is used to synchronize the speed among the controlled motor with its two adjacent motors. The structure of the speed controller of motor i is * * shown as Fig. 1, where iqsi 1 t and iqsi 2 t are the torque
output speed of motor i , respectively. In the synchronization motion, besides ei t 0 , it is also aimed to regulate the
command for speed tracking and speed synchronization, * respectively; iqsi t is the complete torque current command
output speed of the motors to satisfy e1 t ei t en t .
of motor i . (19)
Equation (19) can be divided into n sub-goals as
ei
Speed Tracking Controller i
ei 1 ei 1
* iqsi 1 * iqsi
Speed Synchronization Controller i
* iqsi 2
B. Speed Tracking Controller Using field-orientation principle and considering the uncertainties, the mechanical equation (17) can be represented as t a a t f f b b iqs t , (22) where the terms a , b and f represent the uncertainties of the terms a , b , and f , respectively. The derivative of the speed tracking error of motor i is presented as ei t ai ei t ui t di t , (23)
Fig. 1. Structure diagram of the speed controller
6807
FrA15.2
where the following terms have been collected in ui t : ui t bi iqsi t ai * t fi * t ,
(24)
and the uncertainty terms have been collected in d i t : di t aii t fi bi iqsi t .
(25)
The sliding mode function of ei t is designed as t
Si ei t k ai ei d ,
(26)
0
where k is a constant gain. The switching control law ui t is designed as:
ui t kei t i sign Si ,
(27)
where i denotes the switch gain. Substituting (27) into (24), we obtain: bi iqsi t kei t i sign Si ai * t * t f i . (28) The torque current command for speed tracking of motor i is directly obtained: * * * iqsi 1 t kei t i sign Si ai t t f i bi . (29) C. Speed Synchronization Controller The derivative of the speed coupled error of motor i is: ei* t ai ei* t ui* t d i* t , (30) where ui* t ai 1 ai ei 1 t ai 1 ai ei 1 t
2bi iqsi t bi 1iqs i 1 t bi 1iqs i 1 t
kei* t i* sign Si* 1 * ai 1 ai ei 1 t ai 1 ai ei 1 t . iqsi 2 t bi * t 2ai ai 1 ai 1 2 fi fi 1 fi 1 (36) The complete torque current command of motor i is: * iqsi (37) t iqsi* 1 t iqsi* 2 t .
Remark 1: When sliding mode techniques is utilized, the chattering phenomenon is introduced around the sliding mode surface. This problem can be remedied by replacing the switching function by a smooth continuous function in the sliding surface neighborhood. That is: 1 if S sign S 1 if S , S if S where is a positive constant. D. Stability Analysis The stability analysis for the dynamics behavior of the speed control system based on Lyapunov theory is shown as follows. We define a two-dimensional manifold as Si Si
, (31)
and the uncertain term: di* t 2di t di 1 t di 1 t .
and the switching control laws ui t defined in (27) and ui* t defined in (34), with i di t ,i* di* t , and
k ai 0 , then we can derive the following results: (32)
1) Si 0 is attractive and invariant.
The sliding mode function of e t is designed as the
2) ei t and ei* t converge to zero exponentially.
* i
following form: t
Si* ei* t k ai ei* d ,
(33)
0
where k is the same constant as in (26). The switching control law ui* t is designed as u t ke t sign S * i
* i
* i
,
ai 1 ai ei 1 t * t 2ai ai 1 ai 1
Proof. 1) We select a Lyapunov candidate function as 1 n V t SiT Si . (39) 2 i 1 Differentiating V t with respect to time yields
(34)
where i* is the switch gain. Substituting (34) into (31), we obtain 2bi iqsi t bi 1iqs i 1 t bi 1iqs i 1 t
kei* t i* sign Si* ai 1 ai ei 1 t
(38)
Theorem 1. Considering the sliding surface Si defined in (38)
* t 2ai ai 1 ai 1 2 fi f i 1 f i 1
* i
T
Si* 0 ,
n
V t SiT Si i 1
(40) Si ei t k ai ei t . * i* t k ai ei* t i 1 Si e Substituting the expressions of ei t in (23) and ei* t in n
.
(35)
2 fi fi 1 fi 1
The torque current command for speed synchronization of motor i is directly obtained:
(30) into (40) we obtain n S u t d t ke t i i i i . V t * * * * i 1 Si ui t d i t kei t * Substituting ui t and ui t into (41) yields
6808
(41)
FrA15.2 AC
ei 1 udsi
PI Current Controller u qsi
* idsi
u si
d q to
uai
2-phase to 3-phase
u si
ubi uci
ˆe
idsi
ˆ
e
to d q
iqsi
Flux Observer
0s t 0.4s
3-phase Inverter
0.4s t 1s
ibi
i
Induction Motor i
Fig. 2. Diagram of the controlled system of motor n Si d i t i sign Si V t * * * * i 1 Si t d i t i sign Si
n
i
.
i di t Si i* di* t Si* i 1
The manifold Si 0 is attractive if SiT Si 0 , i.e.,
i di t Si i* di* t Si* 0 ,
that is i di t ,i* di* t . Because of lim SiT Si 0 , the Si 0
sliding surface Si 0 is invariant. 2) When the sliding mode occurs on the sliding surface Si 0 , then S t S t 0 . i
i
Therefore, the dynamic behavior of the speed control system is governed by the reduced-order model: e t k ai ei t Si t 0 *i . (42) * ei t k ai ei t For k ai 0 is satisfied, the convergence to zero of
Parameters PN Kw U SN V I SN A f N Hz
N r/m
TlN Rr Rs Lr Ls M
Motor 4 1.1 380 2.77 50 1475
Wb
0.86
0.86
0.9
0.9
Nm
J kg m2
6.0 5.5 6.7 0.47 0.475 0.45 0.015
6.0 5.5 6.7 0.47 0.475 0.45 0.015
7.0 4.45 5.46 0.492 0.492 0.475 0.008
7.0 4.45 5.46 0.492 0.492 0.475 0.008
B Nms rad
0.01
0.01
0.005
0.005
np
2
2
2
2
* r
Ω Ω H
H H
(a)
speed (rad/s)
* i
TABLE I NOMINAL PARAMETERS OF THE MOTORS Motor 1 Motor 2 Motor 3 1.1 1.1 1.1 380 380 380 2.8 2.8 2.77 50 50 50 1410 1410 1475
Motor 4 3 6
Computer simulations are carried out on a 4-motor system to verify the effectiveness of the proposed control approach. The simulation platform is Matlab/Simulink with ODE 45 solver and variable step size. The schematic structure diagram of the controlled system of motor i is shown as Fig. 2. The nominal parameters of the 4 motors are listed in Table I. The system performance including speed response, speed synchronization errors, sliding mode dynamics, and stator voltages with currents, are evaluated. The load torques Tl of the 4 motors changed at time t 0.4s , their values are shown in Table II. The rotor resistance Rr and the moments of inertia J of the 4 motors changed as twice of their nominal values at time t 0.7s . The other parameters maintained at their nominal values. The speed command is set as U m 1000 r m . The simulation time is set as t 1s . In the simulation results, the speed unit has transformed as rad s . Fig. 3 (a)-(d) illustrate the performance of the output speed of the four motors, respectively. It can be seen the output speed tracks its reference value fast and closely. Moreover, the speed tracking is not affected by the changes of the load torque and the rotor resistance, because when the sliding surface is reached the system becomes insensitive to the boundary external disturbances. Fig. 4 (a)-(d) illustrate the comparisons of the speed synchronization errors using coupling control and decoupling control, respectively. We can see that by using coupling control, the speed synchronization error is smaller at initial state and converges to zero faster than that using decoupling control, which shows that the proposed cross coupling synchronization control strategy is effective to synchronize the output speeds of multiple motors. Fig. 5 (a)-(d) illustrate the stator voltages and the stator (b)
120
120
100
100
80 60 40
* 1
20 0 -20 0
0.2
0.4
t (s)
0.6
0.8
80 60 40
* 2
20 0 -20 0
1
0.2
0.4
(c)
speed (rad/s)
e t , e t readily follows. i
Motor 3 4 1
V. SIMULATION
iai
3-phase to 2-phase
i si i si
Tl Nm PWM &
speed (rad/s)
ei 1
* Speed iqsi Controller
120
100
100
80 60 40
* 3
20 0 -20 0
0.2
0.4
t (s)
t (s)
0.8
1
0.6
0.8
80 60 40
* 4
20 0
1
-20 0
0.2
0.4
t (s)
Fig. 3. Reference Speed and Output speeds
6809
0.6
(d)
120
speed (rad/s)
ei
*
TABLE II LOAD TORQUES Motor 1 Motor 2 5 1 2 3
0.6
0.8
1
FrA15.2 (b)
-5
(V)
-5 -10
0.8
-15 0
1
0.4
0.8
-600 0
1
-5
s1
5
i
0 -5
0.4
t (s)
0.6
0.8
1
-15 0
0.2
0.4
t (s)
0.6
0.8
-600 0
1
0.2
0.4
t (s)
0.6
0.8
1
0.6
0.8
1
(d) 40
20
30
10
20
0 -10
10 0 -10
0.2
0.4
t (s)
0.6
0.8
1
-20 0
0.2
0.4
t (s)
Fig. 5. Stator Voltages and Currents
currents under stationary reference frame of motor 1 using the proposed control method. The figures show that at initial time, the voltage signals and current signals present high values because a high torque is necessary to increment the rotor speed. In the constant speed region, the motor torque only has to compensate the friction and the load torque, so the stator voltages and stator currents remain constant. The voltages and the currents decreased because the load torque has been decreased at time t 0.4s . There is no much overpowering consumption according to the voltages and currents curves, which shows the effectiveness and practicability of the proposed control strategy. The performance of the just variables of motor 2, 3, 4 are similar to motor 1. VI. CONCLUSION A new speed synchronization control strategy of multiple induction motors is proposed in this paper. An adjacent cross coupling control is introduced to multiple motors control field. Sliding mode control is employed into adjacent cross coupling control to obtain simple structure, fast response and high robustness. The proposed control approach can guarantee asymptotic convergence of both speed tracking and speed synchronization by feeding back speed errors and differential speed errors. A sliding mode flux observer is adopted to implement direct field orientation. Global stability of the proposed control scheme has been proven using Lyapunov method. Simulation results have been given to demonstrate the effectiveness of the proposed synchronization control approach. The future work includes the implementation of the control strategy on an experiment platform. REFERENCES
[2]
0.8
30
-30 0
1
Fig. 4. Speed Synchronization Errors
[1]
t (s)
0.6
-20
-10 0.2
0.4
(c)
(A)
10
0 -200 -400
0.2
(d) coupling control decoupling control
15
4 (rad/s)
3 (rad/s)
t (s)
0.6
20
coupling control decoupling control
0
-10 0
-200 -400
0.2
(c) 10
5
200
0
(A)
t (s)
0.6
400
200
s1
0.4
400
i
0.2
600
s1
0
s1
0
5
u
5
(b)
600
(V)
coupling control decoupling control
10
2 (rad/s)
1 (rad/s)
10
-10 0
(a)
15
coupling control decoupling control
u
(a) 15
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