Hamiltonian Modeling and Passivity-based Control ... - Semantic Scholar
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Hamiltonian Modeling and Passivity-based Control of Permanent Magnet Linear Synchronous Motor Zhiping Cheng College of Electric Engineering, Zhengzhou University, Zhengzhou 450001, China Email: [email protected]
Liucheng Jiao College of Electric Engineering, Zhengzhou University, Zhengzhou 450001, China Email: [email protected]
Abstract—In view of the control problem of Permanent magnet linear synchronous motor (PMLSM), applying port-controlled Hamiltonian (PCH) systems and passivity -based control theory, the modeling of Permanent Magnet Linear Synchronous Motor is presented. Using energy-shaping and passivity-based control (PBC) method, the control principle of PMLSM is given. The control laws are presented in load known conditions. The equilibrium stability is analyzed and the feedback controller is designed for the control system of PMLSM. A simulation model has been established in the Simulink/Matlab environment and the simulation results are given. In order to compare the control effect, traditional PID control simulation results are also provided in the same conditions. The simulation results show that the proposed scheme has a better performance than that of conventional PID controller. Keywords—permanent magnet linear synchronous motor, energy-shaping, Hamiltonian system, passivity-based control, stability, modeling
I. INTRODUCTION Permanent magnet linear synchronous motor (PMLSM) is a new technology which has new principle and new theory appeared in electrical engineering in 20th century. Because of the its superior performance and great application value, the researchers and industrial motion control manufacturers show great interested in it at home and abroad[1-2]. Because of direct access to linear motion for PMLSM, it eliminates the intermediate transmission mechanism of elastic deformation, gap, friction and other factors. But at the same time the system parameter perturbation, load disturbances, end effect, cogging, friction and other uncertain factors directly acts on the permanent magnet linear synchronous motor without attenuation, which increases the difficulty of control. For high-precision tools and other applications such as CNC machine, the conflict between high performance control requirements and control algorithm complexity is prominent. So the higher requirements of servo system control technology
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driven by the permanent magnet linear synchronous motor has put forward. At present, in order to solve the difficulties of servo system driven by permanent magnet linear synchronous motor, the research train of thought mainly includes the following: (1) The traditional control technology. Based on this train of thought, appearing the traditional control technology such as PID control [3~4], decoupling control [5], feedback control [6]. However, traditional control technology requirements the identified object mode, which is linear and not change, the operating conditions and the operating environment is uncertain. Therefore, it’s often combined with other control method in the high precision control system. (2) The modern control technology. Modern control technology has been attracted much attention in linear servo motor control research. They are more widely used in permanent magnet linear synchronous motor control research. The conventional control methods are: adaptive control [8~15], sliding mode variable structure control, robust control [16-23], quantitative feedback theory [24], auto-disturbances-rejection [25], model reference inverse methods control [26] and so on. For modern control theory, such as robust control, energy of input signal must be finite, while the actual system does not meet the hypothesis request. The contradiction between control robust stability and robust performance still need further research. Sliding mode variable structure buffeting problem has not been solved very well, which reduce the modern control effectiveness in servo system. (3) Intelligent control. In recent years, intelligent control method such as fuzzy logic control, neural network control is also introduced to control permanent magnet linear motor servo system [13] [27]. But the single fuzzy control needs more control rules and plenty of experience, which makes the precision low, there is not a uniform design method for control system to determine the input number, network layer, threshold selection in the neural network intelligent control system and complex algorithm is not easy to realize, Now, the intelligent control learn from each other with
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the existing mature control method, some new control methods such as the neural network PID [28], fuzzy logic PID [29 ] and fuzzy sliding mode variable structure [30 ~ 33] have appeared , these methods can be part of the realization of nonlinear control and Identification of PMLSM. It can be seen from above that the control strategy used in permanent magnet linear synchronous motor servo system has its specific or starting point, each one has his good points. At the same time, there are also different limitations or insufficient, and some control methods have greater compatibility and complementarity. Therefore, in order to realize the permanent magnet linear synchronous motor servo system, especially the sectional permanent magnet linear synchronous motor servo control system, new control strategy must be found according to the system characteristics. In recent years, the energy-shaping methods get attention [34-35] for nonlinear control system. It’s main characteristic is that the system has a port-controlled Hamiltonian (port-controlled Hamilton, PCH) structure. It is simpler for controller design and stability analysis according to the peculiar feedback stabilization method. Professor Yu Hai-Sheng has done a lot of work for permanent magnet motor by the method [36-37]. The PMLSM with high order, strong coupling nonlinear characteristics causes that, under reasonable hypothesis, it can be considered as a two-port energy conversion device. Therefore, the PCH system model of the PMLSM is necessary by energy-shaping methods and passive control, and the control of PMLSM is realistic significance. It will be a new way for permanent magnet linear driver in the industrial application of AC servo system. II. DRIVE SYSTEM WITH POWER CONVERTER OF PMLSM The driving system schematic diagram of PMLSM with power converter shows as Fig. 1.
voltage symmetry. The high-power resistances are often connected in series in order to reduce the drift which caused by heating and other factors. AC filter capacitor C is 0.47 μF . The NTC22 thermistor is as one of protection circuits. From the diagram can be drawn [37]: ⎡u a ⎤ ⎡2d a − d b − d c ⎤ ⎥ U dc ⎢ ⎥ ⎢ ⎢u b ⎥ = ⎢2d b − d c − d a ⎥ × 3 . ⎢⎣u c ⎥⎦ ⎢⎣2d c − d a − d b ⎥⎦
(1)
where d i ( i = a, b, c ) is the duty ratio. U dc is DC bus voltage. III. BASIC FORMS OF PORT-CONTROLLED HAMILTONIAN (PCH) SYSTEM Port-Controlled Hamiltonian (PCH) system[38] is described as below: •
x = f ( x) + g ( x)u = J ( x)
∂H ( x) + g ( x)u ∂x .
∂H ( x) y = h( x ) = g ( x ) ∂x
(2)
T
Introduce the energy dissipation concept into the PCH system [39] [41] and the PCH dissipative models: •
x = [ J ( x) − R( x)] y = g T ( x)
∂H ( x) + g ( x)u ∂x .
∂H ( x) ∂x
(3)
Where x ∈ R n , u , y ∈ R m , and R ( x) is semipositive symmetric matrix. R ( x) = R ( x) T ≥ 0 , which denotes the accessory resistance structure of port; J (x ) is anti-half symmetric matrices, and J ( x) = − J ( x) T , which denotes the internal system interconnection structure [35-36]. IV. PCH SYSTEM MODEL OF NON-SALIENT POLE OF PMLSM According to the characteristics of Non-salient pole of PMLSM, frequency of 50Hz is considered only, when i d = 0 and voltage is given in the d-q coordinates, the model of non-salient pole of PMLSM can be expressed as [26] [40] :
Fig. 1. Driving system schematic diagram of PMLSM with power converter
Power converter is composed of a three-phase bridge AC-DC rectifier and a three-phase DC-AC voltage type inverter. The input is a three-phase AC voltage. The output is three-phase PWM voltage. It connects to the three-phase stator winding of PMLSM, which is three-phase star-type connection. In practical application, using the series capacitors to reduce the weakness of large volume high price of single capacitor and using the parallel connection resistors to ensure the capacitor © 2013 ACADEMY PUBLISHER
Ld
Lq
di q dt m
di d π = u d − ri d + vLq i q . dt τ = u q − ri q −
π π vψ r − vLd i d . τ τ
dv π = ψ r i q − FL + Fd + f . dt τ
m --rotor quality; L --Armature inductance;
(4)
(5) (6)
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τ --polar pitch; ψ r --permanent magnet working flux; v --motion velocity of rotor;
⎡1 0 0⎤ g (x) = ⎢⎢0 1 0⎥⎥ . ⎢⎣0 0 1⎥⎦
Fx --Electromagnetic thrust; FL --load force; f --friction force; Fd --External disturbance force. Definition is given as follows:
⎡i d ⎤ ⎡ x1 ⎤ ⎡ Ld i d ⎤ ⎡ Ld 0 0 ⎤ ⎡i d ⎤ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x = ⎢ x 2 ⎥ = ⎢ Lq i q ⎥ = ⎢0 L q 0⎥ ⎢i q ⎥ = D ⎢i q ⎥ . ⎢v ⎥ ⎢⎣ x 3 ⎥⎦ ⎢mv ⎥ ⎢0 0 m ⎥ ⎢v ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ (7) ⎡ ud ⎢ u=⎢ uq ⎢F − F + l ⎣ d
⎤ ⎥ ⎥. f ⎥⎦
⎡i d ⎤ ∂H ( x) ⎢ ⎥ T y = g ( x) = ⎢i q ⎥ . ∂ ( x) ⎢v ⎥ ⎣ ⎦
(8)
In order to make system asymptotically stable equilibrium near expectations x 0 , A closed-loop expectations energy function H d (x) will be constructed. The energy function will be take minimum values near x 0 . That is, the feedback control u = α (x) will be found in order to make the closed-loop system as fellow: •
x = [J d ( x) − R d ( x)]
Take the sum of electrical energy and mechanical energy as Hamiltonian function of PMLSM.
R d ( x) --damping matrix. In addition:
•
x = [J d ( x) − R d ( x)]
π ⎤ ⎡ ⎥ ⎢0 0 τ L q i q ⎥ ⎢ π J ( x ) = ⎢0 0 − ( L d i d + ψ r ) ⎥ . ⎥ ⎢ τ ⎥ ⎢ ⎥ ⎢0 π ψ 0 r ⎥⎦ ⎢⎣ τ
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∂H d ( x) . ∂x
(17)
[J d ( x) − R d ( x)]K ( x) = − [J a ( x) − R a ( x)]
. ∂H ( x ) + gα ( x ) ∂x
(18)
and T
∂H ⎡ ∂H ⎤ ( x) = ⎢ ( x)⎥ . x ∂ ∂x ⎣ ⎦
(11) K ( x0 ) = −
⎤ ⎥ ⎥ f ⎥⎦
⎡− r 0 0⎤ R ( x) = ⎢⎢0 − r 0⎥⎥ . ⎢⎣0 0 0 ⎥⎦
(16)
The expected stabilization balance point x 0 of J d (x) , R , H (x) , g is given. If the expression u = α (x) of Feedback control can be found. R a (x) , J a (x) , K (x) satisfy partial differential equations.
Then the system PCH form of PMLSM can be written as follow:
ud ⎡1 0 0⎤ ⎡ ⎢0 1 0 ⎥ ⎢ uq ⎢ ⎥⎢ ⎢⎣0 0 1⎥⎦ ⎢ F − F + L ⎣ d
(15)
J d ( x) --interconnected and damping matrix.
1 1 ⎡ 1 2 1 2 1 2⎤ H ( x) = x T D −1 x = ⎢ x1 + x2 + x 3 ⎥ . (10) Lq Lq 2 2 ⎣⎢ Ld ⎥⎦
⎤ ⎡ • ⎤ ⎡− r 0 π Lq i q ⎢ ⎥ x ⎢ 1⎥ τ ⎥ ⎡i d ⎤ ⎢• ⎥ ⎢ π ⎢ ⎥ ⎢ ⎢ x 2 ⎥ = 0 − r − ( L d i d + ψ r ⎥ ⎢i q ⎥ + ⎢ ⎥ τ ⎢• ⎥ ⎥ ⎢⎣v ⎥⎦ ⎢x ⎥ ⎢ π ⎢ ⎥ 3 . ψ 0 0 ⎢⎣ ⎥⎦ r ⎢⎣ ⎥⎦ τ
∂H d ( x) . ∂x
⎧⎪ J d ( x) = J ( x) + J a ( x) = − J dT ( x) . ⎨ ⎪⎩ R d ( x) = R ( x) + R a ( x) = R dT ( x )
(9)
(14)
∂H ( x0 ) . ∂x
(19)
(20-1)
That is: ∂H d ( x) = 0 . ∂x
(12)
K ( x0 ) > − ∂2Hd ∂x 2
(13)
∂2H ∂x 2
( x0 ) .
(x0 ) = 0 .
(20-2)
(21-1)
(21-2)
Then the closed-loop system (15) is a dissipation PCH systems and the point x 0 is a stable balance for the closed-loop system. H d ( x) − H ( x) = H a ( x) .
(22)
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∂H a ( x) = K ( x) . ∂x
⎡ 0 − J12 ⎢ J a ( x ) = ⎢ J12 0 ⎢⎣ J13 J 23
(23)
H a (x) is undetermined function which means the system energy injected by means of control. If maximum invariant set equals { x 0 }, which is included in the closed-loop system T ⎫⎪ ⎧⎪ ∂H d ⎤ n ⎡ ∂H d ( x)⎥ R d ( x ) = 0⎬ . ⎨x ∈ R | ⎢ ∂x ⎣ ∂x ⎦ ⎪⎭ ⎪⎩
(24)
The system is asymptotically stable. Attract domain estimation is given by maximum limited level set. { x ∈ R n | H ( x) ≤ c }. V. DESIGN AND STABILITY ANALYSIS OF SYSTEM
expectation point of balance can be get in (11) for a given v 0 . There is: ⎡ ⎤ ⎢0 ⎥ ⎡ x10 ⎤ ⎢ ⎥ Lq ( FL − Fd − f ) ⎥ . ⎢ ⎥ x0 = ⎢ x 20 ⎥ = ⎢ ⎢ ⎥ π ψr ⎢⎣ x30 ⎥⎦ ⎢ ⎥ τ ⎢ ⎥ ⎢⎣ mv 0 ⎥⎦
1 ( x − x0 )T D −1 ( x − x0 ) . 2
(25)
(26)
∂H ( x ) = D −1 ( x ) ∂x ∂H a ( x ) ∂H d ( x ) ∂H ( x ) K ( x) = = − = D −1 ( x0 ) ∂x ∂x ∂x
(27)
. (28)
⎧ ⎪ x = x0 ⎪ ⎪ ∂H d ( x ) . =0 ⎨ ∂ x ⎪ ⎪ ∂ 2 H d ( x) >0 ⎪ ⎩ ∂x 2
(29)
(27) and (28) are met. So the closed-loop system designed is stable at the balance point. Take the following expression: © 2013 ACADEMY PUBLISHER
(31)
J12 , J 13 , J 23 , R1 , R2 -undetermined interconnections and damping parameters respectively. Substitute (28) into (18), there is: − [J d ( x) − Rd ( x )]D −1 x0 =
− [J a ( x ) − Ra ( x )]D −1 x + g ( x )α ( x )
(32)
.
The expression of u can be obtained from (13): ⎡u d ⎤ ⎢ ⎥ u = α ( x) = ⎢u q ⎥ . ⎢− τ ⎥ ⎣ L⎦
(33)
Substitute (16) and (30) into (32), (34) can be deduced with (25). − [J d ( x ) − Rd ( x )]D −1 x0 =
.
(34)
Because of:
⎧⎡ ⎪ ⎢0 ⎪⎢ ⎪⎢ ⎪ ⎢0 ⎪⎢ ⎪ ⎢0 ⎪ ⎢⎣ ⎪ ⎨ ⎪ ⎪ ⎪ ⎡− ⎪− ⎢0 ⎪ ⎢ ⎪ ⎣⎢ 0 ⎪ ⎪⎩ ⎡ x 10 ⎢ ⎢ x 20 ⎣⎢ x 30
When (29) is true.
(30)
⎤ ⎥ ⎥. ⎥⎦
⎡u d ⎢ u = α ( x) = ⎢u q ⎢F − F + L ⎣ d
For ∂H d ( x ) ∂ 2 H d ( x) = D −1 ( x − x0 ) = D −1 . 2 ∂x ∂x
0
⎤ ⎥. ⎥ ⎥⎦
− [J a ( x ) − Ra ( x ) ]D −1 x + g ( x )α ( x )
Take the expected Hamiltonian function: H d ( x) =
− J 23
⎡ r1 0 0 ⎢ Ra ( x ) = ⎢0 r2 0 ⎢⎣0 0 0
CONTROLLER
The control essence of PMLSM is to make the error in the permitted range between its actual speed and hope speed. This paper takes a PMLSM as example. Ld and Lq equal in the non-salient PMLSM, make i d =0. The
− J13
0 0
π ψ τ
π ⎤ L q iq ⎥ τ ⎥ π − ( L a i d +ψ r ) ⎥ ⎥ τ r
0
⎥ ⎥ ⎥⎦
⎤ ⎥ ⎥. f ⎥⎦
− J 13 ⎡ 0 − J 12 ⎢ 0 − J 23 + ⎢ J 12 ⎣⎢ J 13 − J 23 0
⎤ ⎡ 1 0 0⎥ ⎢L d ⎥ ⎢ r 0 0 ⎤ ⎡ − r1 0 0 ⎤ ⎥ 1 ⎢ ⎥⎢ 0⎥ − r 0 ⎥⎥ − ⎢ 0 − r2 0 ⎥ ⎢ 0 Lq ⎢ ⎥ 0 0 ⎦⎥ ⎣⎢ 0 0 0 ⎦⎥ ⎢ 1 ⎥ ⎢0 0 ⎥ m ⎦ ⎣
⎤ ⎧ ⎥ ⎪ ⎥ = ⎨ ⎪ ⎦⎥ ⎩
(35)
− J 13 ⎡ 0 − J 12 ⎢ 0 − J 23 ⎢ J 12 J 23 0 ⎣⎢ J 13
⎫ ⎪ ⎤⎪ ⎥⎪ ⎥⎪ ⎦⎥ ⎪⎪ ⎪ ⎪ ⎬× ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭
⎤ ⎡− r 0 0⎤ ⎫ ⎥ ⎢ ⎥⎪ ⎥ − ⎢0 − r 0 ⎥ ⎬ × ⎪ ⎦⎥ ⎢⎣ 0 0 0 ⎦⎥ ⎭
⎡ 1 ⎤ 0 0⎥ ⎢L ⎢ d ⎥ ⎡ x1 ⎤ ⎡1 0 0⎤ ⎡u d ⎤ . (36) ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 1 ⎥ 0 ⎥ × ⎢ x 2 ⎥ − ⎢0 1 0⎥ × ⎢u q ⎢0 ⎥ L q ⎢ ⎥ ⎢ x ⎥ ⎢0 0 1⎥ ⎢ F − F + f ⎥ 3⎦ ⎣ ⎦ ⎣ L ⎣ d ⎦ ⎢ 1 ⎥ ⎢0 0 ⎥ m ⎦ ⎣
There is (37) form (36):
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J13 x10 + Ld
J 23 +
π ψ τ r x = 20
Lq
. (37)
J 13 J x1 + 23 x 2 − ( Fd − Fl + f ) Ld Lq x 0 is substituted into (37): Then:
J 23 = −
Lq x1 Ld ( x2 − x20 )
J 13 = .
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resistance is 3.27 Ω . The d-q axis inductance is 0.0072 H respectively. Carrier frequency is 12kHz. Friction is equivalent to the load resistance. In Fig.2, the starting speed simulation curve is given with r1= r2=1.3. We can see that the maximum speed overshoot is in 3.5% and response time is in 0.1 seconds. The speed response curve of conventional PID controller is given with the same conditions in Fig.3. The maximum speed overshoot is over 30% and response time is in 0.35 seconds.
(38)
Since the equation solution is not only one, we can take the following expression. J13 = −
Ld (x2 − x20 ) Lq
J 23 = − x1
.
(39)
J12 = − Kx3 K --Arbitrary parameters. K does not affect the stability of system. So the controller will be the follow: ⎡ πLq τKLq ( Fl − Fd − f ) ⎤ + u d = −r1id + kmviq − ⎢ ⎥ iq πψ r ⎣ mτ ⎦ . πψ r v0 ⎛ πv0 Ld ⎞ − Ld v0 ⎟id + (Ld − km )vid − r2 iq + uq = ⎜ τ ⎝ τ ⎠
Fig. 2. Starting speed curve of Hamiltonian controller ( r1= r2=1.3)
(40) VI. SIMULATION RESULTS AND ANALYSIS The duty ratio can be got through the d-q coordinate transformation: π π ⎤ ⎡ u cos vt − uq sin vt ⎥ 6 ⎢ d τ τ ⎥ da = ⎢ 3U dc ⎢ ⎥ 2 2 ⎥⎦ ⎢⎣+ ud + uq ⎡ ⎛π ⎢ud cos⎜ vt − 6 ⎢ ⎝τ db = 3U dc ⎢ ⎛ π 2π ⎢sin ⎜ vt − 3 ⎢⎣ ⎝ τ
⎤ 2π ⎞ ⎟−u ⎥ 3 ⎠ ⎥ ⎥ ⎞ 2 2⎥ + + u u ⎟ d q ⎥⎦ ⎠
Fig. 3. Speed curve of conventional PID controller
.(41)
⎡ ⎤ 4π ⎞ ⎛π ⎟− ⎢ud cos⎜ vt − ⎥ 3 ⎠ 6 ⎢ ⎝τ ⎥ dc = ⎢ ⎥ 3U dc 4π ⎞ ⎛π 2 2 ⎢uq sin ⎜ vt − ⎟ + ud + u q ⎥ 3 ⎠ ⎥⎦ ⎝τ ⎣⎢ Fig. 4. Speed curve of Hamiltonian controller with load disturbance
So needed voltage can be calculated according to the formula (41), to control each IGBT turn-on or turn-off. The study of simulation is done in Matlab7.0/Simulink environment. The system includes controller model, inverter model and PMLSM simulation model which adopt package form respectively. System steady/dynamic performance simulation can be achieved by modifying the parameters. The motor rated power is set to 0.75kW. Rated voltage is 220V. Inverter frequency is 30Hz. K = 16. Stator © 2013 ACADEMY PUBLISHER
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to design the control system under load changes is a part of further research content, and in this design, it is lack of science to assume the friction and all other perturbations as a constant load. How to classify the perturbation and how to design a precision control system? There is a lot of work to do. Now variable “load condition control research” is being carried out. VII. CONCLUSION
Fig. 5. Speed curve of conventional PID controller with load disturbance
The modeling of permanent magnet linear synchronous motor is presented in this paper by using energy-shaping and passivity-based control method. And the control principle of PMLSM is given. The control laws are presented in load known conditions. The equilibrium stability is analyzed and the feedback controller is designed for the control system of PMLSM. A simulation model has been established in the Simulink/Matlab environment and the simulation results are given. In order to compare the control effect, traditional PID control simulation results are also provided in the same conditions. The simulation results show that the proposed scheme has a better performance than that of conventional PID controller.
ACKNOWLEDGEMENT
Fig. 6. Speed curve of Hamiltonian controller with speed disturbance
The load disturbance of 60N is added at 2.5 seconds in Fig. 4. The curve is the results of conventional PID controller with the same load disturbance in Fig. 5. The speed disturbance of 0.5 m/s is added at 2 seconds in Fig.6. The results of a conventional PID controller is also given as shown the under curve with the same conditions in Fig.7. We can see that the resist disturbance of PID controller is poor than that which based on passivity. Obviously, the controller of PMLSM based on passivity has good control characteristics which can significantly inhibit system parameter variations and external disturbances. The system has stronger robustness and practicability.
The work is supported by the National Natural Science Foundation of China (61075071), the Natural Science Foundation of Henan Province (2010A470006/ 2011A470010), and Zhengzhou Science and Technology Bureau, Henan Province (10PTGG380-1). REFERENCES [1]
[2]
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[8] Fig. 7. Speed curve of conventional PID controller with speed disturbance
The control system of PMLSM was studied in the assumed load constant known cases; however, the fact is that the load of PMLSM is changed in most cases. How © 2013 ACADEMY PUBLISHER
YE Yunyue, “Review of Development and application of Linear motor technology at home and abroad,” The proceedings of linear motor annual conference 2008, Hangzhou, Zhejiang, 2008, pp.1-6, October 2008. Norhisam, M., Wong, K.C, Mariun, N., Wakiwaka, H., “Double side interior permanent magnet linear synchronous motor and drive system,” Power Electronics and Drives Systems, 2005. PEDS 2005, pp 1370-1373, 2005. MU Hai-hua, ZHOU Yun-fei, YAN Si-jie, “Accurate Control of Linear Motor Based on PID Position/Force Control and Cogging Force Compensation,” Micromotors Servo Technique, No.10, pp48-51, 2007. Ye Yunyue, Lu Kaiyuan, “PID Control and Fuzzy Control in Linear Induction Motor,” Transactions of China Electrotechnical Society, No. 3, pp. 11-15, 2001. ZHI Chang-yi, ZHU Xiao-dong, “Decoupled Control of the Permanent Magnet Linear Synchronous Motor,” Micromotors Servo Technique, No.3, pp11-15, 2001. WANG Li-mei, HUANG Fei, “Research on Synchrodrive Technique of Dual Linear Motors,” Electric Drive, No.6, pp. 51-54, 2009. Men Fu, “Rejection of PMLSM Disturbances Based on ILC,” Shenyang University of Technology, Shenyang, 2010. Srinivasu, B.1, Prasad, P.V.N.2, Ramana Rao, M.V., “Adaptive controller design for permanent magnet linear synchronous motor control system,” 2006 International Conference on Power Electronics, Drives and Energy Systems, PEDES '06, December 12~15, 2006, New Delhi, India, 12-15 Dec. 2006.
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SONG Yi-xu, WANG Chun-hong, YIN Wen-sheng, JIA Pei-fa, “Adaptive-learning Control for Permanent-magnent Linear Synchronous Motors,” Proceedings of the Csee, No. 20, pp151-156, 2005. ZHANG Guo-zhu, CHEN Jie, LI Zhi-ping, “An adaptive robust control for linear motors based on composite adaptation,” Control Theory & Applications, No.5, pp833-837, 2009. GUAN Li-rong, GONG Min, SUN Sheng-bing, “Adaptive Backstepping Sliding Mode Robust Tracking Control for Permanent Magnet Linear Synchronous Motor, “Electric Drive, No. 6, pp. 57-60, 2009. ZHAO Hu, LI Rui, “Adaptive sliding control of precision permanent magnet linear motors, “ Journal of Changchun Post and Telecommunication Institute, No.4, pp318-322, 2004. CAO Rong-min, HOU Zhong-sheng, “Study of model-free adaptive control method in permanent magnet linear motor, “ Computer Engineering and Design, No.6, pp1433-1436, 2007. CAO Rong-min, HOU Zhong-sheng, “Nonparametric model direct adaptive predictive control for linear motor,” Control Theory & Applications, No.3, pp587-591, 2008. CAO Rong-min, HOU Zhong-sheng, BAI Xue-feng, HUANG Jian, “DC Motor Speed Regulating Control System Based on the Model-free Adaptive Control Method, “ Electric Drive, No.7, pp26-30, 2008. Sun yi biao, “Robust Speed Control for Permanent-magnet Linear Synchronous Motor Based on Sliding Mode, “ Shenyang University of Technology, Shenyang, P.R.China, 2007. Sun Yibiao, Guo Qingding; Sun Yanna, “Sliding Mode Variable Structure Control Based on Fuzzy Self-Learning for AC Linear Servo System, “ Transactions of China Electrotechnical Society, No.1, pp52-56, 2001. CAO Rong-min, HOU Zhong-sheng, “Simulation Study on Model-free Control Method in Linear Motor Control System”, Journal of System Simulation, No.10, pp2874-2878, 2006. Lan Yipeng, “Study on Robust Control for Permanent Magnet Linear Motor Servo System, “ Shenyang University of Technology, Shenyang, P.R.China, 2007. Zhao Ximei, “Research on Two-Degree-of-Freedom Robust Tracking Control for High Precision Permanent Magnet Linear Synchronous Motor, “Shenyang University of Technology, Shenyang, P.R.China, 2008. GUAN Li-rong, JIANG Tong, “Based on Adaptive Variable Structure Speed Control for Permanent Magnet Linear Motor, “ Electric Drive, No.9, pp72-74, 2008. GUAN Li-rong, SUN Sheng-bing, “Research on Adaptive Slide Mode Variable Structure Direct Thrust Speed Control for Permanent Magnet Linear Motor, “ Journal of Shenyang Ligong University, No.2, pp20-22, 2009. Song yang, “Variable Gain Cross-coupled Control of XY Table Linear Servo System Based on l_1 Monoaxial Speed Controllers, “ Shenyang University of Technology, Shenyang, P.R.China, 2008. YANG Jin-ming, WU Jie, ZHANG Zhou, DONG Ping, “Auto Disturbance Rejection Controller for Speed Regulation in Permanent-magnet Linear Motors, “ Small & Special Machines, No.5, pp31-34, 2004. FU Zi-yi, ZHAO Wen-tao, WANG Xin-huan, “The Control of Model Reference and Inverse Method for Permanent Magnet Linear Synchronous Motor, “ Inoustry and Mine Automation, No.6, pp17-19, 2005. Zhang Dailin, Chen Youping, Ai Wu, Zhou Zude, “A Precision Fuzzy Control Method of Permanent Megnetic
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Linear Motors, “ Transactions of China Electrotechnical Society , No.4, pp64-68, 2007.Cao wen xia, “The Research on the RBF Neural Network Setting PID Controlling PMLSM, “ Hefei University of Technology, Anhui, P.R.China, 2009. Lu Huacai, Xu Yuetong , Yang Weimin, Chen Zichen, “Fuzzy PID Controller Design for a Permanent Magnet Linear Synchronous Motor Feeding System, “ Transactions of China Electrotechnical Society, No.4, pp59-63 2007. Sun Yibiao, Guo Qingding; Sun Yanna, “Sliding Mode Variable Structure Control Based on Fuzzy Self-Learning for AC Linear Servo System, “ Transactions of China Electrotechnical Society, No.1, pp52-56, 2001. Wang, FaGuang1, Park, Seung Kyu1, Ahn, Ho Kyun1, “Linear matrix inequality-based fuzzy control for interior permanent magnet synchronous motor with integral sliding mode control,” The 12th International Conference on Electrical Machines and Systems, ICEMS 2009, Tokyo, Japan, 15-18 NoV. 2009. ZHENG Xue-mei, ZHAO Lin, FENG Yong, “Neural network control of synchronous motor's dynamic transition phase when load sudden changes,” Electric Machines and Control, No.4, pp365-369, 200. WANG Li-mei, WU Zhi-tao, ZUO Tao, “PMLSM self-constructing fuzzy neural network controller design” Electric Machines and Control, No.5, pp643-648, 2009. Ortega R., Van der Schaftb A. J., Mareels I., etal, “Putting energy backin control,” IEEE Control System Magazine, Vol, 21 No.2, pp18-33, 2001． Wang Yuzhen, Cheng Daizhan, Li Chunwen, et al．, “Dissipative Hamiltonian realization and energy-based l2-disturbance attenuation control of multimachine power systems,” IEEE Transactions on Automatic Control, Vol, 48 No.8, pp1428-1433, 2003． YU Hai-sheng, ZHAO Ke-you, GUO Lei, WANG Hai-liang, “Maximum Torque Per Ampere Control of PMSM Based on Port-controlled Hamiltonian Theory,” Proceedings of the CSEE, 2006, Vol, 26 No.8, pp82-86, 2006. Der Schaft “L2-Gain and Passiving Techniquess in Nonlinear Control, “ London Springer-Verlag , 2000. Ortega R, Vander Scjhaftb A.J, Mareels, etal. “Interconnection and damping assignment passivity-based contron of port-controlled Hamiltionian systems, “ Automatica, Vol, 38 No.4, pp. 585-596, 2002. Cheng Zhiping, “Modeling and Simulation of PLMSM in Vertical transportation system,” Jiaozuo Institute of Technology, Jiao Zuo, P. R. China, 2003. Rongbo Zhu, “Intelligent Rate Control for Supporting Real-time Traffic in WLAN Mesh Networks,” Journal of Network and Computer Applications, vol. 34, no. 5, pp. 1449-1458, 2011. Rongbo Zhu, Yingying Qin and Chin-Feng Lai, “Adaptive Packet Scheduling Scheme to Support Real-time Traffic in WLAN Mesh Networks,” KSII Transactions on Internet and Information Systems, vol. 5, no. 9, pp. 1492-1512, 2011. Ortega R, Van Der Schaft A. J, Maschkec B., et al．“Interconnection damping assignment passivity-based control of port-control Hamiltonian systems,” Automatica, Vol, 38 No.4, pp585-596, 2002.
Zhiping Cheng was born in Shangshui R.P. China in 1974. Major in Control Theory and Control Engineer from September 2000 to June 2003. He received the M.S. degrees in electrical
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engineering from Henan Polytechnic University, Jiaozuo, China, in 2003. Major in Physics from September 1994 to June 1998. Then he received the Bachelor of Science degrees in school of physics electronic engineering from Xinyang Normal University, Xinyang, China, in 1998. The major field of study is linear motor and control. He works in Zhengzhou University from 2003 until now and was promoted to a lecturer in 2004. Now he is responsible for two projects supported by the Natural Science Foundation of Henan Province (2010A470006/2011A470010). Zhengzhou Science and Technology Bureau, Henan Province (10PTGG380-1). He is responsible for technology of the
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project supported by the National Natural Science Foundation of China (61075071).
Liucheng Jiao was born in Jiao zuo (R.P. China)in 1950. Major in Control Theory and Control Engineer from September 1995 to June 1998. Received the Doctor of Science degrees in School of Mechanical Electronic & Information Engineering from China University of Mining & Technology, Beijing, China, in 1998. The major field of study is in linear motor and control. He works in Zhengzhou University as a part-time professor. Now he is responsible for project supported by the National Natural Science Foundation of China (61075071 ).
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Hamiltonian Modeling and Passivity-based Control of Permanent Magnet Linear Synchronous Motor Zhiping Cheng College of Electric Engineering, Zhengzhou University, Zhengzhou 450001, China Email: [email protected]
Liucheng Jiao College of Electric Engineering, Zhengzhou University, Zhengzhou 450001, China Email: [email protected]
Abstract—In view of the control problem of Permanent magnet linear synchronous motor (PMLSM), applying port-controlled Hamiltonian (PCH) systems and passivity -based control theory, the modeling of Permanent Magnet Linear Synchronous Motor is presented. Using energy-shaping and passivity-based control (PBC) method, the control principle of PMLSM is given. The control laws are presented in load known conditions. The equilibrium stability is analyzed and the feedback controller is designed for the control system of PMLSM. A simulation model has been established in the Simulink/Matlab environment and the simulation results are given. In order to compare the control effect, traditional PID control simulation results are also provided in the same conditions. The simulation results show that the proposed scheme has a better performance than that of conventional PID controller. Keywords—permanent magnet linear synchronous motor, energy-shaping, Hamiltonian system, passivity-based control, stability, modeling
I. INTRODUCTION Permanent magnet linear synchronous motor (PMLSM) is a new technology which has new principle and new theory appeared in electrical engineering in 20th century. Because of the its superior performance and great application value, the researchers and industrial motion control manufacturers show great interested in it at home and abroad[1-2]. Because of direct access to linear motion for PMLSM, it eliminates the intermediate transmission mechanism of elastic deformation, gap, friction and other factors. But at the same time the system parameter perturbation, load disturbances, end effect, cogging, friction and other uncertain factors directly acts on the permanent magnet linear synchronous motor without attenuation, which increases the difficulty of control. For high-precision tools and other applications such as CNC machine, the conflict between high performance control requirements and control algorithm complexity is prominent. So the higher requirements of servo system control technology
© 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.2.501-508
driven by the permanent magnet linear synchronous motor has put forward. At present, in order to solve the difficulties of servo system driven by permanent magnet linear synchronous motor, the research train of thought mainly includes the following: (1) The traditional control technology. Based on this train of thought, appearing the traditional control technology such as PID control [3~4], decoupling control [5], feedback control [6]. However, traditional control technology requirements the identified object mode, which is linear and not change, the operating conditions and the operating environment is uncertain. Therefore, it’s often combined with other control method in the high precision control system. (2) The modern control technology. Modern control technology has been attracted much attention in linear servo motor control research. They are more widely used in permanent magnet linear synchronous motor control research. The conventional control methods are: adaptive control [8~15], sliding mode variable structure control, robust control [16-23], quantitative feedback theory [24], auto-disturbances-rejection [25], model reference inverse methods control [26] and so on. For modern control theory, such as robust control, energy of input signal must be finite, while the actual system does not meet the hypothesis request. The contradiction between control robust stability and robust performance still need further research. Sliding mode variable structure buffeting problem has not been solved very well, which reduce the modern control effectiveness in servo system. (3) Intelligent control. In recent years, intelligent control method such as fuzzy logic control, neural network control is also introduced to control permanent magnet linear motor servo system [13] [27]. But the single fuzzy control needs more control rules and plenty of experience, which makes the precision low, there is not a uniform design method for control system to determine the input number, network layer, threshold selection in the neural network intelligent control system and complex algorithm is not easy to realize, Now, the intelligent control learn from each other with
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the existing mature control method, some new control methods such as the neural network PID [28], fuzzy logic PID [29 ] and fuzzy sliding mode variable structure [30 ~ 33] have appeared , these methods can be part of the realization of nonlinear control and Identification of PMLSM. It can be seen from above that the control strategy used in permanent magnet linear synchronous motor servo system has its specific or starting point, each one has his good points. At the same time, there are also different limitations or insufficient, and some control methods have greater compatibility and complementarity. Therefore, in order to realize the permanent magnet linear synchronous motor servo system, especially the sectional permanent magnet linear synchronous motor servo control system, new control strategy must be found according to the system characteristics. In recent years, the energy-shaping methods get attention [34-35] for nonlinear control system. It’s main characteristic is that the system has a port-controlled Hamiltonian (port-controlled Hamilton, PCH) structure. It is simpler for controller design and stability analysis according to the peculiar feedback stabilization method. Professor Yu Hai-Sheng has done a lot of work for permanent magnet motor by the method [36-37]. The PMLSM with high order, strong coupling nonlinear characteristics causes that, under reasonable hypothesis, it can be considered as a two-port energy conversion device. Therefore, the PCH system model of the PMLSM is necessary by energy-shaping methods and passive control, and the control of PMLSM is realistic significance. It will be a new way for permanent magnet linear driver in the industrial application of AC servo system. II. DRIVE SYSTEM WITH POWER CONVERTER OF PMLSM The driving system schematic diagram of PMLSM with power converter shows as Fig. 1.
voltage symmetry. The high-power resistances are often connected in series in order to reduce the drift which caused by heating and other factors. AC filter capacitor C is 0.47 μF . The NTC22 thermistor is as one of protection circuits. From the diagram can be drawn [37]: ⎡u a ⎤ ⎡2d a − d b − d c ⎤ ⎥ U dc ⎢ ⎥ ⎢ ⎢u b ⎥ = ⎢2d b − d c − d a ⎥ × 3 . ⎢⎣u c ⎥⎦ ⎢⎣2d c − d a − d b ⎥⎦
(1)
where d i ( i = a, b, c ) is the duty ratio. U dc is DC bus voltage. III. BASIC FORMS OF PORT-CONTROLLED HAMILTONIAN (PCH) SYSTEM Port-Controlled Hamiltonian (PCH) system[38] is described as below: •
x = f ( x) + g ( x)u = J ( x)
∂H ( x) + g ( x)u ∂x .
∂H ( x) y = h( x ) = g ( x ) ∂x
(2)
T
Introduce the energy dissipation concept into the PCH system [39] [41] and the PCH dissipative models: •
x = [ J ( x) − R( x)] y = g T ( x)
∂H ( x) + g ( x)u ∂x .
∂H ( x) ∂x
(3)
Where x ∈ R n , u , y ∈ R m , and R ( x) is semipositive symmetric matrix. R ( x) = R ( x) T ≥ 0 , which denotes the accessory resistance structure of port; J (x ) is anti-half symmetric matrices, and J ( x) = − J ( x) T , which denotes the internal system interconnection structure [35-36]. IV. PCH SYSTEM MODEL OF NON-SALIENT POLE OF PMLSM According to the characteristics of Non-salient pole of PMLSM, frequency of 50Hz is considered only, when i d = 0 and voltage is given in the d-q coordinates, the model of non-salient pole of PMLSM can be expressed as [26] [40] :
Fig. 1. Driving system schematic diagram of PMLSM with power converter
Power converter is composed of a three-phase bridge AC-DC rectifier and a three-phase DC-AC voltage type inverter. The input is a three-phase AC voltage. The output is three-phase PWM voltage. It connects to the three-phase stator winding of PMLSM, which is three-phase star-type connection. In practical application, using the series capacitors to reduce the weakness of large volume high price of single capacitor and using the parallel connection resistors to ensure the capacitor © 2013 ACADEMY PUBLISHER
Ld
Lq
di q dt m
di d π = u d − ri d + vLq i q . dt τ = u q − ri q −
π π vψ r − vLd i d . τ τ
dv π = ψ r i q − FL + Fd + f . dt τ
m --rotor quality; L --Armature inductance;
(4)
(5) (6)
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τ --polar pitch; ψ r --permanent magnet working flux; v --motion velocity of rotor;
⎡1 0 0⎤ g (x) = ⎢⎢0 1 0⎥⎥ . ⎢⎣0 0 1⎥⎦
Fx --Electromagnetic thrust; FL --load force; f --friction force; Fd --External disturbance force. Definition is given as follows:
⎡i d ⎤ ⎡ x1 ⎤ ⎡ Ld i d ⎤ ⎡ Ld 0 0 ⎤ ⎡i d ⎤ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x = ⎢ x 2 ⎥ = ⎢ Lq i q ⎥ = ⎢0 L q 0⎥ ⎢i q ⎥ = D ⎢i q ⎥ . ⎢v ⎥ ⎢⎣ x 3 ⎥⎦ ⎢mv ⎥ ⎢0 0 m ⎥ ⎢v ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ (7) ⎡ ud ⎢ u=⎢ uq ⎢F − F + l ⎣ d
⎤ ⎥ ⎥. f ⎥⎦
⎡i d ⎤ ∂H ( x) ⎢ ⎥ T y = g ( x) = ⎢i q ⎥ . ∂ ( x) ⎢v ⎥ ⎣ ⎦
(8)
In order to make system asymptotically stable equilibrium near expectations x 0 , A closed-loop expectations energy function H d (x) will be constructed. The energy function will be take minimum values near x 0 . That is, the feedback control u = α (x) will be found in order to make the closed-loop system as fellow: •
x = [J d ( x) − R d ( x)]
Take the sum of electrical energy and mechanical energy as Hamiltonian function of PMLSM.
R d ( x) --damping matrix. In addition:
•
x = [J d ( x) − R d ( x)]
π ⎤ ⎡ ⎥ ⎢0 0 τ L q i q ⎥ ⎢ π J ( x ) = ⎢0 0 − ( L d i d + ψ r ) ⎥ . ⎥ ⎢ τ ⎥ ⎢ ⎥ ⎢0 π ψ 0 r ⎥⎦ ⎢⎣ τ
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∂H d ( x) . ∂x
(17)
[J d ( x) − R d ( x)]K ( x) = − [J a ( x) − R a ( x)]
. ∂H ( x ) + gα ( x ) ∂x
(18)
and T
∂H ⎡ ∂H ⎤ ( x) = ⎢ ( x)⎥ . x ∂ ∂x ⎣ ⎦
(11) K ( x0 ) = −
⎤ ⎥ ⎥ f ⎥⎦
⎡− r 0 0⎤ R ( x) = ⎢⎢0 − r 0⎥⎥ . ⎢⎣0 0 0 ⎥⎦
(16)
The expected stabilization balance point x 0 of J d (x) , R , H (x) , g is given. If the expression u = α (x) of Feedback control can be found. R a (x) , J a (x) , K (x) satisfy partial differential equations.
Then the system PCH form of PMLSM can be written as follow:
ud ⎡1 0 0⎤ ⎡ ⎢0 1 0 ⎥ ⎢ uq ⎢ ⎥⎢ ⎢⎣0 0 1⎥⎦ ⎢ F − F + L ⎣ d
(15)
J d ( x) --interconnected and damping matrix.
1 1 ⎡ 1 2 1 2 1 2⎤ H ( x) = x T D −1 x = ⎢ x1 + x2 + x 3 ⎥ . (10) Lq Lq 2 2 ⎣⎢ Ld ⎥⎦
⎤ ⎡ • ⎤ ⎡− r 0 π Lq i q ⎢ ⎥ x ⎢ 1⎥ τ ⎥ ⎡i d ⎤ ⎢• ⎥ ⎢ π ⎢ ⎥ ⎢ ⎢ x 2 ⎥ = 0 − r − ( L d i d + ψ r ⎥ ⎢i q ⎥ + ⎢ ⎥ τ ⎢• ⎥ ⎥ ⎢⎣v ⎥⎦ ⎢x ⎥ ⎢ π ⎢ ⎥ 3 . ψ 0 0 ⎢⎣ ⎥⎦ r ⎢⎣ ⎥⎦ τ
∂H d ( x) . ∂x
⎧⎪ J d ( x) = J ( x) + J a ( x) = − J dT ( x) . ⎨ ⎪⎩ R d ( x) = R ( x) + R a ( x) = R dT ( x )
(9)
(14)
∂H ( x0 ) . ∂x
(19)
(20-1)
That is: ∂H d ( x) = 0 . ∂x
(12)
K ( x0 ) > − ∂2Hd ∂x 2
(13)
∂2H ∂x 2
( x0 ) .
(x0 ) = 0 .
(20-2)
(21-1)
(21-2)
Then the closed-loop system (15) is a dissipation PCH systems and the point x 0 is a stable balance for the closed-loop system. H d ( x) − H ( x) = H a ( x) .
(22)
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∂H a ( x) = K ( x) . ∂x
⎡ 0 − J12 ⎢ J a ( x ) = ⎢ J12 0 ⎢⎣ J13 J 23
(23)
H a (x) is undetermined function which means the system energy injected by means of control. If maximum invariant set equals { x 0 }, which is included in the closed-loop system T ⎫⎪ ⎧⎪ ∂H d ⎤ n ⎡ ∂H d ( x)⎥ R d ( x ) = 0⎬ . ⎨x ∈ R | ⎢ ∂x ⎣ ∂x ⎦ ⎪⎭ ⎪⎩
(24)
The system is asymptotically stable. Attract domain estimation is given by maximum limited level set. { x ∈ R n | H ( x) ≤ c }. V. DESIGN AND STABILITY ANALYSIS OF SYSTEM
expectation point of balance can be get in (11) for a given v 0 . There is: ⎡ ⎤ ⎢0 ⎥ ⎡ x10 ⎤ ⎢ ⎥ Lq ( FL − Fd − f ) ⎥ . ⎢ ⎥ x0 = ⎢ x 20 ⎥ = ⎢ ⎢ ⎥ π ψr ⎢⎣ x30 ⎥⎦ ⎢ ⎥ τ ⎢ ⎥ ⎢⎣ mv 0 ⎥⎦
1 ( x − x0 )T D −1 ( x − x0 ) . 2
(25)
(26)
∂H ( x ) = D −1 ( x ) ∂x ∂H a ( x ) ∂H d ( x ) ∂H ( x ) K ( x) = = − = D −1 ( x0 ) ∂x ∂x ∂x
(27)
. (28)
⎧ ⎪ x = x0 ⎪ ⎪ ∂H d ( x ) . =0 ⎨ ∂ x ⎪ ⎪ ∂ 2 H d ( x) >0 ⎪ ⎩ ∂x 2
(29)
(27) and (28) are met. So the closed-loop system designed is stable at the balance point. Take the following expression: © 2013 ACADEMY PUBLISHER
(31)
J12 , J 13 , J 23 , R1 , R2 -undetermined interconnections and damping parameters respectively. Substitute (28) into (18), there is: − [J d ( x) − Rd ( x )]D −1 x0 =
− [J a ( x ) − Ra ( x )]D −1 x + g ( x )α ( x )
(32)
.
The expression of u can be obtained from (13): ⎡u d ⎤ ⎢ ⎥ u = α ( x) = ⎢u q ⎥ . ⎢− τ ⎥ ⎣ L⎦
(33)
Substitute (16) and (30) into (32), (34) can be deduced with (25). − [J d ( x ) − Rd ( x )]D −1 x0 =
.
(34)
Because of:
⎧⎡ ⎪ ⎢0 ⎪⎢ ⎪⎢ ⎪ ⎢0 ⎪⎢ ⎪ ⎢0 ⎪ ⎢⎣ ⎪ ⎨ ⎪ ⎪ ⎪ ⎡− ⎪− ⎢0 ⎪ ⎢ ⎪ ⎣⎢ 0 ⎪ ⎪⎩ ⎡ x 10 ⎢ ⎢ x 20 ⎣⎢ x 30
When (29) is true.
(30)
⎤ ⎥ ⎥. ⎥⎦
⎡u d ⎢ u = α ( x) = ⎢u q ⎢F − F + L ⎣ d
For ∂H d ( x ) ∂ 2 H d ( x) = D −1 ( x − x0 ) = D −1 . 2 ∂x ∂x
0
⎤ ⎥. ⎥ ⎥⎦
− [J a ( x ) − Ra ( x ) ]D −1 x + g ( x )α ( x )
Take the expected Hamiltonian function: H d ( x) =
− J 23
⎡ r1 0 0 ⎢ Ra ( x ) = ⎢0 r2 0 ⎢⎣0 0 0
CONTROLLER
The control essence of PMLSM is to make the error in the permitted range between its actual speed and hope speed. This paper takes a PMLSM as example. Ld and Lq equal in the non-salient PMLSM, make i d =0. The
− J13
0 0
π ψ τ
π ⎤ L q iq ⎥ τ ⎥ π − ( L a i d +ψ r ) ⎥ ⎥ τ r
0
⎥ ⎥ ⎥⎦
⎤ ⎥ ⎥. f ⎥⎦
− J 13 ⎡ 0 − J 12 ⎢ 0 − J 23 + ⎢ J 12 ⎣⎢ J 13 − J 23 0
⎤ ⎡ 1 0 0⎥ ⎢L d ⎥ ⎢ r 0 0 ⎤ ⎡ − r1 0 0 ⎤ ⎥ 1 ⎢ ⎥⎢ 0⎥ − r 0 ⎥⎥ − ⎢ 0 − r2 0 ⎥ ⎢ 0 Lq ⎢ ⎥ 0 0 ⎦⎥ ⎣⎢ 0 0 0 ⎦⎥ ⎢ 1 ⎥ ⎢0 0 ⎥ m ⎦ ⎣
⎤ ⎧ ⎥ ⎪ ⎥ = ⎨ ⎪ ⎦⎥ ⎩
(35)
− J 13 ⎡ 0 − J 12 ⎢ 0 − J 23 ⎢ J 12 J 23 0 ⎣⎢ J 13
⎫ ⎪ ⎤⎪ ⎥⎪ ⎥⎪ ⎦⎥ ⎪⎪ ⎪ ⎪ ⎬× ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭
⎤ ⎡− r 0 0⎤ ⎫ ⎥ ⎢ ⎥⎪ ⎥ − ⎢0 − r 0 ⎥ ⎬ × ⎪ ⎦⎥ ⎢⎣ 0 0 0 ⎦⎥ ⎭
⎡ 1 ⎤ 0 0⎥ ⎢L ⎢ d ⎥ ⎡ x1 ⎤ ⎡1 0 0⎤ ⎡u d ⎤ . (36) ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ 1 ⎥ 0 ⎥ × ⎢ x 2 ⎥ − ⎢0 1 0⎥ × ⎢u q ⎢0 ⎥ L q ⎢ ⎥ ⎢ x ⎥ ⎢0 0 1⎥ ⎢ F − F + f ⎥ 3⎦ ⎣ ⎦ ⎣ L ⎣ d ⎦ ⎢ 1 ⎥ ⎢0 0 ⎥ m ⎦ ⎣
There is (37) form (36):
JOURNAL OF COMPUTERS, VOL. 8, NO. 2, FEBRUARY 2013
J13 x10 + Ld
J 23 +
π ψ τ r x = 20
Lq
. (37)
J 13 J x1 + 23 x 2 − ( Fd − Fl + f ) Ld Lq x 0 is substituted into (37): Then:
J 23 = −
Lq x1 Ld ( x2 − x20 )
J 13 = .
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resistance is 3.27 Ω . The d-q axis inductance is 0.0072 H respectively. Carrier frequency is 12kHz. Friction is equivalent to the load resistance. In Fig.2, the starting speed simulation curve is given with r1= r2=1.3. We can see that the maximum speed overshoot is in 3.5% and response time is in 0.1 seconds. The speed response curve of conventional PID controller is given with the same conditions in Fig.3. The maximum speed overshoot is over 30% and response time is in 0.35 seconds.
(38)
Since the equation solution is not only one, we can take the following expression. J13 = −
Ld (x2 − x20 ) Lq
J 23 = − x1
.
(39)
J12 = − Kx3 K --Arbitrary parameters. K does not affect the stability of system. So the controller will be the follow: ⎡ πLq τKLq ( Fl − Fd − f ) ⎤ + u d = −r1id + kmviq − ⎢ ⎥ iq πψ r ⎣ mτ ⎦ . πψ r v0 ⎛ πv0 Ld ⎞ − Ld v0 ⎟id + (Ld − km )vid − r2 iq + uq = ⎜ τ ⎝ τ ⎠
Fig. 2. Starting speed curve of Hamiltonian controller ( r1= r2=1.3)
(40) VI. SIMULATION RESULTS AND ANALYSIS The duty ratio can be got through the d-q coordinate transformation: π π ⎤ ⎡ u cos vt − uq sin vt ⎥ 6 ⎢ d τ τ ⎥ da = ⎢ 3U dc ⎢ ⎥ 2 2 ⎥⎦ ⎢⎣+ ud + uq ⎡ ⎛π ⎢ud cos⎜ vt − 6 ⎢ ⎝τ db = 3U dc ⎢ ⎛ π 2π ⎢sin ⎜ vt − 3 ⎢⎣ ⎝ τ
⎤ 2π ⎞ ⎟−u ⎥ 3 ⎠ ⎥ ⎥ ⎞ 2 2⎥ + + u u ⎟ d q ⎥⎦ ⎠
Fig. 3. Speed curve of conventional PID controller
.(41)
⎡ ⎤ 4π ⎞ ⎛π ⎟− ⎢ud cos⎜ vt − ⎥ 3 ⎠ 6 ⎢ ⎝τ ⎥ dc = ⎢ ⎥ 3U dc 4π ⎞ ⎛π 2 2 ⎢uq sin ⎜ vt − ⎟ + ud + u q ⎥ 3 ⎠ ⎥⎦ ⎝τ ⎣⎢ Fig. 4. Speed curve of Hamiltonian controller with load disturbance
So needed voltage can be calculated according to the formula (41), to control each IGBT turn-on or turn-off. The study of simulation is done in Matlab7.0/Simulink environment. The system includes controller model, inverter model and PMLSM simulation model which adopt package form respectively. System steady/dynamic performance simulation can be achieved by modifying the parameters. The motor rated power is set to 0.75kW. Rated voltage is 220V. Inverter frequency is 30Hz. K = 16. Stator © 2013 ACADEMY PUBLISHER
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to design the control system under load changes is a part of further research content, and in this design, it is lack of science to assume the friction and all other perturbations as a constant load. How to classify the perturbation and how to design a precision control system? There is a lot of work to do. Now variable “load condition control research” is being carried out. VII. CONCLUSION
Fig. 5. Speed curve of conventional PID controller with load disturbance
The modeling of permanent magnet linear synchronous motor is presented in this paper by using energy-shaping and passivity-based control method. And the control principle of PMLSM is given. The control laws are presented in load known conditions. The equilibrium stability is analyzed and the feedback controller is designed for the control system of PMLSM. A simulation model has been established in the Simulink/Matlab environment and the simulation results are given. In order to compare the control effect, traditional PID control simulation results are also provided in the same conditions. The simulation results show that the proposed scheme has a better performance than that of conventional PID controller.
ACKNOWLEDGEMENT
Fig. 6. Speed curve of Hamiltonian controller with speed disturbance
The load disturbance of 60N is added at 2.5 seconds in Fig. 4. The curve is the results of conventional PID controller with the same load disturbance in Fig. 5. The speed disturbance of 0.5 m/s is added at 2 seconds in Fig.6. The results of a conventional PID controller is also given as shown the under curve with the same conditions in Fig.7. We can see that the resist disturbance of PID controller is poor than that which based on passivity. Obviously, the controller of PMLSM based on passivity has good control characteristics which can significantly inhibit system parameter variations and external disturbances. The system has stronger robustness and practicability.
The work is supported by the National Natural Science Foundation of China (61075071), the Natural Science Foundation of Henan Province (2010A470006/ 2011A470010), and Zhengzhou Science and Technology Bureau, Henan Province (10PTGG380-1). REFERENCES [1]
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[8] Fig. 7. Speed curve of conventional PID controller with speed disturbance
The control system of PMLSM was studied in the assumed load constant known cases; however, the fact is that the load of PMLSM is changed in most cases. How © 2013 ACADEMY PUBLISHER
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Zhiping Cheng was born in Shangshui R.P. China in 1974. Major in Control Theory and Control Engineer from September 2000 to June 2003. He received the M.S. degrees in electrical
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engineering from Henan Polytechnic University, Jiaozuo, China, in 2003. Major in Physics from September 1994 to June 1998. Then he received the Bachelor of Science degrees in school of physics electronic engineering from Xinyang Normal University, Xinyang, China, in 1998. The major field of study is linear motor and control. He works in Zhengzhou University from 2003 until now and was promoted to a lecturer in 2004. Now he is responsible for two projects supported by the Natural Science Foundation of Henan Province (2010A470006/2011A470010). Zhengzhou Science and Technology Bureau, Henan Province (10PTGG380-1). He is responsible for technology of the
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project supported by the National Natural Science Foundation of China (61075071).
Liucheng Jiao was born in Jiao zuo (R.P. China)in 1950. Major in Control Theory and Control Engineer from September 1995 to June 1998. Received the Doctor of Science degrees in School of Mechanical Electronic & Information Engineering from China University of Mining & Technology, Beijing, China, in 1998. The major field of study is in linear motor and control. He works in Zhengzhou University as a part-time professor. Now he is responsible for project supported by the National Natural Science Foundation of China (61075071 ).
