Predictive Control of Permanent Magnet Synchronous Motor Based on
Optimization Model Algorithmic Control
1
Chen Junshuo,
2Liu Jinglin,
3Zhang Ying
*1, Chen Junshuo
School of Automation, Northwestern Ploytechnical University,
Xi’an Shaanxi 710129, China, [email protected]
2, Liu Jinglin
School of Automation, Northwestern Ploytechnical University,
Xi’an Shaanxi 710129, China, [email protected]
3, Zhang Ying
School of Automation, Northwestern Ploytechnical University,
Xi’an Shaanxi 710129, China, [email protected]
Abstract
This paper presents an optimization model algorithmic control (OMAC) for permanent magnet synchronous motor drive system. The OMAC controller is designed based on the second order ARMA model of permanent magnet synchronous motor. It optimizes the correction coefficient of calculation error and prediction error coefficient of output as well as the amount of the weighting factor in the traditional model algorithmic control. Then selecting only one future time of predictive values to calculate the optimal control law simplifies the calculation process and so overcomes the shortcomings of traditional control needed to select the number of prediction horizon and required to choose a longer sampling period to calculate appropriate control quantity. The algorithm is simple and robustness of the system can be guaranteed. Therefore, the OMAC is well positioned to meet the dynamic and rapid motor real-time requirements. Simulation is conducted to verify the performance of the proposed control scheme.
Keywords
:Predictive Control Algorithm, Optimization Model Algorithmic Control, Permanent Magnet Synchronous Motor1. Introduction
The permanent-magnet synchronous motor (PMSM) drive has emerged as a top competitor for a full range of motion control applications. For example, the PMSM is widely used in the modern Ac servo systems such as industrial robots, machine tools and is being considered in high-power applications such as vehicular propulsion and industrial drives. It is also becoming viable for commercial/residential applications. The PMSM is known for having high efficiency, high-torque, low torque ripple, superior dynamic performance, high power density, and free maintenance. These drives often are the best choice for high-performance applications and are expected to see expanded use as manufacturing costs decrease [1-3]. However, Many of the control systems of those drives set forth have parameter-dependent formulations. With time, wear, and temperature, the machine parameters can vary significantly, negatively affecting system performance. It is a multivariable, nonlinear and strong coupling system and the servo system’s running situation is more complex. The PI control system is vulnerable to motor parameter variations, load disturbance and other uncertainties. Therefore, the traditional PI control strategy is applied for PMSM to realize closed loop control, which has drawback in high-performance driving system. It can't meet the system requirements both of dynamic response and anti-interference ability at the same time. In order to overcome the drawback of PI control, many control strategies eliminating the influence of uncertainties currently have caused the extensive attention of people [4-6].
Predictive Controllers, which allow us to overcome the drawback of PI control, are known in drive control for quite a long time. In recent years, there are successful examples of predictive control being applied in the complicated industrial process [7-10]. Nevertheless all predictive control schemes proposed for controlling electrical drives until today only precalculate the system behavior for one single future sampling cycle.Powerful control strategies like model algorithmic control (MAC), allow higher prediction horizons, but these strategies have not yet extensive been used for drive control.
In this paper, a new predictive controller for electrical drives based on OMAC is presented. MAC was developed in France during the late 1970s in the chemical process industry. Richalet established the original concept [11]. In contrast to traditional PI, MAC is based on the finite impulse response
models and these models are non-parametric models. Therefore, MAC can drastically reduce the dependence on parameters. However, on-line computation of convolutions is complicated and it needs to set more parameters in MAC. Thus, the sampling period is difficult to choose smaller to adapt to the requirements of the motor’s faster dynamic. General MAC which u(k)is controlling quantity is still proportional control in essence. When the controlling quantity has the constraints, even closed-loop prediction will also lead to static difference. Therefore, this paper presents an OMAC for PMSM drive system. It integrates the advantages of MAC such as online real-time prediction, optimization and feedback correction. By introducing the second order ARMA model of PMSM as predict model of OMAC, the number of optimization variables at every time is greatly reduced so that the on-line computation is simplified. Unlike MAC, the incremental Δu(k) is controlling quantity in OMAC controller. Thus it contains an integral element which can effectively eliminate the static difference of the PI drive system.
The rest of the paper is organized as follows. The mathematical model of PMSM is presented in Section 2. We outline basic steps of MAC and illustrate its application in a PMSM drive system in Section 3. Designing an OMAC controller for PMSM drive system is introduced in Section 4. We demonstrate that the proposed controller performs well by conducting several computer simulations in Section 5, and provide the conclusion of this study in Section 6.
2. A time-continuous model of PMSM
As already mentioned, a model of control object is one of the essential components of a MAC controller. Hence, for drive control a mathematical model suitable for PMSM has to be designed. For high frequency signals the stator resistance and the effects of the permanent magnet flux linkages can be neglected. We also neglect the effects of magnetic saturation. A permanent-magnet synchronous motor is characterised by the following equations in a reference frame, rotating with the speed ωe. The
stator voltage equations of a PMSM in the synchronously rotating reference frame are described as follows: d d e q q s d q q e d d s q
d
u
L i
R i
dt
d
u
L i
R i
dt
(1)Where ud, uq, id, iq,ψd, ψq, and Ld, Lq are the voltage, current, flux linkage and constant inductance,
respectively in d-q axis, Rs represents the resistance of the armature winding, and ωe is an angular
velocity. The stator flux linkage equations are given by:
d d d md f q q q
L i
L i
L i
(2)Where Lmd is the mutual inductance between the stator winding and the rotor magnet, and if denotes an equivalent field current.It is important to note that for the surface-mounted PMSM, Ld = Lq because the stator inductances are independent of rotor position. Therefore, the torque equation can be expressed as:
( )
em d q q d md f q
T p
i
i pL i i (3) Where p is the number of pole pairs. By using (1) and (2), the mechanical motion equation can be derived as follows:e em L
d
J
T
T
p dt
(4) Where J and TL represent the moment of inertia and a cogging torque, respectively.Current-based PMSM drives are typically used in PMSM control systems where a commanded torque is generated. Let the commanded torque be designated as Tem. By using the machine equations that were transformed into the rotor reference frame, other useful formulas can be developed. This method of building the current commands relies heavily on machine parameterization. Advanced methods such as MAC and OMAC, utilize closed-loop methods to eliminate or drastically reduce the dependence on parameters. Indeed, advanced control methods for motor drives is still an active research area [12-13].
3. Model algorithmic control for PMSM drives
The typical structure of a Model algorithmic controller is depicted in Figure. 1. It basically involves: (1) an impulse response model for system representation and prediction, (2) a reference trajectory, (3) an optimality criterion, and (4) the consideration of state and control constraints [14]. The main idea of the MAC strategy is to predict the deviation of the future system outputs from the reference path based on the model, define an optimality criterion which reflects the deviations, and obtain the optimal control strategy to minimize the criterion over a certain horizon in the future.
e k
r
k i
qi k
k
mk
q i k i
kFigure 1. Diagram of conceptual drive system
The MAC controller is designed based on the sampled-data representation of the PMSM. In MAC method the control law is obtained from the minimization of the output error at time k+i.
The prediction output has two main components:
* the error response, being the expected behaviour of the output error vector e(k) between measured value and predicted value
* the forced response, being the additional component of output due to the precalculated set of future actuating values ωm(k)
The total future system behavior is taken to be the addition of the error response and the forced response. This sum is calculated until the prediction horizon Np. A cost function is then used to valuate the resulting total response of the system. Finally, an optimiser determines the best set of future control values iq(k+i), i.e. control actions to minimise the result of the cost function. The choice of this function can be made according to the demands of the process and is not dependent on the controller itself. So it is possible to use linear or quadratic cost functions and penalise e. g. the control error or the control effort with appropriate weighting factors. Hence, the control engineer can design the controller in an optimal way to fit the requirements of the controlled system (process).
An open loop-strategy now would simply assert the set of controls calculated by the optimiser in sequence. The use of the receding horizon approach makes MAC a real closed-loop feedback control. Only the first member of the set, i. e. iq(k), is transmitted to the controlled system, and the whole process of prediction, optimisation and control is repeated at each sample.
3.1. Internal model
For MAC uses a model to precalculate the future system states, a time-discrete model has to be defined at first. Based on the assumption that the sampling value of velocity’s unit impulse response is ω1, ω2, …ωN, internal model is described by using the convolution equation as follows:
1 2 1 1 1 q q N q q k i k i k i k N k z i k k
(5)Where ω(k) is the velocity of PMSM, iq(k) is incoming current, and ξ(k) is an uncertain sequence of impulse response at t = tk = kT with T being the sampling time. N denotes the sequence length of the impulse response. By utilizing difference method for Eq. (5), the increment model could be expressed as:
^ ^ ^ 1 2 ^ 1 1 1 m q q N q q k i k i k i k N z i k (6)
Where the subscript “m” is added to indicate estimates of ωobtained in the model simulation and differentiate the simulated ωm from the measured output ω. ω1^，ω2^，…ωN^ are sampling value of
velocity’s unit impulse response.
3.2. Feedback compensation
Since the drive systems are time-varying nonlinear systems, errors must exist in the practical application. The model (6) can’t response the system precisely.Based on the feedback correction law of the auto-control principle, e(k) is added to the prediction output of the modelfor getting closed-loop prediction output. After correction, the prediction output of PMSM in the future P step can be re-written as:
1 i P m P m N q P i k P k P h k k i k P i h e k
(7)
Where e(k) is the speed error between predictive value and the actual value at time k, and
h
p is an error correction coefficient.3.3. Reference trajectory
Then, the question is the choice of iq(k) to obtain a desirable output response after i time steps. One can request ω(k+i) to be in the right direction and cover a fraction of the “distance” between ω(k+i-1)and the setpoint value. In other words, one can define a desirable value ωr of
the output at the (k+i)th time step by:
/ 1 r iT k i k k e (8)
The expectations output ω(k) of the system gradually reach on the set value ω along a pre-ordained curve. ωr is the curve and reference trajectory. Where α represents a tunable parameter
limited by the bound values: 0<α<1, and T is a sampling period. Clearly, α→0 corresponds to ωr(k+i)→ω, and therefore, will try to force the output to go to the set-point as soon as possible,
whereas α→1 corresponds to ωr(k+i)→ω(k), leaving the output unaffected. An intermediate
choice of a corresponds to a desirable value of the output in between ω and ω(k) that tries to bridge the gap to a certain extent. There is not a criterion to choose α. It depends on the characteristic of the system to be controlled and the control requirement. If the target of control system is to make the system output follow the reference output quickly αshould be chosen as a small value; on the other hand if the target of control system is to make the system output follow the reference output gently with a small input a big αcan be used. Eq. (8) is referred to as the “reference trajectory” in the MAC literature.
3.4. Optimality criterion
Once the reference trajectory has been specified, the question then becomes how to choose the control input iq(k) so that ωp(k)will match ωr(k). This can be formulated as an optimization
problem:
2 2
min P p r q J q k P k P i k (9) Where q is the prediction error coefficient of output, λ is weighting coefficient of control quantity, and ωr(k+P) is the reference value of input velocity at k+P. ConsideringEqs. (7) and(8), this becomes:
2 2
1 min N P i q P m r q i J q
i k P i h
k
k
k P
i k
(10)In the absence of the input constraints, this minimization problem is trivially solvable. Control input iq(k) is the solution of the nonlinear algebraic equation, and Δiq(t) is the increment of control current. Hence, MAC seems to be a promising approach to be applied to drive control. Unfortunately, solving the optimization problem needs a very large amount of calculation power. The time of online calculation has been found to be too long for applying MAC to anonline control of electrical drives.
4. Optimization model algorithmic control controller for PMSM
In this section, for simplifying calculation we optimize MAC frame structure by choosing only one predicted value in the future to calculate the optimal control law. The prediction horizon is involved in the optimization.
Since the model used for MAC has a time-discrete structure (see (5) and (6)), the time-continuous model from Eq. (4) has to be discretized. Using id = 0 vector control, Eq. (4) can be represented in the form:
e n d q e d p i B J dt
(11) Using Laplace transform and zero order hold (ZOH), discrete z-transfer function of PMSM can be written as: 1 1 1 1 ts T k e bz z s Js B az (12) Where kT = pnΨd，a = -e-TB/J, b = kT(1-e-TB/J)/B with T being the sampling time. Velocity differenceequation of PMSM can be derived from Eq. (12):
1
1
e
t
a
et
bi t
qConsidering difference operator Δ = 1-z-1, we can get the following time-discrete velocity model for
the prediction performance of PMSM.
,
1
1
1
e m
t
a
et a
et
b i t
q
(14) Where ωe(t) and Δiq(t) represent an actual velocity and the increment of control current attime t, respectively.
For getting each output of the controlled object in MAC, general according to multiple predicted values to calculate the optimal value,
it
needs to select the prediction step number and the length of the prediction horizon. According to the above optimization idea, for the each output of controlled process, we only choose one predicted value in the future to calculate the optimal control law in OMAC. For example, a predicted value at k+1 time is used to calculate control quantity at k time and control quantity remain unchanged at k+1 time and later, such as iq(t+1) = iq(t+2) = …iq(t+P). That is the size of the control quantity which has nothing to do with prediction steps. Eqs. (7) and (8) can be modified to:
, 1 , 1 , e p t e m t hP t e m t
(15)
1
1
r r y t
t
(16) Where ωe,p(t+1) is the prediction output of a controlled object in the future one step. Here yrdenotes a reference trajectory. Applying the model (15) and (16) to the optimisation problem (10), the quadratic performance index is:
2 2
,
min
1
e e1
q p r e m r1
qJ
q
a
t a
t
b i t h
t
t
y t
i t
(17) For simplicity,the weighting coefficient λ, error coefficient q and error correction coefficient hp
are set to zero, one and one. It is recommended, but not required. Now Eq. (17) has to be minimised,
dJ/dΔiq(t) = 0. The control quantity can be obtained:
1
1 ,
1q r r r r m r
i t
t
t
t
t y t b
(18) According to the above analysis, in order to validate the control performance of the optimization model algorithm control the whole drive system simulation model based on vector control for the permanent magnet synchronous is established in the MATLAB/SIMULINK environment. The system block diagram is shown in Figure. 2.
5. Simulation results
In this section, we evaluate the performance of the proposed control method by applying it to drive system. To assess the performance of OMAC, OMAC is compared with PI. The main parameters of PMSM are shown in Table 1. The results under the OMAC velocity controller and the PI velocity controller are shown in Figure. 3 and Figure. 4, respectively. The sampling period T = 1ms, reference trajectory time constant τ = 6ms, load TL = 2Nm.
Table 1.Basic motor data
Motor parameter Value
d-axis inductances Ld
q-axis inductances Lq
Armature winding resistance Rs
Number of pole pairs P
Moment of inertia J0 Friction coefficient B0 7 mH 7 mH 1.5 Ω 3 1.48×10-3kg·m2 3.1×10-4
(a) (b)
Figure 3. Simulation results with OMAC scheme: 500rpm, 2Nm, model matching (a) Output velocity,
(b) phase current.
(a) (b)
Figure 4. Simulation results with PI: 500rpm, 2Nm, model matching (a) Output velocity, (b) phase
current.
Figure. 3 indicates that the system dynamic response quickly, velocity can be quickly up to expectation and there is no overshoot and steady-state velocity error. In order to compare the results with conventional control, the same experiment has been carried out using PI current
controller. Figure. 4 shows that a PI controller results in a much bigger current distortion when the motor starts up, a few velocity overshot and steady-state velocity error.
To show the robustness of the system, a serious mismatch in the parameters of PMSM model and practical parameters has been simulated. The load is suddenly increased from 2Nm to 4Nm at 0.2s, and reduced from 4Nm to 2Nm at 0.6s.
(a) (b)
Figure 5. Simulation results with OMAC scheme: 500rpm, model mismatching (a) Output velocity, (b)
phase current.
(a) (b)
Figure 6. Simulation results with PI scheme: 500rpm, model mismatching (a) Output velocity, (b)
phase current.
Figure. 5 presents the transient response of the OMAC system when the moment inertia J = 8J0 and the friction coefficient B = 13B0. The simulation results show that velocity is the
constant when load torque changes and the deferent sine wave distortion is smaller than PI system. Although model parameters are serious mismatching, strong robustness is achieved by the effective disturbance estimation. The OMAC guarantees that the model from the uncertainty set leads to a controller with stabilizing the true system. The system is stable and its transient properties are nice. Using PI controller in the same experiment, Figure. 6 shows that there is velocity fluctuation when load torque changes and the deferent sine wave distortion is much bigger.
6. Conclusion
The ideas of MAC in combination with systems, already well know in the field of control engineering and applied to process engineering problems, have been shown to be reasonably applicable to drive control as well. The advantages of MAC can be utilized especially for PMSM control. Since the second order
ARMA
model of the PMSM is considered, an optimalcontrol of PMSM is possible. Due to the high prediction horizon, the proposed OMAC controller is superior performance. In addition to that, OMAC allows simplifying calculating method in a very simple way just by modifying the corresponding weighting factors inthe cost function. The results obtained by an emulational real-time OMAC encourage for further research.
7. Acknowledgments
This paper was supported by National Natural Science Foundation of China under Grant NO. 90716026, Science and Technology Program of Beilin District, Xi’an under Grant NO.GX1110 and Foundation for Basic Research from Northwestern Ploytechnical University under Grant NO.JC201146.
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