Predictive functional control of PMSM based on a composite prediction model

Shihua Li Huixian Liu School of Automation, Southeast University

Nanjing, Jiangsu 210096, China Key lab of Measurement and Control of CSE

Ministry of Education

School of Electrical Engineering Hebei University of Science and Technology

ShiJiaZhuang, 050018, China E-mail:2006guanyue@163.com

Wenshu Fu Estun Automation

Technology Co., LTD Nanjing, 211106, China

E-mail: Ish@seu.edu.cn Telephone: +86 25 83793785

Abstract-Considering the speed-regulation problem for per­manent magnet synchronous motor (PMSM) servo system, a control scheme based on the improved PFC method is designed to ensure an optimal closed loop performance even in the presence of disturbances. In common design of prediction model based control method, usually disturbances are not considered in the prediction model as well as the control design. To improve the disturbance rejection ability of system, an improved predictive functional control(PFC) method is developed in this paper by embedding disturbance information into the prediction model. Here, a composite prediction model is obtained by introducing the estimated value of disturbances, where disturbance observer (DOB) is employed to estimate the lumped disturbances. So the influence of disturbances on system are taken into account in op­timization procedure. Detailed T MS320F2808 DSP experimental

results are provided to verify the effectiveness of the proposed method.

I. INT RODUCTION

Permanent magnet synchronous motor (PMSM) possesses many good characteristics, such as high power density, high torque to inertia ratio and efficiency. Due to existence of nonlinear state coupling effects in such a system, it is not easy for PI and other linear control algorithms to obtain a high closed loop performance [I], [2]. So far various nonlinear control algorithms have been reported, e.g., sliding mode control [3]-[5], adaptive control [6]-[8], internal mode control [8], robust control [9], fractional order control [10], finite-time control [II], predictive control [12], [13], disturbance rejection control [5], [8], [II], [12], [14], [15] and intelligent control [16], [17]. These approaches improve the control performance of the motor from different aspects.

Model predictive control (MPC) is one of the most practical advanced control technique in industrial applications [18], [19], [20]. As the advanced development of computing hard­ware and the convex optimization technique, it is possible to implement MPC controller in real time for the fast-varying dynamical systems, such as power electronics converters and adjustable speed drives. For example, some cheering research reports on the application of MPC method to PMSM servo system [21], [22], [23] show that the MPC method can ensure a satisfying closed loop performance.

In many cases, the computation of common MPC method may still be a burden for practical systems due to economic considerations and hardware limits. Thus a simplified MPC method, i.e., predictive functional control (PFC) method, de­veloped by Richalet [24], may be regarded a solution to this problem. While retaining the benefits of MPC method such as online optimization and constraint handling, PFC method produces an algorithm with a low online computation burden. It achieves computational simplicity by using simpler but more intuitive design guidelines.

Note that in real industrial applications, systems always face with different internal disturbances and external disturbances. In common design of prediction model based control method, usually disturbances are not considered in the prediction model as well as the control design. In the presence of strong disturbances, it is difficult for the prediction model to precisely predict the future outputs. So the future control action thus obtained by optimizing a certain performance index can not react well to resist the disturbances. As shown in [12], this may cause a degradation of the closed-loop control performance in the presence of strong disturbances.

To this end, in [12], a composite controller is developed for the speed loop of PMSM system where a predictive functional feedback control term is combined with a feed­forward compensation term based on the estimated value of lumped disturbances by using an extended state observer. The simulation and experimental results show that the dis­turbance rejection ability is improved. However, since the final speed controller is a combination of both feedback and feed-forward items, one of the advantages of for PFCIMPC feedback methods is lost there, i.e., the capability of handling constraints of manipulated variables. Also the influence of disturbances is not considered in the prediction model which degrades the prediction precision in the presence of strong disturbances. Another thing is that this composite control method is only available for the matched disturbance case. When the disturbances are not in the same channel with control input (so called mismatched case), the control method is not available again.

In this paper, an improved PFC method is developed for

PMSM speed regulation system by embedding disturbance information into the prediction model. Here, a composite prediction model is obtained by introducing the estimated value of disturbances, where disturbance observer (DOB) is employed to estimate the disturbances. So the influence of disturbances on system are introduced in the optimization pro­cedure. Detailed 1MS320F2808 DSP experimental results are provided to verify the effectiveness of the proposed method.

II. IMPROVED PREDICTIVE FUNCTIONAL CONT ROL

MET HOD FOR PMSM

The speed dynamics of PMSM servo system are expressed as:

(I)

where W angular velocity, 'iq q-axis stator currents, np number

of pole pairs, 'ljJf is the flux linkage, Kt = 3n,;Wf torque con­stant, J moment of inertia, d = iq -i� - J!; - �� the lumped disturbances including external disturbances, internal friction and current tracking errors, B viscous friction coefficient and TL load torque,.

A. Disturbance observer design

A disturbance observer (DOB) provides an adequate way to estimate the disturbances for (1), by setting

Q(s) = �; Gn(s) = bos; Y =w; u == i�.

where g > 0 and bo = kt• Substituting (2) to (1), we have:

d = (bosw - i*) -g-= d-g-q s +g s +g

(2)

(3)

From (3), if g -+ 00, then d -+ d, i.e., the observation can track the actual disturbance approximately.

Based on the first equation of (3), the discrete output of disturbance observer is obtained as follows

d(k) = d(k-l)+gbo [w(k )-w(k-l)]-gT( i� (k-l)+d(k-l)) (4)

B. Composite PFC design

(I) Base function Considering (1), the base function is selected as step re­

sponse.

i�(k +i)=IL1' i=I,2,··· ,P. (5)

where ILl is the coefficient to be computed during the opti­mization of the performance index, P the optimization hori­zon.

(2) Prediction model The discrete-time model of (1) is

wm (k + 11k) = amWm (k) + Kmi� (k) + Km d(k) (6)

where d(k) is the estimation of d, am = e-BT,/ J and Km = T, �I (1 - am) I e -B AI J dA, Ts is the sample time, Wm is the o

output of predictive model. At (k + P)Ts:

Wm (k + Plk) = a;;'wm (k) + Km (1 + ... + a;;'-l)i� (k) + Km (1 + ... + a;;'-l) d(k)

(7) (3) Error correction

e(k + P - 1) = ... = e(k + 1) = e(k) = w(k) - wm(k), (8)

where W (k) is the actual output of system. (4) Reference trajectory A first-order exponential reference trajectory is used here.

wr(k + i) = w*(k + i) - a�[w*(k) - w(k)], i = 1,2"" , P, (9)

where Wr is the reference trajectory, w* is the set-point value, T

ar = e - r; is the reference time constant, and Tr is the desired response time of closed loop system.

(5) Cost function The quadratic performance index is set as:

J(PX1) = IIWr (k) - Wm (k) - E (k)112Q + Ri�2. (10)

2 wherellXl1 A = XT AX, Wr(k)(PX1) = [wr(k + 1), ... , wr(k + p)]T; Wm(k)(PX1) = [ wm(k), ,wm(k + P) ]T ; E(k)(PX1) = [ e(k + 1), ... , e(k + p)]T;

. [ 2 2 ] Q(PXP) = dwg Q1, ... , Qp ; R(lX1) = r2. Finally, by setting l{� = 0, the controller is obtained as

follows

i� = (W!QW3 +R)-lW!Q (W1 (w* - w) + W2wm - w3 d ) where W1(k)PX1 = [ (1 - ar), , (1 - an f; W2(k)PX1 = [ (1 - am), , (1 - a;;') f;

(11)

W3(k)PX1 = [ Km, ,Km(1 + ... + a;;'-l) f. The diagram of speed controller can be seen in Fig. 1. Remark l:Here we give another solution is to directly add

feed-forward compensation term on the obtained predictive model feedback term when designing a speed controller for PMSM system, i.e., PFC+DOB:

i�l = (W!QW3 + R) -lW!Q (W1 (w* - w) + W2wm) - d (12)

L ____________________ �

Fig. 1. Diagram of PMSM speed regulation system based on composite PFC method.

III. EXPERIMENTAL RESULTS

In order to verify the robustness of the controllers, the PMSM runs at speed reference signal as 2000rpm. A compari­son of performance indices on the two methods are also shown in Table I. The comparison experiment results are shown in Fig.2. Tests have been performed to evaluate the performances of the proposed control system under dynamic, steady state, and sudden load disturbance impact. There are some indexes introduced in this work, such as overshoot (os %), settling time (ts), speed fluctuation (/r-pm), speed fluctuation rate (%), standard deviation (/rpm) of steady speed and speed decrease caused by the load torque TL = 1.2N . m.

From these results, it shows that the composite PFC method has an overall better performance improvement than that of the PFC+DOB method.

TABLE I COMPARISON OF PERFORMANCE INDICES (2000rpm).

�me I ndex

Composite PFC PFC+DOB

os(%) 3.58 :l.60

t, I (ms) 116 120

speed fluctuation/(lpm) 7.76 7.HH

speed fluctuation rate(%) 0.39 0.39

standard deviation/(rpm) 3.94 4.08

speed decrease/(rpm) 90 9H

IV. CONCLUSIONS

In this paper, an improved PFC method has been developed for PMSM speed-regulation system by embedding disturbance information into the system model. A composite prediction model has been obtained by introducing the estimated val­ue of disturbances, where disturbance observer (DOB) has been employed to estimate the lumped disturbances. So the influence of disturbances on system has been taken into account in optimization procedure. Experimental results based on TMS320F2808 DSP have been provided to verify the effectiveness of the proposed method.

REFERENCES

[I] G. J. Wang, C. T. Fong, K. J. Chang. "Neural-network-based self tuning PI controller for precise motion control of PMAC motors", IEEE

Transactions on Industrial Electronics, vol.48, no.2, pp. 408-415, 2001.

[2] H. H. Choi, N. T.-T. Vu, 1.-W. Jung. "Digital implementation of an

adaptive speed regulator for a PMSM", IEEE Transactions on Power

Electronics, vol. 26, no. I, pp. 3-8, 2011.

[3] C. J. Xi a, X. C. Wang, S. H. Li, and X. S. Chen. "Improved integral sliding mode control methods for speed control of PMSM system", International Journal of Innovative Computing, InjiJrmation and Control,

vol. 7, no. 4, pp. 1971-1982,2011.

[4] 1. C. Baik, K.-H. Kim, and M. J. Youn. "Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode

control technique", IEEE Transactions on Control Systems Technology,

vol. 8, no. I, pp. 47-54, 2000.

[5] S. H. Li, M. M. Zhou, X. H. Yu. "Design and implementation of terminal sliding mode control method for PMSM speed regulation system", IEEE

Transactions on Industrial Inf(Jrmatics, 2013, published online

[6] S. H. Li, and Z. G. Liu. "Adaptive speed control for permanent magnet synchronous motor system with variations of load inertia", IEEE Trans­

actions on Industrial Electronics, vol. 56, no. 8, pp. 3050-3059, 2009.

[7] H. H. Choi, N. T.-T. Vu and J.-w. Jung. "Digital Implementation of an Adaptive Speed Regulator for a PMSM, IEEE Transactions on Power

Electronics, vol. 26, no. I, pp. 3-8, 2011

[8] S. H. Li, and H. Gu. "Fuzzy adaptive internal model control schemes for PMSM speed-regulation system", IEEE Transactions on Industrial

Infi)rmatics, vol. 8, no. 4, pp. 767-779, 2012

[9] F. F. M. El-Sousy."Hybrid H-infinity-based wavelet-neural-network track­ing control for permanent-magnet synchronous motor servo drives", IEEE Transactions on Industrial Electronics, vol. 57, no. 9, pp. 3157-3166, 2010.

[10] Y. Luo, Y. Q. Chen, H.-S. Ahnc and Y. G. Pi. "Fractional order robust control for cogging effect compensation in PMSM position servo systems:

stability analysis and experiments", Control Engineering Practice, vol. 18, no. 9, pp. 1022-1036, 2010.

[11] S. H. Li, H. X. Liu and S. H. Ding. "A speed control for a PMSM using finite-time feedback control and disturbance compensation", Transactions

of'the Institute of'Measurement and Control, vol. 32, no. 2, pp. 170-187, 2010.

[12] H. X. Liu, S. H. Li, "Speed control for PMSM servo System using predictive functional control and extended state observer", IEEE Trans­

actions on Industrial Electronics, vol. 59, no. 5, pp. 1171 - 1183, Feb.

2012.

[13] P. Cortes, M. P. Kazmierkowski, R. M. Kennel, D.E. Quevedo, and 1. Rodriguez. "Predictive control in power electronics and drives", IEEE

Transactions on Industrial Electronics, vol. 55, no. 12, pp. 4312-4324, 2008.

[14] S. H. Li, C. J. Xia, and X. Zhou. "Disturbance Rejection Control Method for Permanent Magnet Synchronous Motor Speed-regulation System", Mechatronics, vol. 22 no. 6, pp. 706-714, 2012.

[15] K. H. Kim, 1. C. Baik, G.-w. Moon and M.-J. Youn. "A current control for a permanent magnet synchronous motor with a simple disturbance

estimation scheme", IEEE Transactions on Control Systems Technology,

vol. 7, no.5, pp. 630-633, 1999.

[16] Y. S. Kung, and M. H. Tsai. "FPGA-based speed control IC for PMSM drive with adaptive fuzzy control", IEEE Transactions on Power

Electronics, vol. 22, no. 6, pp. 2476-2486, 2007.

[17] F. J. Lin, and C. H. Lin. "A permanent magnet synchronous motor servo drives using self-constructing fuzzy neural network controller", IEEE

Transactions on Energy Conversion, vol. 19, no. I, pp. 66-72, 2003.

[18] 1. M. Maciejowski, "Predictive control with constraints", Prentice Hall, London, Jun. 2001.

[19] W. H. Chen, D. 1. Ballance, and P. J. Gawthrop, "Optimal control of

nonlinear systems: a predictive control approach", Automatica, vol. 39,

no. 4, pp. 633-641, Apr. 2003.

[20] M. Rivera, Y. Yaramasu, J. Rodriguez, B. Wu, "Model predictive current control of two-level four-leg inverters-part IT experimental implementation

and validation", IEEE Transactions on Power Electronics vol. 28, no. 7, pp. 3469-3478, July 2013.

[21] F. Morel, X. F. Lin-Shi, J. M. Retif, B. Allard, and C. Buttay, "A comparative study of predictive current control schemes for a permanent­magnet synchronous machine drive", IEEE Transactions on Industrial

Electronics, vol. 56, no. 7, pp. 2715-2728, Jul. 2009.

[22] S. Bolognani, S. Bolognani, L.Peretti, and M. Zigliotto, "Design and implementation of model predictive control for electrical motor drives", IEEE Transactions on Industrial Electronics, vol. 56, no. 6, pp. 1925-1936, Jun. 2009.

[23] S. Chai, L. Wang, E. Rogers, "A cascade MPC control structure for PMSM with speed ripple minimization", IEEE Transactions on Industrial

Electronics, online, 2013 [24] J. Richalet, "Industrial applications of model based predictive control",

Automatica, vol. 29, no. 5, pp. 1251-1274, Sept. 1993.

[25] R. Krishnan, "Electric motor drives: modeling, analysis, and control", Unite States: Prentice Hall. 2001.

[26] K. Ohnishi. "A new servo method in mechatronics", Transaction of'the

Japanese Society of' Electrical Engineers, 107-D, pp.83-86, 1987. [27] L. Guo, W. H. Chen. "Disturbance attenuation and rejection for a class of

nonlinear systems via DOBC approach", International Journal of'Robust and Nonlinear Control, vol. IS, no. 3, pp.109-125, 2005.

[28] X. J. Wei, H. F. Zhang, L. Guo. "Composite disturbance observer based control and variable structure control for nonlinear systems with disturbances", Transactions of'the Institute of'Measurement and Control,

vo1.31, no.5, pp.401-423, 2009.

[29] M. Chen, W. H. Chen. "Sliding mode control for a class of uncertain nonlinear system based on disturbance observer", International Journal of'Adaptive Control and Signal Processing, vo1.24, no.I, pp.51-64, 2010.

[30] L. Olivier, 1. Craig, Y. Chen. "Fractional order and BICO disturbance observers for a run-of-mine ore milling circuit", Journal of' Process

Control, vol. 22, no. I, pp. 3-10, 2012.

2000

1500 E e-:0 Q) 1000 Q) c. Cf)

500

0

2020

2010

0

-,-, -,

0.1

� 2000 :0 Q) Q) c. Cf)

1980

1970 1.7

E e- 1960 :0 Q) Q) c. 1940 Cf)

1920

1900 0.8

2000

1980 E e- 1960 ......

'0 Q) Q) c. 1940 Cf)

1920

1900 0.8

,-,-,-,-,-,

--- Composite PFC PFC+ DOB

0.2 0.3 0.4 0.5 Vs

(a)

--- Composite PFC PFC+ DOB

1.8 1.9 2 tis (b)

--- Composite PFC I 1.2 1.4 1.6 1.8

Vs (c)

I -- PFC + DOB I 1.2 1.4 1.6 1.8

Vs (d)

Fig. 2. Experimental speed responses at 2000rpm. (a) Dynamic response. (b) Steady state response.( c) Composite PFC controller in the case of load disturbance. (d) PFC + DOB controller in the case of load disturbance.