EXPLICIT MODEL PREDICTIVE CONTROL OF A PERMANENT MAGNET SYNCHRONOUS MOTOR DRIVE M. T. Cychowski†, R. Nalepa‡ & T. O’Mahony† † ‡
Cork Institute of Technology, IRELAND. Moog Ltd., Co. Cork, IRELAND.
ABSTRACT This paper considers the application of an explicit approach to the design of model predictive control (MPC) for a permanent-magnet synchronous motor (PMSM) drive. The resulting explicit MPC controller achieves the same level of performance as the conventional MPC, but requires only a fraction of the real-time computational machinery, thus leading to fast and reliable implementation. The results are compared to the current state of the art demonstrating the potential for notable performance improvements. Keywords: permanent magnet motor, model predictive control, explicit solutions.
INTRODUCTION IN recent years, advancements in magnetic materials, semiconductor power drives, and control theories have made the permanent-magnet synchronous motor (PMSM) drive play a vitally important role in motioncontrol applications in the low-to-medium-power range. Relative to the induction motor the AC PM motor offers the advantages of high-power density, high efficiency and improved dynamic performance [1]. In addition, the maintenance of AC PM motors is minimal because of the brushless rotor construction and its decoupling control performance is far less sensitive to the parameter variations of the motor [2]. However, the torque ripple generation in AC PM motor systems limits the applications of AC PM motors in high-performance speed and position control systems. Traditionally, the proportional plus integral (PI) controller has been utilized to achieve fast four-quadrant operation, smooth starting, and acceleration in the design of the PMSM drives [3, 4]. With the current availability of relatively cheap digital signal processors there has been renewed research interest in more advanced control techniques to achieve reduced torque ripple, greater efficiency, insensitivity to parameter variations and load disturbances – albeit at the expense of higher controller complexity. Proposals in the advanced category include, but are not limited to, robust LQ state feedback [5], fuzzy logic control [4], sliding mode control [2], adaptive control [6], and intelligent techniques [7]. This paper considers the application of an alternative control scheme, model predictive control (MPC), to the problem of enhancing the performance of permanent magnet synchronous motors. The MPC paradigm is widely extolled in the petrochemical, and to a lesser extent, the process industries because constraints can be
implicitly included in the problem formulation. At each sampling time, based on current measurements, MPC uses an explicit process model and information about operating and physical constraints to compute optimal process inputs so as to optimize future plant behaviour over a prediction horizon. The first portion of this optimal control sequence is then implemented to the plant and the procedure is repeated at the next sampling time [8]. The control objectives are directly expressed in an objective function. Using a 2-norm objective function and linear inequality constraints, leads to a quadratic programming (QP) problem. Despite the fact that QP problems can be solved efficiently using offthe-shelf solvers, the computational effort required for the implementation of the MPC algorithms on-line can be quite prohibitive for many real-time applications. In Bemporad et al. [9] it was recognized that the constrained linear MPC problem is a multi-parametric quadratic program (mp-QP), when the state is viewed as a parameter to the problem. They show that the solution (the control input) has an explicit representation as a piecewise affine (PWA) function defined on a (polyhedral) partition of the state space and develop an algorithm to compute this function. The main motivation behind the explicit MPC is that the on-line optimization simply reduces to a function evaluation problem. The explicit MPC is therefore potentially useful for applications with fast sampling where MPC has not traditionally been used. While considerable advances have been made on the theoretical front, few real-time industrial applications have been reported. The main contribution of this paper is therefore to develop a novel advanced control algorithm, explicit model predictive control, for an AC permanent magnet synchronous motor. While the primary motivating factor for considering this research problem is to demonstrate that the model
predictive control paradigm can be applied to systems with fast dynamics, the authors also believe that MPC offers a number of attractive features for the control of AC drives. It is recognized that standard predictive current control schemes for ac drives [10] offer the benefits of enhanced static and dynamic performance but at the cost of sensitivity to model uncertainty and increased real-time computational complexity. The predictive control formulation advocated in this paper surmounts both of these disadvantages. In MPC the prediction is performed over several samples which results in reduced sensitivity to plant-model mismatch [8] while the explicit nature of this algorithm implies that the real-time controller can be implemented as a simple look-up table. These considerable benefits are the main reasons for adopting the MPC framework.
MODEL PREDICTIVE CONTROL In this section, the model predictive strategy will be briefly introduced (see e.g. [8] for a comprehensive overview). MPC refers to a class of computer control algorithms that utilize an explicit process model to predict the future response of a plant. In the research literature, a linear discrete-time state-space model is typically assumed: x(k + 1) = Ax(k ) + Bu ( k ),
(1)
where x(k ) ∈ \ n and u (k ) ∈ \ n denote the state and control input, respectively. At each control interval an MPC algorithm attempts to optimize future plant behavior by solving the following optimization problem [8]: Hp
min u (⋅)
∑ ||x(i) − r (i)||
2 Q
i =1
Hu
+ ∑ ||u (i ) − u (i − 1)|| , i =0
2 R
(2a)
(2b) (2c)
where x(i ), u (i ) and r(i) denote respectively the values of the state, control and reference trajectories at a future time i, and Q ≥ 0 and R > 0 are the weighting matrices. In addition, Hp, Hu denote the prediction and control horizons, respectively and the norm in (2a) is a standard weighted Euclidean norm, i.e. ||w||Q = wT Qw.
Let {u o (i )}iH= 0 be the minimizing sequence of (2a) subject to system dynamics (1) and process constraints (2b), (2c). The MPC algorithm proceeds by sending only the first input u o (0) of the optimal sequence to the plant, and the entire calculation is repeated at the next u
Because problem (2) depends on the initial state x(0), the implementation of the MPC algorithm can be performed either by solving a quadratic program on-line for a given x(0), or as shown by Bemporad et al. [9], by solving problem (2) off-line for all x(0) within a polyhedral set of values using multi-parametric quadratic programming. Parametric programming techniques systematically subdivide the parameter space into characteristic regions where the optimal value and the optimizer are given as explicit functions of the parameters. These methods result in PWA functions represented as ⎧ F1 x + g1 , if x ∈ X 1 ⎪⎪ F x + g 2 , if x ∈ X 2 u ( x) = ⎨ 2 # ⎪ ⎪⎩ FN x + g N , if x ∈ X N
(3)
with a polyhedral partition P = { X 1 ,..., X N } where the polyhedral sets are represented by linear inequalities X i = {x ∈ \ n |H i x ≤ di ,} for i = 1,..., N . The explicit MPC controller is completely characterized by the following set of parameters: {Fi , g i , H i , d i }iN=1 . As a result, the implementation of the MPC control law amounts to the evaluation of the PWA mapping (3) in the control unit without the need of expensive numerical optimization. MOTOR MODEL
The machine model of a PMSM can be described in the d-q reference frame as follows [11], [12]: did (t ) − ω e (t ) Lq iq (t ), dt diq (t ) + ωe (t )( Ld id (t ) + ψ m ), vq (t ) = Rs iq (t ) + Lq dt 3p Te (t ) = [ψ m iq (t ) + ( Ld − Lq )iq (t )id (t )] 2 dω m (t ) J = Te (t ) − Tl (t ) − ω m (t ) B, dt vd (t ) = Rs id (t ) + Ld
subject to x(0) = x(k ) and umin ≤ u (i ) ≤ umax , ∀i ∈ {0,..., H u }, xmin ≤ x(i ) ≤ xmax , ∀i ∈ {1,..., H p },
time step based on new measured (or estimated) state x(k+1).
(4a) (4b) (4c) (4d)
where the parameters and variables have the following meanings: Rs Ld, Lq p J B
ψm
vd(t), vq(t)
stator resistance, d-q frame stator inductances, number of poles, moment of inertia, viscous damping, constant magnetic flux, d-q frame stator voltages,
VDC
ωref
velocity compensator
iqref idref
q-axis current compensator
vq
d-axis current compensator
vd
cross coupling terms
inv. Park tr. d,q
vα vβ
Space Vector PWM
3 - Phase Inverter
α,β
td tq
iq d,q
iα α,β
iA
id
iβ
iB
α,β Park tr.
A,B,C Clarke tr. VA VB VC PMSM
Θe
ω
position & speed
sensors signals
Figure 1 Closed-loop cascade control system of the PMSM.
id(t), iq(t) Te(t) Tl(t) ωm(t) ωe(t)
d-q frame stator currents, electromagnetic torque, load torque, rotor mechanical velocity, rotor electrical velocity,
The configuration of a conventional field-oriented drive system [12] with cascade current and velocity control is depicted in Figure 1. The PMSM used in this drive system is a three-phase twelve-pole machine, the nominal parameters of which are reported in Table 1. For the continuous-time model (4), a linear discretetime model corresponding to a sampling interval Ts = 0.25 msec is obtained by applying the exact discretization using zero-order hold and assuming id(t)=0 in (4c) according to the vector control policy [13]. Furthermore, in order to avoid the steady-state offset of the MPC controller, the PMSM model (4) is augmented with additional states x1, x2, and x3 which take into account the integral error [8]. The discretetime model of a PMSM then becomes: ⎡ id (k +1) ⎤ ⎡ 0.8821 0 ⎤ ⎡ id ( k ) ⎤ ⎡0.0462 ⎤ v ( k ), + ⎢ x1 (k +1) ⎥ = ⎢⎣ 0 1 ⎥⎦ ⎢⎣ x1 (k ) ⎥⎦ ⎢⎣ 0 ⎥⎦ d ⎣ ⎦
|vd , q | ≤ 187.63 (V),
|id , q | ≤ 12.7 (A),
|Te | ≤ 3.85 (Nm),
|ω e | ≤ 600 (rad/sec).
(9) (10)
It should be emphasized that the design of PI compensators for the PMSM will be based on the discrete-time equivalent of (4), while the discrete-time model (5)-(8) is build only for the purposes of constructing the explicit MPC controllers, as will be shown in the next section.
SIMULATION RESULTS
(5)
⎡ iq (k +1) ⎤ ⎡ 0.8816 0 ⎤ ⎡ iq (k ) ⎤ ⎡ 0.0464 ⎤ + v (k ), (6) ⎢ x (k +1) ⎥ = ⎢ 0 1 ⎥⎦ ⎢⎣ x2 ( k ) ⎥⎦ ⎢⎣ 0 ⎥⎦ q ⎣ 2 ⎦ ⎣ ⎡ω m ( k +1)⎤ ⎡0.999 −2476⎤ ⎡ω m ( k )⎤ ⎡1.562⎤ T (k ) (7) + ⎢ x3 (k +1) ⎥ = ⎢⎣ 0 1 ⎥⎦ ⎢⎣ x3 ( k ) ⎥⎦ ⎢⎣ 0 ⎥⎦ e ⎣ ⎦ Te ( k ) = 0.303iq ( k ),
Table 1 Machine data Max DC bus voltage 325 V Number of poles: p 12 Stator resistance: Rs 2.55 Ω d-axis inductance: Ld 5.08 mH q-axis inductance: Lq 5.06 mH 0.0337 V sec/rad Magnetic flux linkage: ψm Motor inertia: J 0.00016 kg m2 Viscous damping: B 0.0001 kg m2/s Nominal max speed 5800 rpm
(8)
The following constraints, which correspond to physical limitations of the motor, are imposed on (5)–(8):
The explicit MPC approach described in Section 2 is applied to design the current and velocity compensators for the PMSM drive. The controller minimizes the cost function (2a) subject to plant dynamics (5)-(8) and the motor constraints (9)-(10). In (2a), the horizons and the weighting matrices are chosen as: Curent loop:
H p = H u = 2, Q = I , R = 0.0001,
Velocity loop: H p = 10, H u = 2, Q = I , R = 10. Computing the explicit control laws for the current and velocity loops yields four-dimensional partitions of 19
400
50
300
Velocity [rad/sec]
Velocity [rad/sec]
40 30 20
no load
half load
200
no load
100
10 0
0 0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.02
0.04
Time [sec] 4
3
3
Torque [Nm]
Torque [Nm]
0.08
0.1
0.12
0.14
0.16
0.1
0.12
0.14
0.16
0.1
0.12
0.14
0.16
2
2 1 0 -1 -2
1 0 -1 -2 -3
-3
-4
-4
-5 0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.02
0.04
Time [sec] 200
200
150
150
100
100
50
vd(t)
0
vq(t)
-50
50
-150
-150 0.04
0.06
0.08
0.1
0.12
Time [sec]
vd(t) vq(t)
-50 -100
0.02
0.08
0
-100
0
0.06
Time [sec]
Stator voltages [V]
Stator voltages [V]
0.06
Time [sec]
4
-200
half load
-200
0
0.02
0.04
0.06
0.08
Time [sec]
Figure 2 Closed-loop simulation of the PI (dashed) and MPC (solid) schemes in the low-velocity operating region.
Figure 3 Closed-loop simulation of the PI (dashed) and MPC (solid) schemes in the high-velocity operating region.
and 71 polyhedral regions, respectively. In Figure 4 and Figure 5, a two-dimensional cut through the polyhedral partitions of both controllers is depicted for illustration.
Table 2 Industrial PI controller settings d-axis q-axis velocity Proportional gain 17.8 25.4 0.3 Integral gain 9.3 3.3 30
The performance of the proposed explicit MPC strategy has been compared to the existing industrial PI control scheme, the settings of which are reported in Table 2. Figure 2 and Figure 3 demonstrate the closed-loop simulation results of both controllers corresponding to the low- and high-velocity operating conditions. Initially, the motor is running in no-load mode. After the first cycle has elapsed, a load torque of -2 (Nm) is applied and the procedure repeated. It can be seen from
Figure 2 that smooth tracking and superior regulatory behavior is achieved with the proposed strategy. On the other hand, long settling times and slight overshoots are experienced with the PI control, particularly, in the presence of sudden load changes. In the high-velocity operating region, the discrepancy in velocity responses between both controllers is almost indistinguishable, but still significantly larger torque ripple generation is
Figure 4 State-space partition of the explicit MPC current compensator. experienced with the PI-based compensation scheme. The on-line computational complexity of the resulting explicit MPC controller mainly comes from the sequential search in a look-up table (3). In the worstcase scenario (all regions are inspected), the control law of the considered example is found after a total of 7568 arithmetic operations (946 comparisons, 3784 multiplications and 2838 sums). Alternatively, binary search techniques proposed in [13] can be used to accelerate the evaluation of the explicit MPC control law. Using this method, the on-line computational burden of the proposed controller can be reduced to a total of 240 arithmetic operations per sample making the approach particularly useful in low-cost control units.
CONCLUSIONS This paper addresses the design and validation of explicit model predictive control for the permanent magnet synchronous motor. The approach demonstrates the potential for improving the performance of PMSM drives with respect to state of the art industrial control strategies, as the comparison with PI control emphasizes. Results also indicate that new advanced control algorithms such as MPC can be successfully applied to PMSM drives with sampling times in the micro-second scale, and implemented in low-cost processing units with no floating point arithmetic capacity.
Figure 5 State-space partition of the explicit MPC velocity compensator. [4] Z. Ibrahim and E. Levi, “A comparative analysis of fuzzy logic and PI speed control in high-performance AC drives using experimental approach”, IEEE Trans. on Industry Applications, Vol. 38, pp. 1210–1218, 2002. [5] K.-K. Shyu, C.-K. Lai, Y.-W. Tsai, and D.-I. Yang, “A Newly Robust Controller Design for the Position Control of Permanent-Magnet Synchronous Motor”, IEEE Trans. on Industrial Electronics, Vol. 49, pp. 558–565, 2002. [6] T.-S. Lee, C.-H. Lin, and F.-J. Lin, “An adaptive H∞ controller design for permanent magnet synchronous motor drives”, Control Eng. Practice, Vol. 13, pp. 425– 439, 2005. [7] F.-J. Lin, and C.-H. Lin, “A permanent-magnet synchronous motor servo drive using self-constructing fuzzy neural network controller”, IEEE Trans. On Energy Conversion, Vol. 19, pp. 66–72, 2004. [8] J. M. Maciejowski, “Predictive control with constraints”, Prentice Hall, UK, 2002. [9] A. Bemporad, M. Morari, V. Dua and E. N. Pistikopoulos, “The explicit linear quadratic regulator for constrained systems”, Automatica, Vol. 38, pp. 3–20, 2002. [10] H. Le-Huy, K. Slimani, and P. Viarouge, “Analysis and implementation of a real-time predictive current controller for permanent-magnet synchronous servo drives”, IEEE Trans. On Industrial Electronics, Vol 41, pp. 110–117, 1994. [11] J.F. Moynihan, “Aspects of Digital Current Control for AC Drives”, PEI Technologies, 1996. [12] P. Vas, Electrical Machines and Drives: A Space-Vector Theory Approach. Oxford, U.K.: Clarendon, 1992. [13] P. Tondel, T.A. Johansen, and A. Bemporad, “Evaluation of piecewise affine control via binary search tree”, Automatica, Vol. 39, pp. 945–950, 2003. [14] G. O’Donovan, Personal consultations. MOOG Ltd., Ireland, 2005.
REFERENCES [1] J. K. Gieras, and M. Wing, Permanent Magnet Motor Technology: Design and Applications. New York: Marcel-Dekker, 1997. [2] F. J. Lin, R. F. Fung, and Y. C. Wang, “Sliding mode and fuzzy control of toggle mechanism using PM synchronous servomotor drive”, Proc. IEE—Control Theory Applications, Vol. 144, pp. 393–402, 1997. [3] L. Harnefors, and H-P Nee, “Model-based current control of AC machines using the Internal Model Principle”, IEEE Trans. on Ind. Appl., Vol. 34, pp. 133–141, 1998.
AUTHOR’S ADDRESS The corresponding author can be contacted at Department of Electronic Engineering Cork Institute of Technology Rossa Avenue, Cork, IRELAND. E-mail:
[email protected]
