IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 7, JULY 2012
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Robust Nonlinear Predictive Controller for Permanent-Magnet Synchronous Motors With an Optimized Cost Function Rachid Errouissi, Mohand Ouhrouche, Senior Member, IEEE, Wen-Hua Chen, Senior Member, IEEE, and Andrzej M. Trzynadlowski, Fellow, IEEE
Abstract—A robust nonlinear predictive controller for permanent-magnet synchronous motors is proposed. The nonlinear predictive control law is formulated by optimizing a novel cost function. A key feature of the proposed control is that it does not require the knowledge of the external perturbation and parameter uncertainties to enhance the robustness. A zero steadystate error is guaranteed by an integral action of the controller. The stability of the closed-loop system is ensured by convergence of the output-tracking error to the origin. The proposed control strategy is verified via simulation and experiment. High performance with respect to speed tracking and current control of the motor has been demonstrated. Index Terms—Digital signal processors, nonlinear optimization problem, permanent-magnet synchronous motor (PMSM), robust nonlinear predictive control (RNPC).
I. I NTRODUCTION
D
URING recent years, considerable attention has been paid to the permanent-magnet synchronous motors (PMSMs). They are characterized by high values of efficiency, power density, and the torque-to-current ratio. PMSMs are used widely in many variable-speed drive applications such as robotic actuators, machine tools, and automobiles. Several control methods for PMSMs have been proposed in the literature, but still, many issues remain. This is due to the fact that the PMSM is subjected usually to unknown disturbances and it constitutes a nonlinear multivariable system whose parameters change during operation. In order to improve the performance of the PMSM, various robust control techniques have been developed. The widely
Manuscript received December 10, 2009; revised August 2, 2010 and February 7, 2011; accepted April 19, 2011. Date of publication May 19, 2011; date of current version February 17, 2012. This work was supported by the Natural Sciences and Engineering Research Council of Canada. R. Errouissi and M. Ouhrouche are with the Department of Applied Sciences, University of Quebec, Chicoutimi, QC G7H2B1, Canada (e-mail:
[email protected];
[email protected]). W.-H. Chen is with Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough LE11 3TU, U.K. (e-mail:
[email protected]). A. M. Trzynadlowski is with the Department of Electrical and Biomedical Engineering, University of Nevada, Reno, NV 89557-0260 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2011.2157276
used approach is based on the use of linear control theory with an estimator of the disturbance [1]–[3]. To take into account the nonlinearities of the PMSM, other approaches have been endeavored, such as nonlinear control [4] and sliding mode control [5]. Recently, many approaches have been developed in order to improve the robustness of the PMSM control system under parameter uncertainty and load perturbation. An adaptive control scheme is developed in [6] by considering that only the load inertia varies, which is identified using an extended state observer, whereas a fuzzy controller is designed to tune the feedforward compensation gain with the identified inertia. An adaptive approach can also be used to improve the robustness of the control scheme against all parameter variations [7], [8]. Other methods based on deadtime compensation have been adopted to ensure good performance. Notably, a simple disturbance observer is developed in [9] to estimate the voltage disturbance induced by the deadtime, and its effect is compensated by adding the estimated disturbance to the current control loop. Model predictive control (MPC) is now regarded as one of the most robust control strategies. Several variants of MPC have been proposed in the technical literature. They are based on the optimization of a cost function consisting of the difference between the actual output and the trajectory to be tracked [10]. Many applications have employed the discrete-time linear model (DTLM) for predictive control. It allows a fast analytical solution of the optimization problem. The predictive control of the PMSM based on the DTLM has been described in [11], where the load torque is considered as a known disturbance. Clearly, this is seldom true in practice. In [12], constrained MPC based on the DTLM for the PMSM is proposed, but the control strategy suffers from a heavy computational burden. In the case of slow nonlinear systems, a predictive control technique based on the DTLM can be used with satisfactory results. However, it is not applicable for systems having fast dynamics such as adjustable speed drives. More recently, various MPC schemes have been proposed for power converters, induction motors, and PMSMs [13]–[18]. Control theorists have made substantial efforts in applying predictive control to nonlinear systems with fast dynamics. Several nonlinear predictive control laws have been proposed to reduce the computational effort [19]–[26]. In [19] and [20], an optimal predictive control law for continuous-time systems has been presented. Chen [21] describes a nonlinear generalized
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predictive controller (NGPC) based on the Taylor series expansion for a multiple-input–multiple-output system (MIMO), where the control order is considered different from zero to analyze the stability of the closed-loop system when the input relative degree is higher than four. In [22], this method is extended to a MIMO system with various relative degrees. As pointed out in [23], high performance can be achieved with NGPC only when the external disturbance is known. Therefore, in order to guarantee the offset-free performance under external perturbation, the NGPC must be combined with a disturbance observer [23], [24]. The application of a disturbance observer is limited to nonlinear systems whose input and disturbance have the same relative degree. However, when the disturbance relative degree is less than the input’s one, which is the case for PMSMs, the design of the load-torque observer is nontrivial and the analysis of the closed-loop system becomes very complicated. In this paper, the performance index proposed in [25] and [26] is used, resulting in a novel robust nonlinear predictive control (RNPC) scheme. The cost function is based on the predicted integral action of the tracking error, and the controller is developed under the assumption that the system is free from any disturbance and mismatched parameters. It is shown that this leads to an integral action in the controller, which is exploited to improve the disturbance attenuation without using an offset observer. This paper is organized as follows: In Section II, the PMSM model is presented. Sections III and IV give the classical NPC and the proposed RNPC scheme applied to a PMSM, respectively. Sections V–VII give the simulation results, the description of the experimental setup, and the experimental results, respectively. Section VIII concludes this paper.
and f (x) is given by ⎤ ⎡ ⎡ f1 (x) ⎢ f (x) = ⎣ f2 (x) ⎦ = ⎣ f3 (x)
x(t) ˙ = f (x) + g1 u(t) + g2 b(t) i = 1, 2 yi (t) = hi (x),
(1)
u = [ud
uq ]T
(3)
(φv iq +(Ld −Lq )iq id )− B J ωr
(4)
The variables to be controlled are the components of the output vector y i.e., the rotor speed ωr and the d-axis component of the armature current id , which is forced by the control action to zero. III. C LASSICAL NPC FOR A PMSM A. Design of the Classical NPC The objective of the NPC is to find the best input such that the predicted plant output y(t + τ ) tracks a future reference trajectory yr (t + τ ) for 0 ≤ τ ≤ T in the presence of a perturbation, where T > 0 is a prediction horizon. This requirement is tantamount to the minimization of the cost function, as defined by =
1 2
T (yr (t+τ )−y(t+τ ))T (yr (t+τ )−y(t+τ )) dτ
(5)
0
where
y(t) = [y1 (t) y2 (t)]T = [id (t) ωr (t)]T yr (t) = [yr1 (t) yr2 (t)]T = [idr (t) ωrr (t)]T .
Since the dynamics of the current is faster than that of the speed, the predictive time T1 for the currents can be chosen different to the predictive time T2 for the rotor speed. As a result, the quadratic performance index given by (5) can be modified to T1 (yr1 (t + τ ) − y1 (t + τ ))2 dτ 0
x = [id iq ωr ]T y = [id ωr ]T
⎥ ⎦.
db = 0. dt
1 = 2
where
p J
⎤
In (2) and (3), R, Ld , and Lq , are the per-phase armature resistance and the d-axis and q-axis inductances, respectively; φv is the permanent-magnet flux, p is the number of pole pairs, J is the moment of inertia, and B is the coefficient of friction. The disturbance is assumed to change slowly, so that
II. M ATHEMATICAL M ODEL OF THE PMSM The mathematical model of the PMSM in the (d − q) rotor reference frame can be expressed in the bilinear form as in [3]
L
− LRd id + Ldq pωr iq φv pωr R d −Lq iq − L Lq pωr id − Lq
b = [fd fq fω ]T +
1 2
T2 (yr2 (t + τ ) − y2 (t + τ ))2 dτ.
(6)
0
and x, u, y, and b are the vectors of the states (the d-axis and q-axis components of the armature current and rotor speed), the inputs (d-axis and q-axis components of the armature voltage), the outputs (the d-axis component of the armature current and the rotor speed), and the disturbances that represent offsets caused by the parameter variations and the load torque TL . For a PMSM, g1 and g2 are defined as ⎡ g1 = [ gd
1 Ld
gq ] = ⎣ 0 0
⎤
⎡ 1 − Ld 1 ⎦ g =⎣ 0 2 Lq 0 0 0
0 − L1q 0
⎤ 0 0 ⎦ − J1
(2)
To solve the nonlinear optimization problem (6), the predicted output yi (t + τ ) and the predicted reference yri (t + τ ) are expanded into a ρi th-order Taylor series using the Lie derivative h(x) along a field of vectors f (x). Here, ρi represents the relative degree for each output. In the case of the outputs y1 (id ) and y2 (ωr ) of the PMSM, the following are given: ⎧ ⎨ y˙ 1 (t) = Lf h1 (x) + Lg1 h1 (x)u(t) + Lg2 h1 (x)b(t) y˙ 2 (t) = Lf h2 (x) + Lg h2 (x)b(t) ⎩ y¨ (t) = L2 h (x) + L 2 L h (x)u(t) + L L h (x)b(t). 2 g1 f 2 g2 f 2 f 2 (7)
ERROUISSI et al.: ROBUST NONLINEAR PREDICTIVE CONTROLLER FOR PMSMs WITH COST FUNCTION
Hence, the relative degrees of y1 and y2 are ρ1 = 1 and ρ2 = 2, respectively, and the relative degree of the PMSM outputs is ρ = ρ1 + ρ2 = 3.
(8)
The relative degree of the system is equal to the system’s order. Consequently, there is no zero dynamics. In addition, the currents and the speed are assumed to be available for measurement, which implies that the nonlinear system (1) is input–output feedback linearizable [27]. The necessary condition for the optimal control is given by d = 0. du
(9)
As in [22], using the Taylor series expansion to approximate output yi (t + τ ) and reference yri (t + τ ) and invoking (6) and (7) with (9) yields ⎡ 1 ⎤
1 i (i) L σ h (x)−y (t) +L b(t) 1 1 r1 f ⎢ i=0 i ⎥ ⎥ u(t) = −G1 (x)−1 ⎢ 2 ⎣ ⎦ (i) 2 i σi Lf h2 (x)−yr2 (t) +L2 (x)b(t) i=0
where
σ01 = σ02 =
(10)
3 2T1 ; 10 ; 3T22
σ11 = 1 σ12 = 2T5 2 ;
Matrix G1 (x) is defined by 1 Ld G1 (x) = p JLd (Ld −Lq )iq
p JLq
σ22 = 1.
0 (φv +(Ld −Lq )id )
(11)
q
B. Asymptotic Stability Analysis
The characteristic equation has the following poles: s1ω, 2ω =
t→∞
−1.25 ± j1.3307 . T2
(14)
0
IV. P ROPOSED RNPC S CHEME FOR THE PMSM The disadvantage of the previously described NPC is that it requires the knowledge of the disturbance. To overcome this drawback, it is wise to use an integral action in the controller when the disturbance is ignored in the synthesis of the controller. The integral action can be embedded in the controller by choosing a novel cost function as in [26] as follows: 1 = 2
T1
1 I1 (t + τ ) dτ + 2
T2
2
I2 (t + τ )2 dτ
(16)
0
where t Ii (t) =
ei (τ )dτ,
i = 1, 2
0
ei (τ ) = (yri (τ ) − yi (τ )) ,
i = 1, 2.
In this case, the predicted term Ii (t + τ ) is expanded into a (ρi + 1)th-order Taylor series expansion. Thus, the NPC minimizing the performance index (16) can be formulated as ⎡ t ⎤ 2
i−1 1 1 (i−1) y K e (τ)dτ + K (t)−L h (x) 1 i r1 f ⎢ 00 1 ⎥ i=1 ⎥ u(t) = G1 (x)−1⎢ t 3 ⎣ ⎦
i−1 2 2 (i−1) K0 e2 (τ)dτ + Ki yr2 (t)−Lf h2 (x) 0
In order to examine the asymptotic stability of the closedloop system, the stability of output-tracking errors at the origin must be investigated. Substituting (10) into (7) gives the characteristic equation of the closed-loop system as follows: 2 σ1 s + σ01 = 0 (13) σ22 s2 + σ12 s + σ02 = 0.
3 2T1
t→∞
0
In the controller (10), the speed reference must be twice differentiable. This can be accomplished by passing the reference speed signal through a second-order linear filter (see Appendix C).
s1d = −
As the predictive time is positive, the closed-loop system is stable asymptotically when the perturbation is known. It can also be shown that a small predictive time results in a fast control response. Even if the load torque is known, the model uncertainties are difficult to determine. This is why the disturbance should be estimated by an observer. However, as the relative degree of disturbance is less than that of the input, designing an estimator for the perturbation is nontrivial [23]. Furthermore, it can be shown that, when the disturbance is neglected in the controller, a steady-state error appears in the stable closed-loop system. Indeed ⎧ ⎨ lim e1 (t) = lim −L11b(t) σ0 t→∞ t→∞ (15) −L 2 (x(t))b(t) ⎩ lim e2 (t) = lim . σ2
N
and condition (φv + (Ld − Lq )id ) = 0 allows G1 (x) to be invertible. Matrices L1 and L2 (x) are given by − 1 0 0 L1 Ld = −p(Ld−Lq )iq p(φv+(Ld−Lq )id ) B σ12 . (12) L2 (x) JL JL J2 − J d
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where
K01 = K02 =
10 ; 3T12 21 ; 2T23
K11 = K12 =
i=1
5 2T1 ; 42 ; 5T22
(17)
K21 = 1 K22 =
7 2T2 ;
K32 = 1.
The controller given by (17) is easy to implement as the perturbation is not included. Furthermore, it should be noted that the obtained controller contains an integral action. Hence, if the closed-loop system is stable, the proposed controller eliminates the steady-state error despite the presence of the unknown perturbation (see Appendix A).
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A. Asymptotic Stability Analysis This section will investigate the asymptotic stability of the closed-loop system under controller (17). The main result is stated in the following theorem. Theorem: Consider the nonlinear system (1), and suppose that disturbance b(t) is bounded. Under the RNPC law (17), the output of the system tracks the desired output with an error converging to zero as t goes to ∞., i.e., lim e1 (t) = 0
t→∞
lim e2 (t) = 0.
t→∞
(18)
As the proposed controller contains an integral action, the result is established using the Barbalat lemma [28] (see Appendix B). B. Limitation of the Armature Current and the Control Effort It should be noted that the predictive time must be chosen to be very small to guarantee the validity of the Taylor series expansion. A smaller predictive time will increase the control effort and may amplify excessively the undesirable effects of the measurement noise. Increasing the predictive time to reduce the control effort may cause high oscillations to appear or, even worse, render the system unstable. This is because the approximation using the Taylor series expansion becomes invalid. In the literature, there have been many suggested ways of implementing the control limitation within the predictive control. Linear programming is used in [29] to give, at any instant, the smallest predictive time that does not saturate the control, and the state constraints are converted, i.e., at each sampling time, into additional control constraints. This strategy may cause heavy online computational burden, particularly for nonlinear systems with fast dynamics. On the other hand, when the trajectory to be tracked is changing very fast, the control input of form (17) may exceed its limit value, and the current will go above its maximum value during the transients. This phenomenon occurs, in particular, when large set-point changes are made, i.e., when the slope of speed reference is bigger than the maximum acceleration of the drive. As the d-axis and q-axis components of the armature current are the states and cannot be limited directly by saturation blocks for example, current limitation can be achieved by setting in an adequate way the speed reference to be tracked. V. C OMPUTER S IMULATIONS Fig. 1 shows the block diagram of the proposed RNPC for a PMSM. Computer simulations have been carried out using Matlab/Simulink software package to verify its performance and effectiveness. The predictive times T1 and T2 are set to 1 and 5 ms, respectively. The sampling time of the controller Tc is chosen to be equal 0.1 ms, whereas the sampling time Ts in the mathematical model of the PMSM is set to 0.001 ms. A. Comparison Between the RNPC and the Classical NPC The tracking performance was investigated under unknown load torque and parameter uncertainties. The load torque depends on the rotor speed, and its value is 0.25 Nm at 100 rad/s.
Fig. 1.
Block diagram of the proposed RNPC for a PMSM.
Fig. 2.
Speed response with classical NPC.
Fig. 3.
Speed response with RNPC.
The variation of the parameters was performed in the mathematical model of the PMSM at t = 0.5 s. The values of the moment of inertia, the viscous coefficient, the armature resistance, and the q-axis inductance were stepped up to 200%, whereas the value of the permanent-magnet flux was decreased by 20%. Fig. 2 illustrates the effect of the load torque and mismatched parameters on both the dynamic response and the steady-state error, when the disturbance is ignored in the classical NPC. A small steady-state error is observed in the speed tracking. However, it can be seen in Fig. 3 that the RNPC eliminates the steady-state error. Figs. 4 and 5 show the d-axis and q-axis components of the armature current with the classical NPC and the RNPC. It can be seen that, even when the parameters are changed, the d-axis component is maintained at zero when the RNPC is employed.
ERROUISSI et al.: ROBUST NONLINEAR PREDICTIVE CONTROLLER FOR PMSMs WITH COST FUNCTION
Fig. 4. d-axis and q-axis components of the armature current with NPC.
Fig. 5. d-axis and q-axis components of the armature current with RNPC.
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Fig. 7. Speed response with RNPC under slow reference trajectory.
Fig. 8. Armature phase current under fast reference trajectory.
Fig. 9. Armature phase current under slow reference trajectory. Fig. 6.
Speed response with RNPC under fast reference trajectory.
VI. L ABORATORY T EST S ETUP B. Implementation of the Armature-Current Limitation This test shows the effect of the speed-reference trajectory on the armature phase current during transients. Fig. 6 gives the speed response when the slope of the reference trajectory is chosen to be bigger than the maximum acceleration, whereas Fig. 7 shows the speed response when the dynamics of the trajectory is slow. Figs. 8 and 9 give the armaturecurrent waveforms corresponding respectively to the fast and slow dynamics of the speed-reference trajectory. As shown, the dynamic behavior of the overall system depends on the speed reference, and a good dynamic performance can be obtained with the proposed RNPC scheme. As shown in Fig. 8, it is clear that, if the dynamics of the speed-reference trajectory is chosen to be very fast, the peak value of the instantaneous armature current may exceed the maximum permissible during transients. Therefore, the speedreference trajectory must be chosen accordingly to keep the armature instantaneous current within the limits, as in Fig. 9.
The experimental test bed for testing the validity of the proposed controller scheme for a PMSM is given in Fig. 10. It consists of a ten-pole 0.25-kW 42-V PMSM coupled to a permanent-magnet direct-current generator, an insulated-gate bipolar transistor-based inverter, and a dSPACE DS1104 board. This board is a popular tool for the rapid prototyping of control systems and the hardware-in-the-loop applications. It is equipped with two processors: a Motorola MPC8240 processor with the PPC 603e core and on-chip peripherals and a Texas Instruments TMS320F240 digital-signal-processing controller. The first is acting as the master, and the second is acting as the slave. The motor speed is measured by an encoder, which can generate 1000 lines per revolution. The proposed RNPC algorithm was implemented in the main processor, with the sampling frequency of 10 kHz. The slave unit was dedicated to pulsewidth-modulation signal generation at the switching frequency of 50 kHz and the management of the input and output signals.
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Fig. 13.
Speed response under inaccurate value of rotor-flux linkage.
Fig. 14.
Speed response under an inaccurate value of armature resistance.
Fig. 15.
Speed response under an inaccurate value of the d-axis inductance.
Fig. 10. Experimental setup.
Fig. 11. Rotor speed trajectory tracking.
Fig. 12. d-axis and q-axis components of the armature current.
VII. E XPERIMENTAL R ESULTS The load torque depends on the rotor speed. Specifically, at 100 rad/s of speed of the motor, the load torque was 0.32 Nm. As in the simulations, the predictive times T1 and T2 are set to 1 and 5 ms, respectively, and the control period Tc is chosen to be equal to 0.1 ms. The time dedicated to the proposed RNPC algorithm is equal practically to 0.06 ms. The experimental results are illustrated in Figs. 11–21 for four different scenarios: the tracking performance of the proposed RNPC under unknown load torque, robustness testing under electrical-parameter uncertainties, mechanical-parameter uncertainties, and performance testing during an abrupt change of the load torque.
Fig. 16. d-axis and q-axis components of the armature current under an inaccurate value of rotor-flux linkage.
A. Tracking Performance Under Unknown Load Torque Bidirectional speed control under variable load torque was tested first, with the speed reference generated by a secondorder linear filter (see Appendix C). Figs. 11 and 12 show respectively the speed tracking response and the d and q
ERROUISSI et al.: ROBUST NONLINEAR PREDICTIVE CONTROLLER FOR PMSMs WITH COST FUNCTION
Fig. 17. d-axis and q-axis components of the armature current under an inaccurate value of the armature resistance.
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Fig. 21. Speed response under a rapid and abrupt change of the load torque.
B. Electrical-Parameter Uncertainties
Fig. 18. d-axis and q-axis components of the armature current under an inaccurate value of the d-axis inductance.
To verify the robustness of the drive using the proposed control scheme, three electrical parameters were set incorrectly in the controller. In comparison with the real values, the value of rotor-flux linkage was increased by 20%, the value of armature resistance was decreased by 50%, and the value of the d-axis inductance was increased by 50%. In addition, the load torque is unknown. As shown in Figs. 13–15, the motor speed quickly converged to the reference value. Figs. 16–18 show how the d-axis component of the armature current is driven to zero even if the electrical parameters are varied in the controller. The current ripples are caused by the inverter and its control. To deal with this problem, the configuration of the power converter must be chosen adequately [18]. C. Mechanical-Parameter Uncertainties
Fig. 19. Speed response under an inaccurate value of the viscous coefficient.
To verify the robustness of the proposed approach against the variations of the mechanical parameters, the moment of inertia and the viscous coefficient were set in the controller to 50% of the real values. Figs. 19 and 20 show the speed response. It can be seen that the steady-state error is eliminated, and precise speed tracking is achieved. The d-axis and q-axis components of the armature current are not plotted as the current control is not affected by the mismatched mechanical parameters. D. Abrupt Change of the Load Torque To verify the performance of the proposed controller with respect to its disturbance-rejecting capability, the load torque is increased suddenly by 60% at t = 1 s. Fig. 21 shows the speed reference and the actual speed. The steady-state error is driven quickly to zero as the predictive time T2 is very short. VIII. C ONCLUSION
Fig. 20. Speed response under an inaccurate value of the moment of inertia.
components of the armature current. As shown in Fig. 11, the dynamic performance of the proposed RNPC is very good, and there is no steady-state error in spite of the presence of variable load torque. Fig. 12 shows that the d-axis component of the armature current is maintained at zero value.
A novel RNPC for a PMSM with an optimized cost function has been presented. The objective of the controller is to track the reference speed profile while maintaining the d-axis component of the armature current at zero. To ensure robustness against external perturbations and parameter uncertainties, an integral action is introduced in the control loop. The disturbance rejection and robustness against uncertainties are improved significantly in the proposed RNPC without using a disturbance
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observer. It would be difficult to adapt the existing disturbance observer-based control techniques for a PMSM as the perturbed relative degree is less than that of the input. For this, a newly design cost function is proposed to design the RNPC. To the best of our knowledge, this achievement is a novel contribution. The stability of the proposed RNPC was investigated in details, and it was proven using the Lyapunov theory. A laboratory prototype was developed to test experimentally the validity of the proposed control scheme. Experimental results have shown its effectiveness. The proposed controller has been also tested under mismatched parameters and external perturbation. A high degree of robustness regarding parameter variations and load-torque changes has been demonstrated. A PPENDIX A. Solving the Optimization Problem in Section IV We first rewrite the predicted integral I(t + τ ) in the following matrix notation as I(t + τ ) = Λ(τ )Υ(t)
(19)
where Δ1 = − Lg2 h1 (x) = [ L1d 0 0 ] Δ2 (x) = − K22 Lg2 h2 (x) + Lg2 Lf h2 (x) −Lq )iq −p(φv +(Ld −Lq )id ) = p(LdJL JLq d
0
I2 (t + τ ) ]T .
When the disturbance is neglected, expression (7) leads to ⎧ ⎨ e˙ 1 (t) = y˙ r1 (t) − Lf h1 (x) − Lg1 h1 (x)u(t) e˙ 2 (t) = y˙ r2 (t) − Lf h2 (x) (20) ⎩ e¨ (t) = y¨ (t) − L2 h (x) − L L h (x)u(t). 2 r2 2 g f 2 1 f Based on the above approximation, the cost function given by (16) becomes 1 (21) N = Υ(t)T Ψ (T1 , T2 ) Υ(t) 2 where T1
T2 Λ1 (τ )T Λ1 (τ )dτ +
Ψ (T1 , T2 ) = 0
Λ2 (τ )T Λ2 (τ )dτ. 0
Now, substitute (20) into (21), and invoke the necessary condition for the optimal control to yield the RNPC (17).
ξ(t) = [ I1 e1 ]T ; η(t) = [ I2 e2 e˙ 2 ]T ⎡ 0 1 0 1 ⎣ A1 = = 0 0 ; A 2 −K01 −K11 2 2 −K −K 1 0 01x3 02x3 M1 = ; M2 (x) = . Δ1 Δ2 (x)
Substituting (17) into (7), we obtain the following closedloop error: t K01 t K02
.
⎤ 0 1 ⎦ −K22
The eigenvalues of matrix A1 are λ1 =
−1.25 ± j1.3307 T1
(24)
and those of matrix A2 are λ21 = −
1.9523 ; T2
λ22 =
−0.7739 ± j2.1862 . T2
(25)
As all the eigenvalues have negative real parts, matrices A1 and A2 are Hurwitz matrices. Consequently, there exist symmetric and positive-definite matrices Pi and Qi , where i = 1, 2, such that ATi Pi + Pi Ai = −2Qi ;
i = 1, 2.
(26)
Thus, the Lyapunov function candidate for the ξ and η subsystems can be chosen as V1 (ξ) =
1 T ξ P1 ξ 2
V2 (μ) =
1 T η P2 η. 2
(27)
Let λm (X) and λM (X) represent the minimum and the maximum eigenvalues of matrix X, respectively. The time derivative of the Lyapunov function V1 is 1 1 V˙ 1 (ξ) = −ξ T Q1 ξ + b(t)T M1T P1 ξ + ξ T P1 M1 b(t) 2 2 λM (P1 ) 2 ≤ −λm (Q1 ) ξ + b(t) ξ. Ld Note that
B. Proof of the Theorem
B J2
where
0
I(t + τ ) = [ I1 (t + τ )
−
The equations above can be written in the new (ξ, η) coordinates as ξ˙ = A1 ξ + M1 b(t) (23) η˙ = A2 η + M2 (x)b(t)
where
1 τ τ 2 /2! 0 0 0 0 Λ (τ ) = Λ(τ ) = 1 Λ2 (τ ) 0 0 0 1 τ τ 2 /2! τ 3 /3! T t t Υ(t) = e1 (τ )dτ e1 e˙ 1 e2 (τ )dτ e2 e˙ 2 e¨2
K22 J
M1 = L1d ξ = I12 + e21 .
(28)
(29)
From (28), a sufficient condition for the derivative of V1 being negative is
e1 (τ )dτ +K11 e1 (t)+K21 e˙ 1 (t) = Δ1 b(t) 0
ξ >
e2 (τ )dτ +K12 e2 (t)+K22 e˙ 2 (t)+K32 e¨2 (t) = Δ2 (x)b(t) 0
(22)
λM (P1 ) b(t) . λm (Q1 )Ld
(30)
Following (29) and the fact that b(t) is bounded, there exists a finite time t > 0 such that trajectory ξ reaches the region where
ERROUISSI et al.: ROBUST NONLINEAR PREDICTIVE CONTROLLER FOR PMSMs WITH COST FUNCTION
the derivative of V1 is negative. Then, the trajectory in question will remain bounded despite the perturbation. This means that the ξ subsystem is stable uniformly and globally. Furthermore, if the predictive time T1 is chosen to be very small, such that the poles of matrices A1 are placed sufficiently far in the left halfplane, the trajectory will converge rapidly to the steady state. The Barbalat lemma presented in [28] states that, if t → Z(t) is a differentiable function with a finite limit at t → ∞ and if Z˙ is uniformly continuous, then Z˙ → 0 when t → ∞. As I1 (t) is bounded and continuous, the Barbalat lemma allows to conclude that lim e1 (t) = 0.
t→∞
(31)
Regarding the η subsystem in (23), it follows from (1)–(3) that iq =
(J ω˙ r + Bωr + fω ) . p (φv + (Ld − Lq )id )
(32)
Then, the d-axis and q-axis components of the armature current can be rewritten as id = idr − e1 (33) i = (J(ω˙ rr −e˙ 2 )+B(ωrr −e2 )+fω ) . q
p(φv +(Ld −Lq )(idr −e1 ))
M2 (x) ≤ α1 e2 +α2 e˙ 2 +α3 idr +α4 ωrr +α5 ω˙ rr (34) where α1 , α2 , α3 , α4 , and α5 are positive numbers depending on the physical parameters. Differentiating the Lyapunov function candidate V2 along the trajectories of (26) yields 1 1 V˙ 2 (η) = −η T Q2 η + b(t)T M2T (x)P2 η + η T P2 M2 (x)b(t) 2 2 ≤ −λm (Q2 ) η2 + M2 (x) λM (P2 ) b(t) η. (35)
η =
I22 + e22 + e˙ 22 .
(36)
Combining (34)–(36) and noting that M2 (x) does not depend on I2 , it can be concluded that, for any bounded disturbance b(t), there exists a sufficiently large I2M , such that V˙ 2 (η) < 0 for any
I2 ≥ I2M .
(37)
With the assumption that the disturbance is bounded, the stability of the η subsystem can be proven easily in the same way as that of the ξ subsystem. It follows from (37) that I2 is also bounded. Taking into account the definition of I2 , the Barbalat lemma yields lim e2 (t) = 0
t→∞
lim e˙ 2 (t) = 0.
t→∞
TABLE I PARAMETERS OF THE PMSM
Therefore, under the RNPC (17), the asymptotical tracking is achieved. Remark: The uniform continuity of the error signal e(t) is required to show that the steady-state error approaches zero when using the Barbalat lemma. The system considered here is continuous and smooth, and all the input signals are continuous. It follows from the properties of nonlinear systems that all the signals, including the error signals, are continuous. C. Second-Order Linear Filter
By substituting (33) into M2 (x) in (23), it can be shown that M2 (x) only depends on errors e1 , e2 , and e˙ 2 and references idr , ωrr , and ω˙ rr but not on I1 and I2 . The convergence rate of ξ depends on the predictive time T1 . The electric current loop is designed so as to have much faster response than the mechanical loop, which is achieved by choosing T1 < T2 . Therefore, the tracking error e1 approaches to zero in a finite time. Under this condition, M2 (x) is bounded by
Note that
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(38)
The reference speed signal is passed through a second-order linear filter, which is given by F (s) =
ωn2 s2 + 2ξωn s + ωn2
(39)
where ξ = 1 and ωn = 30 rad/s for slow speed-reference trajectory, and ξ = 1 and ωn = 100 rad/s for fast speed-reference trajectory. D. Parameters of the PMSM The parameters of the PMSM are shown in Table I. R EFERENCES [1] Y. Zhang, C. M. Akujuobi, W. H. Ali, C. L. Tolliver, and L.-S. Shieh, “Load disturbance resistance speed controller design for PMSM,” IEEE Trans. Ind. Electron., vol. 53, no. 4, pp. 1198–1208, Jun. 2006. [2] K.-H. Kim and M.-J. Youn, “A simple and robust digital current control technique of a PM synchronous motor using time delay control approach,” IEEE Trans. Power Electron., vol. 16, no. 1, pp. 72–82, Jan. 2001. [3] Y. Abdel-Rady and I. Mohamed, “Design and implementation of a robust current-control scheme for a PMSM vector drive with a simple adaptative disturbance observer,” IEEE Trans. Ind. Electron., vol. 54, no. 4, pp. 1981–1988, Aug. 2007. [4] J. Solsona, M. I. Valla, and C. Muravchik, “Nonlinear control of a permanent magnet synchronous motor with disturbance torque estimation,” IEEE Trans. Energy Convers., vol. 15, no. 2, pp. 163–168, Jun. 2000. [5] Y. Abdel-Rady and I. Mohamed, “A newly designed instantaneous-torque control of direct-drive PMSM servo actuator with improved torque estimation and control characteristics,” IEEE Trans. Ind. Electron., vol. 54, no. 5, pp. 2864–2873, Oct. 2007. [6] S.-Y. Kim, W. Lee, M.-S. Rho, and S.-Y. Park, “Effective dead-time compensation using vectorial disturbance estimator in PMSM drives,” IEEE Trans. Ind. Electron., vol. 57, no. 5, pp. 1609–1614, May 2010. [7] H.-H. Choi, N. T.-T. Vu, and J.-W. Jung, “Digital implementation of an adaptive speed regulator for a PMSM,” IEEE Trans. Power Electron., vol. 26, no. 1, pp. 3–8, Jan. 2011. [8] S. Li and Z. Liu, “Adaptive speed control for permanent-magnet synchronous motor system with variations of load inertia,” IEEE Trans. Ind. Electron., vol. 56, no. 8, pp. 3050–3059, Aug. 2009.
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Rachid Errouissi received the B.Sc. degree from École Mohammedia d’ingénieurs, Rabat, Morocco, in 2001; the M.Sc. degree from Université Claude Bernard, Lyon, France in 2004; and the Ph.D. degree from the University of Quebec at Chicoutimi, QC, Canada, in 2010. His research interests include advanced control, electric machines, and adjustable speed drives.
Mohand Ouhrouche (M’99–SM’10) received the Ingéniorat d’État (with honors) in electrical engineering from the University of Béjaia, Béjaia, Algeria, in 1988, and the M.Sc. and Ph.D. degrees in electrical engineering from École Polytechnique de Montréal, Montréal, QC, Canada, in 1992 and 1998, respectively. From 1993 to 1995, he worked as an Assistant Researcher with École Polytechnique de Montréal on the integration of nonutility generators into the Hydro-Quebec’s distribution system. In 1997, he joined the Hydro-Quebec Research Institute, Montréal, QC, as a Scientific Researcher to carry out studies on power quality. Since December 1998, he has been with the University of Quebec, Chicoutimi, QC, where he is currently a Full Professor of electrical engineering and the Director with the Electric Machines Identification and Control Laboratory. He is the author or coauthor of over 70 journal and conference papers, a chapter for Wind Turbines (Intech 2011), and of Electric Circuits: Analysis and Applications (in French, Presses internationales Polytechnique, 2008). His research interests include advanced control, electric machines and drives, and renewable-energy conversion systems. Dr. Ouhrouche is a Registered Engineer in the province of Quebec.
Wen-Hua Chen (M’00–SM’08) received the M.Sc. and Ph.D. degrees from Northeast University, Shenyang China, in 1989 and 1991, respectively. From 1991 to 1996, he was a Lecturer and then Associate Professor with the Department of Automatic Control, Nanjing University of Aeronautics and Astronautics, Nanjing, China. From 1997 to 2000, he held a research position and then a Lecturer in control engineering with the Centre for Systems and Control, University of Glasgow, Glasgow, U.K. He is currently a Senior Lecturer in fight control systems with the Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, U.K. He is the author of about 120 papers, including 50 journal papers. His research interests include the development of advanced control strategies and their applications in aerospace engineering.
Andrzej M. Trzynadlowski (M’83–SM’86–F’99) received the Ph.D. degree in electrical engineering from the Technical University of Wroclaw, Wroclaw, Poland. He became a Faculty Member with the Technical University of Wroclaw and then worked with the University of Salahuddin, Arbil, Iraq; the University of Texas–Arlington, Arlington; and the University of Wyoming, Laramie. Since 1987, he has been with the University of Nevada, Reno, where he is a Professor of electrical engineering. In 1997, he spent seven months with Aalborg University, Aalborg, Denmark, as the Danfoss Visiting Professor. In 1998, he was a Summer Faculty Research Fellow with the Naval Surface Warfare Center, Annapolis, MD. In 1998 and 1999, he delivered short courses with the University of Padova, Padova, Italy. He is the author of over 160 journal and conference papers, which are mostly on power electronics and electric drive systems. He is the holder of 12 patents. He is the author of The Field Orientation Principle in Control of Induction Motors (Kluwer, 1994), Introduction to Modern Power Electronics (Wiley, 1998 and 2010), and Control of Induction Motors (Academic Press, 2001). He wrote chapters for Modern Electrical Drives (Kluwer, 2000) and Control in Power Electronics (Academic Press, 2002). Dr. Trzynadlowski serves as an Associate Editor of the IEEE T RANSAC TIONS ON P OWER E LECTRONICS and IEEE T RANSACTIONS ON I NDUS TRIAL E LECTRONICS . He is a member of the Industrial Drives Committee and Industrial Power Converters Committee of the IEEE Industry Applications Society. He was the recipient of the 1992 IAS Myron Zucker Grant. He has been listed in Who’s Who in the World, Who’s Who in America, and Who’s Who in Science and Engineering.
