Predictive Variable Gain Iterative Learning Control for PMSM

A predictive variable gain strategy in iterative learning control (ILC) is introduced. Predictive variable gain iterative learning control is constructed to improve the performance of trajectory tracking. A scheme based on predictive variable gain iterative learning control for eliminating undesirable vibrations of PMSM system is proposed.The basic idea is that undesirable vibrations of PMSM system are eliminated from two aspects of iterative domain and time domain. The predictive method is utilized to determine the learning gain in the ILC algorithm. Compression mapping principle is used to prove the convergence of the algorithm. Simulation results demonstrate that the predictive variable gain is superior to constant gain and other variable gains.


Introduction
Due to their high efficiency, high power density and low noise, low loss, small size, and so forth, permanent magnet synchronous motor (PMSM) is used widely in various industrial fields.Furthermore, the applications of PMSM are expanding rapidly.
However, PMSM performance at low speed is bad because of the existence of torque ripple which deteriorates the accuracy and repeatability of PMSM and undermines potentially its suitability in precision electromechanical device.Thus, eliminating torque ripple is very important for improving PMSM performance.
Many control methods have been utilized to suppress the torque ripple.They include PID control scheme [1,2], predictive control [3], adaptive fuzzy control [4], robust control [5], sliding-mode control [6], and so on.These control means improve the performance of PMSM system from different aspects [7], but applying conventional PID controller and modern control techniques mentioned in the above to deal with torque ripple cannot attain desired levels; moreover, some of them are too complex to employ in practice.
Because of the periodic feature of PMSM on some applications and the simplicity of iterative learning control, a large number of learning control schemes were developed to remove torque ripple in PMSM.Those learning schemes applied to permanent magnet synchronous motors can be divided into two categories according to the learning gain: the first class is fixed gain iterative learning control [8] and the other is variable gain iterative learning control.The second category has obvious advantages in instantaneous characteristics and robustness of system when compared with fixed gain iterative learning control, so designing a reasonable, objective, and effective algorithm in iterative learning controller to determine the value of learning gain at each moment is a vital important factor in solving the problem of instantaneous error growth which has aroused a strong interest of the researchers.
An iterative learning algorithm with a variable gain in iteration domain to remove measurement disturbances and guarantee that the tracking error converges to zero was developed by Zhang et al. [9], but time domain uncertainty is not considered; under this circumstance, Xu et al. [10] proposed a variable PID gain with iterative learning control scheme applied to nonlinear system to tracking the desired output; although this PID gain takes into account disturbances in both time domain and iterative domain, the choice of coefficients for PID gain is subjective.
Unlike [9,10], we propose to use predictive control to determine the gain of iterative learning control during in both 2 Journal of Control Science and Engineering time index and iterative index.The main contributions of this method include three aspects: the first is superiority in control performance compared with constant gain and PID variable gain.The second is that this scheme can be applied not only to the linear system but also to the nonlinear system, so it has a wider application.Finally, although predictive control has been successful applied to the motor control [11][12][13][14], it is the first time, as far as the author known, that it is used to determine the gain of iterative learning control; the proposed scheme in this paper overcomes the subjectivity of the gain value choice.

Material and Methods
2.1.Process Description.We assume that the PMSM motion model is described as [15] in the above,  is the load weight,  is a dampen coefficient,  1 ,  2 represent the position and velocity of mover, respectively,   is a motor load disturbance,  fric is the friction,  cog is a alveolar thrust ripple,  rel is a reluctance thrust ripple,   is the reluctance motor electromagnetic force to remove part of the thrust fluctuations, and  fric can be expressed as where   is the maximum static friction;   is a Coulomb friction,  2 and  are empirical parameters used to describe the Stribeck effect, and sgn() is a switching function; generally it is believed that  cog is a periodic function which is described as is polar distance.It is assumed that motor electromagnetic force   =   ( 1 )  , where we assume that   is the -axis mover current in the vector control mode: 0 > 0 is an average thrust constant,   is reluctance thrust constant, and   =  0   : rel is a periodic function to satisfy In (1), current   denotes the input signal,  1 ,  2 represent the states of system, and  denotes the output; generally, the exact input produces the desired output, but in practice, the torque ripple is unavoidable; as a result, the real output is not the desired.
In order to resolve above question, we develop a control scheme which merges the iterative learning and predictive control to eliminate the bad effect of the torque ripple.
The main target of this paper is to find a proper learning gain such that the output error is minimized, and arbitrary high precision output tracking is achieved.

Iterative Learning Control Law.
In this section, we develop a scheme to get the appropriate learning gain by minimizing the performance function in the predictive control process.
For simplicity, we consider an open-loop iterative learning law.During the iterative index  + 1, the learning update is given by where  is the learning control gain and   is the output error; that is, is a realizable desired output trajectory,  +1 ,   are the system inputs in the  + 1st iterative and th iterative,  is a filter, and  is the iterative learning index.

Predictive Gain.
In order to find out the appropriate learning gain by predictive control, we firstly set up the predictive model where the learning gain acts as a system input.
At sampled time  in the iterative index  + 1, combining (1) with (5), we can write the equation for the PMSM: Inserting ( 9) into (7), we get For the sake of getting an updating learning gain in next sample time  + 1, we make use of predictive control method.The predictive model is given by ,   ,  1, , and  2, denote the input, output, and states of system in the th iterative index, respectively; the system in the above is a nonlinear system.Secondly, we turn the nonlinear PMSM model ( 11) decouples into a new linear system via the input-output feedback linearization scheme.According to the exact linear theory, calculating Lie derivative of the output variable, we get the relative degree which is equal to the number of the system state variables; as a result, (11) satisfies the exact linear condition and can be linearized as where , and V  is a new control input.We can discretize (10): where  1 =   ,  2 = ∫  0   ,  = 0.1 ms is the sampling period, and ΔV = V  − V  , where V  denotes desired input.We assume that the predictive horizon is   and the control horizon is  V ; ΔV can be calculated by the following criterion: In the light of the least square formula, the expression of the ΔV is obtained as where Θ = [ ], () = (  − Ψ() − V( − 1)),   and   are square roots of eigenvalues of matrix  and , respectively;   is the desired state; according to the exact linear theory, we get The open-loop learning gain   () is given by Considering ( 5), we have It can be rewritten in the form 2.4.Convergence Analysis.In this section, we give the condition under which the system output error converges to zero.Consider (1) and it can be written as where ] sgn( 2 )+  ) ], and () =   .We assume the following properties for system (20): (1) The functions , , and ℎ are assumed to satisfy the following conditions: ∀ 1 ,  (2) It is assumed that, at each iterative process, the initial state error sequence {  (0)},  ≥ 0, converges to zero.
(3) For any realizable output trajectory and an appropriate initial condition, there exists a unique control input generating the trajectory for the plant.
Proof.Let system satisfy assumptions (1)-( 3) and ( 7) be applied; define the state, input and the output errors as According to (20), we get Then, Define the operator is a unit matrix; we make an estimate for Υ  : According to lemma 2 in [16], we have the convergence conditions 0 <  < 1,  4 > 1, and  3 and  4 are constant.2.5.Simulation.In this section, we will make a comparison between the iterative learning method based on the predicted variable gain and two kinds of efficient iterative learning control method which in recent years have been widely used for permanent magnet synchronous motors: constant gain of iterative learning control strategy and the variable PID gain iterative learning control strategy.PMSM system preferences are as follows.
The simulation results are shown in Figures 1-7.
Figure 1 shows a comparison between the desired PMSM output and the actual PMSM output in the 1st, 2nd, and 3rd iteratives with the fixed gain iterative learning controller, although the actual system output will approach the desired output with the increase of the number of iterations, instantaneous vibration bandwidth is sometimes dramatic large which is in fact detrimental to system.Figures 2 and 3 give a more intuitive description about the instantaneous vibration growth which show that max vibration bandwidth increases from 0.158 mm to 1.4 mm just after four iteratives.These phenomena also give full explanation of the importance of predictive gain learning control.Figures 4-6 show the actual system output in different iterative indices with constant gain learning controller, PID variable gain learning controller, and predictive gain learning controller, respectively.We can see from Figures 4-6 that the predictive gain is significantly better than the fixed gain and the PID gain.Firstly, the error vibration amplitude of predictive gain is less than the other two gains; secondly, error descent rate of predictive gain in time domain is also faster than fixed gain and PID gain iterative learning control.
For the sake of getting a more intuitive understanding for the advantages of the scheme proposed by this paper, Figure 7 is given.It can be seen that when the initial state error is zero, on the premise of the convergence conditions of iterative learning control, the predictive variable gain iterative learning has a higher convergent rate; meanwhile tracking accuracy also has been greatly improved.

Conclusion
A variable gain iterative learning control strategy based on predictive control for PMSM has been developed.This method not only increases the iterative convergence rate, but also eliminates error in the time domain.The convergence of this technique is proved by the contractive operator theory.
Future work will consider the choosing of predictive horizon to get a better control performance.

1 Figure 5 :Figure 6 :
Figure 5: Errors of the constant gain, PID gain, and predictive gain in the 6th iterative.