A High Order Periodic Adaptive Learning Compensator for Cogging Effect in PMSM Position Servo System Ying Luo† , YangQuan Chen‡ and Hyo-Sung Ahn§ and Youguo Pi¶ Abstract—In this paper, a simulation model of PMSM position servo system is presented briefly first, we then propose a high order periodic adaptive learning compensation (HO-PALC) method for cogging effect on PMSM position and velocity servo tasks. The cogging force is considered as a position-dependent disturbance that is periodic. The key idea of the implemented cogging disturbance compensation method is to use past information of more than one position period along the state axis to update the current adaptation learning law. Simulation results are presented to illustrate the effectiveness of the high order periodic adaptive cogging compensation scheme. Furthermore, the advantage of the HO-PALC is demonstrated through comparing with the first order periodic adaptive learning compensation. Index Terms—Cogging force, permanent magnet synchronous motor (PMSM), adaptive control, high order periodic adaptive learning control, state-dependent disturbance.
I. I NTRODUCTION Permanent magnet motors are the most popularly used electromechanical devices for high performance industrial servo applications of accurate speed and position control of the rotary or linear system. However, the cogging force as the main disadvantage disturbs the PM motor speed and limits the position servo control performance of application especially in a high-precision tracking applications. So, In parallel with the popularity of PM motors in industrial applications, the suppression of cogging effect has been paid much attention to. As the cogging force is considered as a position-dependent disturbance that is periodic [1], so the leaning control is suggested to compensate the cogging. Adaptive learning compensators of cogging and coulomb friction in permanent-magnet linear motors were designed in [2] and [3]; [4] and [5] proposed an iterative learning control algorithm and a variable step-size normalized iterative learning control scheme to reduce periodic torque ripples from cogging and other effects of PMSM, respectively; in [6], a periodic adaptive learning compensation method for cogging was performed on the PMSM servo system. However, all these articles didn’t represent the high order learning control scheme for cogging compensation where the information of more than one previous periods is used in the learning control updating law. At the same time, we are facing the fact that, the memory is becoming cheaper and cheaper, so that the cost of recording the history information in the course of control is not critical nowadays. There are †
[email protected]; Ying Luo is a Ph.D. candidate on leaving from Department of Automation Science and Technology in South China University of Technology, Guangzhou, P.R.China, as a visiting scholar currently in Center for Self-Organizing and Intelligent Systems (CSOIS), Dept. of Electrical and Computer Engineering, Utah State University, Logan, UT, USA. ‡
[email protected]; Tel. 01(435)797-0148; Fax: 01(435)797-3054; Electrical and Computer Engineering department, Utah State University, Logan, UT 84341, USA. URL: http://www.csois.usu.edu/people/yqchen. §
[email protected]; Department of Mechatronics, Gwangju Institute of Science and Technology (GIST), Gwangju, Korea. ¶
[email protected]; Dept. of Automation Science and Engineering, South China University of Technology, China.
c 2008 IEEE 1-4244-2384-2/08/$20.00
also many papers to introduce the high order iterative learning control [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], but how to use the high order information in adaptive learning control to compensate the cogging effect more efficiently is what we are focusing on here. In this paper, we first present a simulation model of PMSM position servo system as in [6] briefly. We then propose a high order periodic adaptive learning compensation (HOPALC) method for cogging effect on PMSM position and velocity servo tasks. The cogging force is considered as a position-dependent disturbance that is periodic. The key idea of the implemented cogging disturbance compensation method is to use past information of more than one position period along the state axis to update the current adaptation learning law. Simulation results are presented to illustrate the effectiveness of the high order periodic adaptive cogging compensation scheme. Furthermore, the advantage of the HOPALC is demonstrated through comparing with the first order periodic adaptive learning compensation (FO-PALC). The major contributions of this paper include 1)A new high order periodic adaptive learning compensation method for cogging effect; 2) Simulation testification of the high order periodic adaptive learning compensation method for multiharmonic cogging effect on the PMSM position servo system simulation model; 3) Demonstration of the advantage of the HO-PALC by performing the simulation comparison with the FO-PALC. The rest of this paper is organized as follows. In Sec. II, PMSM position servo system is introduced briefly and in Section III, a new high order periodic adaptive learning compensator for cogging effect is designed. Simulation tests are implemented in Sec. IV, the performances of using the second order PALC are compared with the first order PALC for multiharmonic cogging effect. Conclusion is given in Sec. V. II. PMSM P OSITION S ERVO S YSTEM A. Model of PMSM Permanent magnetic (PM) synchronous motor consists of three-phase stator windings and a PM rotor. The variables and parameters represented in the stator frame can be transformed into those in the synchronous reference frame. Therefore, the well-known mechanical equations and electrical equations of a permanent-magnet synchronous motor are given: dθ = ω, dt
dω 1 = (Tm − Tl ), dt J
Tm = Kt iqs =
(1)
3P ψdm iq , 22
(2)
did 1 = (Vd + ωLq iq − Rid − ωψqm ), dt Ld
(3)
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III. H IGH O RDER P ERIODIC A DAPTIVE LEARNING C OMPENSATION OF C OGGING E FFECT In this section, a high order state-dependent periodic adaptive learning compensator for cogging is designed. The cogging force of (6) can be written as: −a(θ), where a(θ) is the function of θ. In this paper, to present our ideas clearly, without loss of generality, The motion control system is modeled as follows M P ³
Fig. 1.
V
dT dt
T
Block diagram of the PMSM position servo system model
diq 1 = (Vq − ωLd id − Riq − ωψdm ), dt Lq
(4)
where (1) represents the mechanical subsystem, and equations (3) and (4) represent the electrical subsystem. θ and ω are motor rotor angular position and speed, respectively; J is the moment of inertia of the rotor; Tm is the motor electromagnetic torque generated, and Tl is the load torque applied; id and iq are stator currents along the d and q axes, respectively; Vd and Vq are the voltages along d and q axes, respectively; R is the stator resistance; Ld and Lq are the stator self-inductances in the d and q axes, respectively, it has been assumed that as the surface mounted PMSM is non-salient, Ld and Lq are the same denoted by L. By using the concept of field oriented control of the PMSM, the d-axis current is controlled to be zero to maximize the output torque. Under this assumption, the motor electromagnetic torque is given in (2), where Kt is the actually torque coefficient and P is the number of poles in the motor. However, in practice the motor torque can be expressed as 3P ψdm iq + Fcogging , (5) Tm = 22 where Fcogging is the periodic torque pulsation due to cogging. Cogging force is produced by the magnetic attraction between the rotor mounted permanent magnets and the stator [6]. In this paper, we also consider the cogging force as the general multi-harmonic form as considered in [6] Fcogging =
∞ X
Ai sin(ωi x + ϕi ).
(6)
i=1
where Ai is the amplitude, ωi is the state-dependent cogging force frequency, and ϕi is the phase angle. In order to compensate the cogging force of general signal shape, it is suggested to make use of the periodicity of the position-dependent cogging disturbance. B. PMSM Position Servo System Simulation Model Figure 1 shows the control structure of system model with three closed-loops for position, speed and current as in [6]. The position reference is given, we can get the rotor angular information as the position feedback from the PMSM module, and the derivative of the rotor angle is the speed feedback, then the position and speed closed-loops controls can be performed. More details about the position, speed and current closed-loops controls, and the SVPWM algorithm are introduced in [6].
˙ θ(t)
= v(t), (7) a(θ) v(t) ˙ = u− − Tl 0 , (8) J 1 1 u = Tm , T l 0 = Tl , J J where θ is the periodic rotor angle position; v is the velocity; u is the control input and a(θ) is the unknown positiondependent cogging disturbance which is repeating in every pole-pitch, at the same time a(θ) should be bounded and denoting |a(θ)| ≤ b0 . (9) First, before proceeding our main results, the following definitions and assumptions are necessary. Definition 3.1: The total passed trajectory is given as: Z t Z t dθ s(t) = | |dτ = |v(τ )|dτ, 0 dτ 0 where θ is the angle position, and v is the velocity. Physically, s(t) is the total passed trajectory, hence it has the following property: s(t1 ) ≥ s(t2 ), if t1 ≥ t2 . With notation s(t), the position corresponding to s(t) is denoted as θ(s(t)) and the cogging force corresponding to s(t) is denoted as a(s(t)). In our definition, since s(t) is the summation of absolute position increasing along the time axis, just like t, s(t) is a monotonous growing signal, so we have a(θ(s)) = a(s(t)) = a(t).
(10)
Definition 3.2: Since cogging force is periodic with respect to position, so, based on Definition 3.1, the following relationship is derived: θ(s(t)) = θ(s(t) − sp ), a(s(t)) = a(s(t) − sp ).
(11)
where sp is the periodicity of the trajectory. Definition 3.3: In Definition 3.2, sp was defined as periodic trajectory. So, s(t)−sp is one trajectory past point from s(t) on the s axis. Let us denote the time corresponding to s(t)−sp with τ . Then, t−τ is the time-elapse to complete one periodic trajectory from the time τ to time t. This time-elapse is called “cycle”. Particularly, it is called “k-th trajectory cycle” at time t and denoted as Pk . So, τ =t−Pk . It is called “the search process” to find Pk at time instant t (note: the search process can be performed by interpolation). Definition 3.4: The first trajectory cycle P1 is the elapsed time to complete the first repetitive trajectory from the initial starting time t0 . In other words, P1 is the time corresponding to the total passed trajectory when s(t) = sp .
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From now on, for accurate notation, the rotor angle position corresponding to time t is denoted as θ(t) and its total passed trajectory by the time t is denoted as s(t). Henceforward, one trajectory past time from the time instant t is denoted as τ , and its corresponding k-th cycle is denoted as Pk . Assumption 3.1: Throughout the paper, it is assumed that the current position and current time of PMSM are measured. Let us denote the current position as θ(t) at time t. Then, τ is always calculated, hence Pk is calculated at time instant t. With the above definitions and assumption, the following property is observed. Remark 3.1: As will be shown in the following theorem, the actual state-dependent cogging force a is not estimated on the state axis. In our adaptation law, a is estimated on the time axis. So, to find a(s(t) − sp ), the following formula is used: a(s(t) − sp ) = a(t − Pk ).
(12)
Here, Pk is calculated in Assumption 3.1 (recall that Pk can be used to indicate exactly one-trajectory past position). From Definition 3.2, Remark 3.1 and (10), we also have the following property: Property 3.1: The current cogging force is equal to onetrajectory past cogging force. From the relationship: a(t)
= a(s(t)) = a(s(t) − sp ) = a(t − Pk ),
(13)
a ˆ(t) + αm(t) + γev (t), J
(15)
with m(t) := γeθ (t) + ev (t),
(16)
where α and γ are positive gains; a ˆ(t) is an estimated cogging force from an adaptation mechanism to be specified later; v˙ d (t) is the desired acceleration; and eθ (s(t)) = eθ (t); and m(s(t)) = m(t). Our adaptation law is designed as follows: δˆ a(t − Pk ) + K s ≥ sp J S(t) if a ˆ(t) = (17) z − µv if s < sp with S(t) :=
N X
βi mi (t),
(18)
i=1
mi (t) = m(t −
i X
Pk+1−j ),
(i = 1, 2, ..., N )
(19)
j=1
ea (s(t)) = a(s(t)) − a ˆ(s(t)), where a ˆ(s(t)) is the estimated cogging force, a ˆ(s(t)) = a ˆ(t) (note: t is the current time corresponding to the current total passed trajectory s(t)). Here, let us change ea (s(t)) = a(s(t)) − a ˆ(s(t)) into time domain such as: = a(s(t)) − a ˆ(s(t)) = a(t) − a ˆ(t) = ea (t).
u(t) = v˙ d (t) + Tl0 +
where
the following equality is derived: a(t) = a(t − Pk ). Then we defined the notation
ea (s(t))
reasonable to assume that θd (t), vd (t) and v˙ d (t) are all bounded. From now on, based on relationship: a(θ(t)) = a(t−Pk ) = a(t), a(θ(t)) is equalized to a(t) as done in (10); So, a(θ) is replaced by a(t) in the following theorems. The feedback controller is designed as:
and a ˆ(t − Pk ) = a ˆ(s − sp ) (note: Pk is the trajectory cycle defined in Definition 3.2); P1 is the first trajectory cycle specified in Definition 3.4; δ is the weighting coefficient and 0<δ<1, K is a positive design parameter (it is called the periodic adaptation gain); µ is also a positive parameter; βi are the coefficients of high order feedback errors, they are chosen to be the bounded and their upper bound denoted by bβ is defined as below bβ = max |βi |; 1≤i≤N
(14)
In the same way, the following relationships are true: vd (s(t)) = vd (t), v(s(t)) = v(t), and the following notations are also defined eθ (t) = θd (t) − θ(t), ev (t) = vd (t) − v(t). Now, based on the above discussions, the following stability analysis is performed. Our N -th order periodic adaptive learning compensation approach is summarized as follows: • When s(t)
(20)
in our analysis part, the following tuning mechanism is required for z: ev (t) . (21) J Consider two cases: 1) when 0 ≤ t < P1 (0 ≤ s < sp ) and 2) when t ≥ P1 (s ≥ sp ). The key idea is that, for case 1), it is required to show the finite time boundedness of equilibrium points; for case 2), it is necessary to show the asymptotic stability of equilibrium points. First, let us consider the case 1) when t < P1 (s < sp ). Our major results are summarized in the following theorems with Remark 3.2. Remark 3.2: From the relationship (14), it can be said that if ea (t)→0 as t→∞, then ea (s)→0 as s→∞. Thus, in what follows, the stability analysis of a ˆ(θ) is performed on the time axis. Theorem 3.1: If |∂a(θ)/∂θ| < ba (bound of changes in a(θ)), µ > 41 J(1 + b2a ) and (α + γ) > 1, the equilibrium points of eθ , ev , and ea are bounded, when t < P1 (s < sp ). z˙ = µ[v˙ d (t) + αm(t) + γev (t)] +
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Proof: Omitted due to the limitation of pages. Now, let us investigate the case 2) when t ≥ P1 (s ≥ sp ). First of all, the following lemma is needed for the proof of Theorem 3.2. Lemma 3.1: Suppose a real position series [an ]∞ 1 satisfies an ≤ ρ1 an−1 + ρ2 an−2 + · · · + ρN an−N + , (n = N + 1, N + 2, · · ·), where ρi ≥ 0, (i = 1, 2, · · · , N ), ≥ 0 and ρ=
N X
A. Case-1: FO-PALC In this simulation, we use the first order PALC (N =1), so the adaptation law (17) is presented as: a ˆ(t − Pk ) + K s ≥ sp J S(t) if a ˆ(t) = (24) z − µv if s < sp with
ρi < 1,
(22)
S(t) = m1 (t).
i=1
(23)
n→∞
The proof of Lemma 3.1 see Chapter 2 in [17]. Theorem 3.2: When t ≥ P1 (s ≥ sp ), the control law (15) and the periodic adaptation law (17) guarantee the asymptotically stability of the equilibrium points eθ (t), ev (t) and ea (t), as t → ∞ (s → ∞), with Pithe initial condition, eθi (t) = evi (t) = eai (t) = 0, as (t − j=1 Pk+1−j ) ≤ 0, where i X ηi (t) = η(t − Pk+1−j ), i = 1, 2, ..., N, j=1
Figures 3(a) and 3(b) show the position/speed tracking errors with compensation of using the FO-PALC. We can observe that, as time increases, the positive/speed tracking errors become smaller and smaller. The FO-PALC works efficiently comparing with the tracking errors without compensation in Figures 2(a) and 2(b). We also show the frequency spectrums of the position/speed errors of using FO-PALC in Figures 6(a) and 7(a). 40
0.03
30 0.02
20 0.01
Velocity error (rpm)
lim an ≤ /(1 − ρ).
Position error (rad)
then the following holds:
(25)
0
10
0
−10
−0.01
η ∈ {θd , θ, vd , v, a, a ˆ, eθ , ev , ea , m, S}.
−20 −0.02
−30
(This is a very important notation in this paper). Proof: Omitted due to the limitation of pages.
sd (t) vd (t)
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(a) Position Fig. 2.
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Tracking errors without compensation.
40
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0.01 Position error (rad)
IV. S IMULATION I LLUSTRATIONS The suggested method is verified on the PMSM position servo control system simulation model. In order to testify the system performances of using the HO-PALC and compare the rapidity of convergence between the HO-PALC and the FOPALC of cogging effect, two cases of tests are performed. • Case-1: Simulation demonstration of first order PALC of multi-harmonics cogging effect; • Case-2: Simulation demonstration of feedback second order PALC of the same cogging effect as in Case-1; For our simulation tests, the following reference trajectory and low velocity signals are used:
−0.03 0
0
10
0
−10
−0.01
−20 −0.02
−30
−0.03 0
2
4
6 Time (seconds)
8
10
−40 0
12
2
(a) Position
= t(rad), = 1(rad/s) = 9.55(rpm).
Fig. 3.
The control gains in (15) were selected as: α = 50, γ = 20 and µ = 3. The periodic adaptation gain K was selected as 0.2. The motor parameters are given in Table I and the Tl = 0.5[N m]
4
6 Time (seconds)
(b) Velocity
Tracking errors with compensation using FO-PALC.
40
0.03
30 0.02
20
TABLE I PMSM S PECIFICATIONS Rated speed Stator resistance Magnet flux Moment of Inertia
2000 rpm 2.125 Ω 0.387 0.00289 kgm2
0
10
0
−10
−0.01
−20 −0.02
−30
−0.03 0
In the simulation, the actual cogging force is modeled as the state-period sinusoidal signal of multiple harmonics below:
Velocity error (rpm)
1.64 Kw 8 Nm 11.6 mH 6
Position error (rad)
Rated power Rated torque Stator inductance Number of poles
0.01
2
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6 Time (seconds)
(a) Position Fig. 4.
8
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−40 0
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6 Time (seconds)
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(b) Velocity
Tracking errors with compensation using SO-PALC.
Fcogging = 2 cos(6θ) + cos(12θ) + 0.5 cos(18θ).
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with
1 0.9
0.8
0.8
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|FFT(Velocity)|
1 0.9
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(a) FO-PALC
S(t) = 0.5m0 (t) + 0.5m1 (t).
−3
x 10
16
FOPALC SOPALC
10
15
20
25 30 Frequency (Hz)
35
40
45
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(b) SO-PALC
(27)
Figures 4(a) and 4(b) show the positive/speed tracking errors with using the SO-PALC. From the figures, we observe that the servo system of using the HO-PALC for cogging effect is asymptotic stable. At the same time, comparing with figures 3(a) and 3(b), the convergence speed of the position/speed tracking errors of using the SO-PALC is faster than that of using the FOPALC. In order to illustrate the comparison more clearly, figures 5(a) and 5(b) are presented, the red lines represent the root mean squires (RMS) of position/speed tracking errors of the FO-PALC, and the blue lines are for the RMS of position/speed tracking errors of the SO-PALC. It is obvious that using the SO-PALC method makes the system obtain a faster convergence speed than using the FO-PALC method. Furthermore, figures 6(a) - 7(b) show the frequency spectrums comparison of the position/speed errors between the FO-PALC and SO-PALC. From the figures,we also observe that the magnitudes of the position/speed errors signals frequency with the SO-PALC is smaller than that with the FO-PALC. 4.5
|FFT(Velocity)|
B. Case-2: SO-PALC In this simulation, we choose N =2, namely, use the second order PALC to test the HO-PALC. At the same time, in order to compare with the FO-PALC fairly, we design β1 =β2 =0.5, so the adaptation law (17) is presented as: a ˆ(t − Pk ) + K s ≥ sp J S(t) if a ˆ(t) = (26) z − µv if s < sp
FOPALC SOPALC
4
Fig. 7. Tracking velocity errors frequency spectrum comparison between FO-PALC and SO-PALC.
V. C ONCLUDING R EMARKS In this paper, a new cogging force compensation method is proposed for PMSM position and velocity servo system. The key idea of this method is to use past information of more than one position period along the state axis to update the current adaptation learning law. From the simulation results, we can conclude that, the proposed HO-PALC method for cogging works effectively, and performs better than the FO-PALC method, the convergence speed of the position/speed tracking errors with the HO-PALC is faster than that with the FOPALC. Furthermore, although the suggested HO-PALC method is developed for the cogging force compensation, our method also can be used to compensate other nonlinear disturbance. Along with the development of the whole society, especially as the memory become cheaper and cheaper, it will be very easy to record the history information in the course of control and realize the high order adaptive leaning control method in many real applications.
14 3.5
ACKNOWLEDGEMENT
Velocity error RMS
Position error RMS
12 3
2.5
2
10
6
1.5
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0.5
Ying Luo would like to thank to to Dr. Huifang Dou for her expertise in cogging compensation of PMLM (permanent magnet linear motor) and to the China Scholarship Council (CSC) for the financial support.
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R EFERENCES
(b) Velocity
Fig. 5. Tracking errors RMS comparison between the FO-PALC and SOPALC; the red lines represent the RMS of position/speed tracking errors of FO-PALC, and the blue lines are for the RMS of position/speed tracking errors of SO-PALC.
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