A Method to Improve the Response of a Speed Loop by Using ... - MDPI

energies Article

A Method to Improve the Response of a Speed Loop by Using a Reduced-Order Extended Kalman Filter Tao Liu 1 , Qiaoling Tong 1 , Qiao Zhang 2, *, Qidong Li 1 , Linkai Li 1 and Zhaoxuan Wu 1 1

2

*

School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] (T.L.); [email protected] (Q.T.); [email protected] (Q.L.); [email protected] (L.L.); [email protected] (Z.W.) School of Automation, Wuhan University of Technology, Wuhan 430074, China Correspondence: [email protected]; Tel.: +86-136-560-37870

Received: 9 October 2018; Accepted: 21 October 2018; Published: 24 October 2018







Abstract: In servo systems, encoders are usually used to measure the position and speed signals of electric machines. But in a low speed range, the traditional M/T method has a larger time delay, which will cause an increase of the speed loop order and degradation of the speed loop performance. A method employed to reduce the delay of speed feedback by using a reduced-order Extended Kalman Filter (EKF) is introduced in this paper. The speed of the permanent magnet synchronous motor is estimated by the reduced-order EKF in a low speed range, which reduces the delay of speed feedback and extends the cutoff frequency of the speed loop to improve the dynamic performance of the servo system. In order to solve the issues that the traditional full-order EKF is sensitive to the inertia of the system and computationally complex, a composite load torque observer (CLTO) is proposed in this paper. The load torque and the friction torque are simultaneously observed by the CLTO. Additionally, the CLTO is used to reduce the order of the EKF, which reduces the sensitivity of EKF on inertia to enhance the robustness of the algorithm and simplifies the computational complexity. The feasibility and effectivity of the above method are verified by simulations and experiments. Keywords: reduced-order EKF; response of speed loop; composite load torque observer; PMSM

1. Introduction In recent years, the permanent magnet synchronous motor (PMSM) has been widely used in servo systems because of its high-power density, controllability, and ability to easily achieve a high-precision speed and position control. As the inner loop of position servo systems, the speed loop plays an essential role in the whole system performance. How to reduce the sampling delay of position and speed signals under the premise of ensuring accuracy is a key factor, which will affect the dynamic response and stability of the speed loop. Incremental encoders are widely used in position and speed measurement because they are accurate and easy embed into the system. However, in a low speed range, the M/T method [1] needs a large time delay to obtain more accurate speed information, which inevitably introduces a large feedback delay and reduces the dynamic performance of the speed loop. The M/T method is a method which works as follows: 1. 2. 3.

Counting the number of encoder pulses m1 in a fixed period Tc . Recording the time interval ∆T of the next pulse edge after the fixed period. The calculated speed Nf (rpm) can be obtained as: Nf =

60(m1 + 1) 2πN ( Tc + ∆T )

(1)

where N is the total number of encoder pulses per rotation. Energies 2018, 11, 2886; doi:10.3390/en11112886

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Therefore, a large number of research has focused on velocity estimation methods based on sampled position signals only. However, due to discrete position signals and measurement noises, most linear observers cannot perform well in terms of both the steady-state performance and dynamic response [2–5]. Reference [2] proposed a velocity observer based on Phase Locked Loop (PLL), which reduces the fluctuation of steady-state speed, but its dynamic response is not significantly improved. Reference [3] proposed a linear observer based on an asynchronous pulse sampling method to improve the accuracy of steady-state measurement, but its linear gain cannot balance both steady-state and dynamic characteristics. With the development of control theory, the nonlinear observers have been extensively studied [6–10]. Although the nonlinear observer is used to improve the steady-state and dynamic performance, its dynamic response is not fast enough. Furthermore, its dynamic and steady-state parameters are not easy to determine. Reference [6] used the fal-function to construct a nonlinear observer. Compared with the linear observer, its steady-state characteristics are greatly improved, but the dynamic response characteristics still present a large feedback delay. Reference [7] proposed a method for constructing the acceleration observer by using a proportional-derivative (PD) controller to estimate speed, but the parameters of the PD controller are not easy to determine. Reference [8] proposed a method to construct a nonlinear velocity observer using mechanical models, but its gain parameters are higher in number and difficult to determine. At the same time, it requires too many motor parameters and the dynamic response is not significantly improved. Reference [9] proposed a position observer and a velocity estimator of a nonlinear actuator for an application in a sensorless control system for engines. The Extended Kalman Filter (EKF) is a filtering method based on model prediction, which helps to improve the response of the speed. Meanwhile, the measurement noise and process noise are considered in the EKF method, which helps to improve the steady-state performance [11–25]. Reference [11] constructed a third-order EKF for speed feedback, and its velocity loop has a good dynamic response, but requires precise inertia. A second-order EKF method which was constructed by using the voltage equation and mechanical equations of the motor is suggested in [12]. The q-axis current and the rotational speed of the motor are filtered together, but it requires the knowledge of many motor parameters. In [13], a fourth-order EKF is used for sensorless control by using the full model of a motor. Reference [14] constructed a fifth-order EKF to observe motor current and speed at the same time. Reference [18] proposed the unscented Kalman filter and the resilient extended Kalman filter-based sensorless direct torque control technique for permanent magnet synchronous motors. Reference [19] proposed a full-order Kalman Observer and Recursive Least Squares for online inertia identification and load torque observation. Reference [20] proposed an improved square root unscented Kalman filter for rotor speed estimation of permanent magnet synchronous motor sensorless drives. Reference [21] proposed a two-stage Kalman filter combining a robust and optimal algorithm for state and disturbance estimation. However, high-order EKFs are computationally complex and values of the covariance matrix are not easy to determine. They need more motor parameters, and they easily deteriorate or even fail to work properly when the parameters are mismatched, which limits their application. In this paper, a reduced-order EKF is proposed to reduce the delay of speed feedback and extend the cutoff frequency of the speed loop to improve the dynamic performance of the servo system. In order to reduce the computational complexity of the traditional third-order EKF, a composite load torque observer (CLTO) is proposed to reduce the order of EKF. Additionally, the reduced-order EKF is easy to embedd in different occasions. At the same time, the load torque and the friction torque are observed together by the CLTO, which avoids the problem that the system friction coefficient is not easy to obtain in practical applications. Compared with other load observers, the CLTO with a PI controller displays a better adaptability and faster convergence. Meanwhile, the problem that full-order EKFs are sensitive to the inertia of systems can be solved by the CLTO based on the PI adaptive convergence method, which improves the robustness of the algorithm. In this paper, the improved speed loop control strategy, the CLTO, and reduced order EKF equations are introduced in Section 2. In Section 3, the robustness of the system is analyzed. The choice

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order EKFs are sensitive to the inertia of systems can be solved by the CLTO based on the PI adaptive convergence method, which improves the robustness of the algorithm. Energies 2886 the improved speed loop control strategy, the CLTO, and reduced order3 EKF of 16 In2018, this11,paper, equations are introduced in Section 2. In Section 3, the robustness of the system is analyzed. The choice of the values of the covariance matrix, Q, P0, and R, is introduced in Section 4. Simulations and of the values of the covariance matrix, Q, P0 , and R, is introduced in Section 4. Simulations and experiments are given in Sections 5 and 6. Through the Simulink/MATLAB (9.1.0.441655 (R2016b)) experiments are given in Sections 5 and 6. Through the Simulink/MATLAB (9.1.0.441655 (R2016b)) and the experimental platform, the feasibility and effectivity of the above algorithm are verified by and the experimental platform, the feasibility and effectivity of the above algorithm are verified by simulations and experiments. The final section presents the conclusions of this paper. simulations and experiments. The final section presents the conclusions of this paper. 2. Speed Control Strategy Based on Filtered Speed Feedback by EKF 2. Speed Control Strategy Based on Filtered Speed Feedback by EKF 2.1. Overview Overview of of the the Proposed Proposed Speed Speed Loop Loop Control Control Strategy Strategy 2.1. According to to the the previous previous analysis, analysis, in in order order to to improve improve the the speed speed loop response, the the time According loop response, time delay delay of the speed feedback must be reduced. A reduced-order EKF which will be introduced in Section 2.4 2.4 of the speed feedback must be reduced. A reduced-order EKF which will be introduced in Section is used used as as the the speed speed closed-loop closed-loop control, control, as as shown shown in in Figure Figure 1. 1. The The control control strategy strategy is is aa dual dual closed closed is loop control based on Field Oriented Control (FOC). The inner loop is the current loop and the outer loop control based on Field Oriented Control (FOC). The inner loop is the current loop and the outer loop is the speed loop. The observed load torque is added to the output of the speed loop as the input loop is the speed loop. The observed load torque is added to the output of the speed loop as the of the of current loop. The angle filtered the EKF used coordinate transformation angle, and input the current loop. The angle by filtered by is the EKFasisthe used as the coordinate transformation the filtered speed is used as speed feedback. The filtered angle is used for coordinate transformation, angle, and the filtered speed is used as speed feedback. The filtered angle is used for coordinate which reduces the discretization the position of signal and the signal measurement to the system transformation, which reduces theofdiscretization the position and the noise measurement noise interference. Due to observing the load torque as feedback, the system has a certain inhibitory effect to the system interference. Due to observing the load torque as feedback, the system has a certain on the load disturbance and it improves the system’s anti-load disturbance ability. The construction inhibitory effect on the load disturbance and it improves the system’s anti-load disturbance ability. of the CLTO and the reduced-order EKF based on the mechanical model of PMSM will be introduced The construction of the CLTO and the reduced-order EKF based on the mechanical model of PMSM in the next section. will be introduced in the next section.

DC

i

*

* d

Ud

PI

i q*

Uq

PI

U

dq

PI



U

SVPWM

id

Inverter

ia ib

dq

iq abc

1/Kt

ˆ

TL Load Torque Observation

Kt

EKF

PMSM



Encoder

ˆ Figure 1. 1. The block diagram of the speed loop system. Figure

2.2. 2.2. Mechanical Mechanical Model Model of of PMSM PMSM According According to to mechanical mechanical equations equations of of PMSM: PMSM: J

dωmdm J − Bω − Bm= =TTe − −TLL m e dt dt

Te =

3 n p [ ϕ f i q − ( L d − L q )i d i q ] 2

(2) (2) (3)

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3 n [ i - ( Ld - Lq )id iq ] 2 p f q

(3)

J is the inertia, B is coefficient the coefficientofoffriction, friction, ω is the mechanical angel speed, Te is the where J iswhere the inertia, B is the m m is the mechanical angel speed, Te is the  f is the electromagnetic torque,torque, TL is Tthe load torque, thenumber number of pairs, pole pairs, ϕ fpermanent is the permanent electromagnetic L is the load torque, n npp isisthe of pole magnet flux linkage value,value, iq is the current, thed-axis d-axis current, Lq q-axis is theinductance, q-axis inductance, magnet flux linkage iq is q-axis the q-axis current, iidd isisthe current, Lq is the andd-axis Ld is theinductance. d-axis inductance. a surfacemount mount PMSM, thethe d-q d-q axis axis inductance is equal.is Therefore, and Ld is the As aAssurface PMSM, inductance equal. Therefore, (3) can be simplified as follows: (3) can be simplified as follows: 33 Te = n n  f iiqq (4) Te = (4) p pϕ 22 f Considering the sampling period can be be obtained that: Considering the sampling period is Tiss ,Tsit, itcan obtained that:

(



 −

= T +

1

aT 2

1 2 1 m s θk −θkk−1 k −= ωm Ts + 2 2s aTs  TeT− T − Bω − T − B  0 m L m a = ωam= =m = e JL

 

(5)

(5)

J

Therefore,Therefore, the state of the system obtained (4),(5), and (5), as follows: theequation state equation of the systemcan can be be obtained by by (2), (2), (4), and as follows: (



Ts

 m= + m T+s ( T(eTe− − TTL − − BBω ) ) m θ 0 = ω m L 2J 2 J  1 0 ωm = ( T − T − Bω ) 1 m L  J  e 

(6)

m = (Te − TL − B m )

(6)

J

2.3. Composite Load Torque Observer 2.3. Composite Load Torque Observer

State variables such as the load torque, the mechanical angle, and the speed are required in the State variables such as the load torque, the mechanical angle, and the speed are required in the traditional third-order EKF. In order to reduce the order of a traditional EKF, the load torque needs to traditional third-order EKF. In order to reduce the order of a traditional EKF, the load torque needs be known.toTherefore, a CLTO is used to estimate the load torque. The ofthe theCLTO CLTO be known. Therefore, a CLTO is used to estimate the load torque. Thestructure structure of is is shown in Figure 2. Theinspeed the electromagnetic torquetorque are used as as thetheinput shown Figure and 2. The speed and the electromagnetic are used input variables variables of of thethe CLTO. CLTO. A proportional-integral adaptive convergence in CLTO. Therefore, A proportional-integral (PI) adaptive(PI) convergence method method is usedisinused CLTO. Therefore, thethe load torque load torque can be obtained by the PI controller according to the error of the estimated speed and the can be obtained by the PI controller according to the error of the estimated speed and the input speed. input speed.

Te

ˆ

PI

TL

1/J





Figure 2. The structure diagram of CLTO. Figure 2. The structure diagram of CLTO.

According to (2), (7) can be obtained as follows:

According to (2), (7) can be obtained as follows:

d TˆL = TL + Bm = Te − J m dtdωm

Tˆ L = TL + Bωm = Te − J

(7)

dt torque and the friction torque. A where TˆL is the composite load torque, which includes the load

(7)

is constructed to estimate the composite load torque and obtain the discretization where Tˆ L velocity is the observer composite load torque, which includes the load torque and the friction torque. equation as: A velocity observer is constructed to estimate the composite load torque and obtain the discretization equation as: ( ωk = ωk−1 + TJs ( Te(k) − Tˆ L(k−1) ) (8) Tˆ L(k) = (K p + Ki )(ωˆ k − ωk ) + Uk−1

where ωˆ k is the input speed. Here, the EKF output estimated speed is used as the input speed of the CLTO. ωk is the estimated speed of the speed observer. Kp and Ki are parameters of the PI controller. Uk−1 is the integral value of Ki at time k − 1. The composite load torque is calculated by the PI adaptive convergence method, which reduces the dependence of the observer on the inertia and improves the robustness of the system.

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2.4. Reduced-Order EKF Equation According to the mentioned torque observer, the traditional third-order EKF can be reduced to a second-order EKF. The mechanical angle θ and mechanical speed ωm are set as state variables, the electromagnetic torque Te and the composite load torque Tˆ L are set as input variables, and the mechanical angle θ is set as the observational variable. Then, (6) can be rewritten as: "

θ0 0 ωm

#

"

=

0 0

1 0

#"

θ ωm

#

"

+

Ts 2J 1 J

Ts − 2J − 1J

#"

Te Tˆ L

# (9)

Discretizing (9) by the Euler method, (10) can be obtained: (

xk = Ak−1 xk−1 + Bk−1 uk−1 yk = Hk xk

(10)

Considering the prosses noise is ω and measurement noise is v, (10) can be rewritten as: (

# " 2 Ts 1 Ts 2J where A = ,B= Ts 0 1 J and Q and R are the covariance which can be expressed as: "

xk = Ak−1 xk−1 + Bk−1 uk−1 + wk−1 yk = Hk xk + vk

(11)

2 # h iT h iT h i − T2Js ˆL , x = , u = , y = θ, H = , θ ω T T 1 0 m e − TJs matrices of prosses noise and measurement noise, respectively,

" #    Q = cov(w) = q0 0 0 q1   R = cov(v) = r

(12)

The recursive progress of the reduced-order EKF can be separated by three steps: 1.

Calculating the load torque using the load observer Tˆ L(k) .

2.

State variables prediction and Kalman gain calculation:

3.

xek+1 = Ae xk + Buk

(13)

Pek+1 = A Pˆk A T + Q

(14)

e T ( H PH e T + R ) −1 Kk+1 = PH

(15)

Calculating the optimal estimate values and update matrix: xek+1 = xek + Kk+1 (yk − H xek )

(16)

Pˆk+1 = ( I − Kk+1 H ) Pek

(17)

There are a lot of matrix operations in the EKF method. The third-order matrix operations are simplified by the reduced-order EKF, which reduces the computational complexity. Meanwhile, the EKF is a filter method based on a model which is sensitive to the inertia of the system. However, the problem can be suppressed by the CLTO, which will be discussed in the next section.

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3. Robustness Analysis Since the system adopts a new control strategy, we need to discuss the robustness of the system. Assuming the inertia input error is ∆J at time k, and the time k − 1 data are all correct values, then the changed inertia can be obtained as follows: e J = J + ∆J = αJ

(18)

where α is the coefficient of the inertia error. eL as the estimated load torque e is recorded as the estimated speed and T In this approach, ω obtained from the CLTO after the inertia is changed. Then, the calculated values from (8) at time k are as follows: Ts ωk = ωk−1 + ( Te(k−1) − Tˆ L(k−1) ) (19) J e k = ω k −1 + ω

Ts (T − Tˆ L(k−1) ) αJ e(k−1)

(20)

According to (19) and (20), the speed estimation error due to the inertia error is obtained as: e k − ω k = β k −1 ( ∆ωk = ω

1 − 1) α

(21)

where β k−1 = TJs ( Te(k−1) − Tˆ L(k−1) ). Also, the observed load torque using the PI controller from (8) can be expressed as: Tˆ L(k) = (K p + Ki )(ωˆ k − ωk ) + Uk−1

(22)

eL(k) = (K p + Ki )(ωˆ k − ω e k ) + Uk−1 T

(23)

According to (22) and (23), the torque estimation error due to the inertia error can be obtained as: eL(k) − Tˆ L(k) = (K p + Ki ) β k−1 (1 − 1 ) ∆TL(k) = T α

(24)

The result of load torque is then incorporated into (13) to calculate the predicted speed of EKF. Here, it declares that ωˆ k+1 as the predicted speed of EKF using the correct inertia J, with ω k+1 as the eL(k) , and ^ predicted speed of EKF using the changed inertia and the uncorrected load torque T ω k+1 as the eL(k) , respectively. So: predicted speed of EKF using the changed inertia and the corrected load torque T ωˆ k+1 = ωˆ k +

Ts − Tˆ L(k) ) (T J e(k)

(25)

ω k+1 = ωˆ k +

Ts (T − Tˆ L(k) ) αJ e(k)

(26)

Ts eL(k) ) (T −T αJ e(k)

(27)

^

ω k+1 = ωˆ k +

Then, the predicted speed error caused by the inertia error can be obtained from (25), (26), and (27) as: 1 (28) ∆ω k+1 = ω k+1 − ωˆ k+1 = β k ( − 1) α ^ ^ 1 Ts ∆ ω k+1 = ω k+1 − ωˆ k+1 = β k ( − 1) + ∆TL(k) (29) α αJ Considering β k is approximately equal to β k−1 , (29) can be rewritten as: ^

∆ ω k+1 = k∆ω k+1

(30)

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(K +K )T

where k = (1 − p αJ i s ). According to (30), the predicted speed error of EKF using the corrected load torque is reduced by k times compared to the uncorrected predicted speed error. From this, it can be seen that appropriately increasing the PI parameter values of the load observer helps to reduce the speed estimation error due to the inertia error. However, when the parameter values of the PI controller are too large, the steady-state fluctuation of the observed load torque will increase. So, it needs to compromise the choice of the PI parameter. It is shown that the load observer using the PI convergence method can reduce the dependence of the system on the inertia and improve the robustness of the system. 4. Design and Tuning of Q and R A critical point during the design of EKF is the choice of the elements of the covariance matrix Q, R and the initial values of the covariance matrix P0 , which will affect the performance and the convergence [13]. In this paper, values of the covariance matrixes are obtained by trials according to the requirement of the actual system. Matrix Q is related to the process noise. Increasing Q would indicate the presence of either heavy process noise or increased parameter uncertainty. As shown in Figure 3, increasing the value of q0 will cause the value of k0 to be increased and the value of k1 to be decreased; where k0 is the element of gain matrix K which is related to the angle error correction. The k1 is the element of gain matrix K which is related to the speed error correction. Furthermore, as shown in Figure 4, increasing the value of q1 will cause both the value of k0 and the value of k1 to be increased. Moreover, the value of k1 is a major factor affecting the loop performance. It can be seen from Figures 5 and 6 that when the value of q0 decreases or the value of q1 increases, it will cause the value of k1 to be increased. This leads to an improvement in the dynamic performance of the loop, but a large steady-state fluctuation. On the other hand, matrix R is related to measurement noise. Increasing the value of the element of matrix R means the measurements are affected by noise and thus they are of little confidence. Increasing the value of r will cause both the value of k0 and the value of k1 to be decreased. When the values of the gain matrix K decrease, the system correction weight decreases and the state performance of the system is improved. Therefore, the comprehensive consideration has been chosen as: Energies 2018, 11, x FOR PEER REVIEW

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" #   0  Q = 0.1  00.1 12000 0   Q =   0.1   R=  0r = 12000  R = r =0.1 

k0

(31) (31)

k1

q0

q0

(b)

(a)

ofof EKF gain k1 kvaries with q0 (q Figure 3. (a) The curve curve of of EKF EKFgain gainkk00varies varieswith withq0q,0and , and(b) (b)the thecurve curve EKF gain with q01 1 varies = 112000, r = 0.1) (simulation). (q = 12000, r = 0.1) (simulation).

k0

k1

q0 q0

q0 q0

(b) (b)

(a) (a)

Figure 3. (a) The curve of EKF gain k0 varies with q0, and (b) the curve of EKF gain k1 varies with q0 (q1 Figure 3. (a) The curve of EKF gain k0 varies with q0, and (b) the curve of EKF gain k1 varies with q0 (q1 8 of 16 = 12000, r = 0.1) (simulation).

= 12000, r =2886 0.1) (simulation). Energies 2018, 11,

k1 k1

k0 k0

(a) (a)

q1 q1 (b) (b)

q1 q1

Speed(rpm) Speed(rpm)

Speed(rpm) Speed(rpm)

Figure 4. (a) The curve of EKF gain k0 varies with q1, and (b) the curve of EKF gain k1 varies with q1 (q0 Figure varies with with qq11, ,and and(b) (b)the thecurve curveofofEKF EKFgain gaink1kvaries with Figure4.4.(a) (a)The Thecurve curve of of EKF EKF gain gain k00 varies with q1 q(q1 0 1 varies = 0.1, r = 0.1) (simulation). (q=00.1, = 0.1, r = 0.1) (simulation). r = 0.1) (simulation).

time(sec) time(sec) (b) (b)

time(sec) time(sec)

(a) (a)

(a) The The speed speed response response at different different values of q00,, and and (b) (b) partial partial amplification amplification at a steady state Figure 5. (a) 12000, rr==0.1) 0.1)(simulation). (simulation). (q11 ==12000, (q1 = 12000, r = 0.1) (simulation).

Speed(rpm)

Speed(rpm)

EnergiesFigure 2018, 11, FOR REVIEW 9 of 17 5. x(a) ThePEER speed response at different values of q0, and (b) partial amplification at a steady state

time(sec)

time(sec)

(a)

(b)

Figure Figure 6. 6. (a) (a) The The speed speed response response at at different different values values of of qq11,, and and (b) (b) partial partial amplification amplification at at aa steady steady state state (q (q00 ==0.1, 0.1,rr==0.1) 0.1)(simulation). (simulation).

As mentioned before, it can be seen from Figure 7 that the initial value of the covariance matrix P0 has a slightly influence on the loop performance. So, the initial value of the covariance matrix P0 is selected as:

0 0  P0 =   0 0 

(32)

time(sec)

time(sec)

(a)

(b)

Figure 6. (a) The speed response at different values of q1, and (b) partial amplification at a steady state (q02018, = 0.1,11,r 2886 = 0.1) (simulation). Energies

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As mentioned before, it can be seen from Figure 7 that the initial value of the covariance matrix Asa mentioned before,on it can be seen from Figure 7 that the initial the covariance matrix P0 has slightly influence the loop performance. So, the initial valuevalue of theofcovariance matrix P0 is P has a slightly influence on the loop performance. So, the initial value of the covariance matrix P 0 0 is selected as: selected as: " # 0 00  0 P0 =  (32) P0 = (32)  00 00 

Speed(rpm)

From the the above above results, results, the thementioned mentionedalgorithm algorithmcan canwork workstably stably within a large range within a large range of of parameters. parameters. feedback of the encoder, so that the filtered speed can be kept This is due to the actual position feedback within a reasonable range. Therefore, Therefore, the the EKF EKF algorithm algorithm relying relying on position feedback feedback has has a good algorithm is working properly within, but but is not stability and and convergence. convergence.ItItisisverified verifiedthat thatthe theabove above algorithm is working properly within, is limited to, this range of parameters: not limited to, this range of parameters:  10−-44 ,,1]  ×10 1]  q0 q∈0 [1[1  (33) q1q1∈[[4000,60000] 4000, 60000] (33)    r ∈ [0.01, 1]  r  [0.01,1]

time(sec) Figure The speed speed response response at (simulation). Figure 7. 7. The at different different initial initial values values of of P P00 (simulation).

5. Simulation and Analysis 5.1. Step Excitation Simulation After the theoretical and robustness analysis, a simulation system was built by Simulink/ MATLAB. Parameters of PMSM and PI parameters of the torque observer used in simulations and experiments are shown in Table 1. The sampling time is selected depending on the control period of the speed loop and the current loop. Generally speaking, the control frequency of the current loop is about 12 kHz, which is an empirical value. In practical applications, the requirement of the power module in a hardware circuit and the current harmonic waves of the motor should be considered. Taking the Intelligent Power Module (FSBB30CH60F) as an example, the switching frequency is up to 20 kHz. Considering the need to minimize current harmonics and switching losses, the current loop control frequency chosen is 12 kHz. Since the mechanical response speed of the motor is much smaller than the electrical response speed, the control frequency of the speed loop selected is 4 kHz. Moreover, the sampling frequency of speed feedback and the control frequency of the speed loop are consistent for easy control. Therefore, the sampling time selected is 0.25 ms.

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Table 1. Parameters of PMSM and PI parameters of the torque observer. Parameters

Values

Stator resister R (Ω) Stator inductance L (mH) Pole-pairs number J (kg m2 ) ϕf (Wb) Kp Ki Ts (ms) Encoder (1/rev)

1.86 2.8 4 2.45 × 10−4 0.109 0.03 0.005 0.25 10000

Speed(rpm)

Figure 8 shows the simulation result of 1000 rpm step excitation. From the result, it can be seen that the delay between the estimated speed by a reduced-order EKF and the actual speed is smaller than the delay between the speed calculated by an encoder and the actual speed. This shows that the speed loop which estimated speed by the reduced-order EKF has a faster step response. Therefore, this improved method is likely to follow changes quickly when a set speed is changed rapidly. Thus, the improved method extends the cutoff frequency of the speed loop to improve the dynamic performance of the servo system. In the initial stage of the simulation, the EKF estimated the speed curve slightly ahead of the real speed curve, which is caused by the dynamic response of the CLTO not being fast enough. When the speed reaches the set value, there is no significant difference Energies 2018, x FOR PEER REVIEW 11 of 17 between the11,steady-state speed of the two methods.

time(sec) Figure 8. Simulation result under 1000 rpm step excitation (simulation). Figure 8. Simulation result under 1000 rpm step excitation (simulation).

m)

m)

5.2. Load Torque Simulation 5.2. Load Torque Simulation In order to verify the dynamic performance of the above algorithm when the load changes, In order when to verify performance of the above algorithm when themotor load changes, the the situation the the loaddynamic torque increases and the load torque decreases during operation is situation when the load torque increases and the load torque decreases during motor operation is simulated, as shown in Figures 9 and 10. When the load torque suddenly increases or decreases at simulated, as shown in Figures 9 and 10. Whenbut theitload increases decreases 0.5 0.5 s, the speed will drop or overshoot slightly, willtorque return suddenly to the set speed veryorquickly in a at short s, the speed will drop or overshoot slightly, but it will return to the set speed very quickly in a short time, indicating that the system is highly resistant to load disturbance. This is because the CLTO has a time, indicating the system is highly to load disturbance. This isremain becausestable the CLTO certain negative that feedback effect when the resistant load changes so that the system can whenhas the aload certain negative effect when the load changes so that the system can remain changes. The feedback load torque estimated by the load observer contains the friction torque stable and thewhen load the loadso changes. The load torque estimated by the load value observer the friction torqueincreases. and the torque, it is slightly larger than the actual load torque andcontains will increase as the speed load torque, so it is slightly larger than the actual load torque value and will increase as the speed increases.

Speed(rpm)

Torque(Nm)

time, indicating that the system is highly resistant to load disturbance. This is because the CLTO has a certain negative feedback effect when the load changes so that the system can remain stable when the load changes. The load torque estimated by the load observer contains the friction torque and the load torque, so it is slightly larger than the actual load torque value and will increase as the speed increases. Energies 2018, 11, 2886 11 of 16

time(sec)

time(sec)

(b)

(a)

Figure (a) Speed curve loadsuddenly suddenlyincreases increasesfrom from0.5 0.5Nm Nmtoto1 1Nm Nmatat0.5 0.5s,s,and and(b) (b)observed observed Figure 9.9.11, (a) curve atatload Energies 2018, xSpeed FOR PEER REVIEW 12 of 17

Speed(rpm)

Torque(Nm)

loadtorque torque(simulation). (simulation). load

time(sec)

time(sec)

(b)

(a)

Figure 10.10. (a)(a) Speed curve at at load suddenly decreases from 1 Nm to to 0.50.5 Nm at at 0.50.5 s, and (b)(b) observed Figure Speed curve load suddenly decreases from 1 Nm Nm s, and observed load torque (simulation). load torque (simulation).

orque(Nm)

Speed(rpm)

5.3. Inertia Simulation 5.3. Inertia Simulation Figure 11 shows the response of the system under different inertia errors. From this, it can be Figure 11 shows the response of the system under different inertia errors. From this, it can be seen that when the load inertia has a certain input error, the above algorithm can still run stably with a seen that when the load inertia has a certain input error, the above algorithm can still run stably with good steady-state and dynamic performance, which proves that the system has a good robustness. a good steady-state and dynamic performance, which proves that the system has a good robustness. It can be seen from Figure 6b that when the input inertia is smaller than the actual inertia, the observed It can be seen from Figure 6b that when the input inertia is smaller than the actual inertia, the load torque becomes much larger. In this way, the speed error due to the inertia is compensated for observed load torque becomes much larger. In this way, the speed error due to the inertia is by the larger observed torque. Therefore, the CLTO can reduce the dependence of the system on the compensated for by the larger observed torque. Therefore, the CLTO can reduce the dependence of inertia. When the system reaches a steady-state, the observed load torque will converge to the true the system on the inertia. When the system reaches a steady-state, the observed load torque will load torque, ensuring the steady-state balance. This is consistent with our theoretical analysis. converge to the true load torque, ensuring the steady-state balance. This is consistent with our theoretical analysis.

Speed(rpm)

Torque(Nm)

It can be seen from Figure 6b that when the input inertia is smaller than the actual inertia, the observed load torque becomes much larger. In this way, the speed error due to the inertia is compensated for by the larger observed torque. Therefore, the CLTO can reduce the dependence of the system on the inertia. When the system reaches a steady-state, the observed load torque will Energies 2018,to 11, the 2886 true load torque, ensuring the steady-state balance. This is consistent with 12 ofour 16 converge theoretical analysis.

time(sec)

time(sec)

(a)

(b)

Figure11. 11.(a) (a)Speed Speed step under different inertia error, andsimulation (b) simulation result of observed load Figure step under different inertia error, and (b) result of observed load torque torque (simulation). (simulation).

6.6.Experiment Experimentand andAnalysis Analysis 6.1. Experimental Platform Introduction 6.1. Experimental Platform Introduction Energies 2018, 11, x FOR PEER REVIEW 13 of 17 After simulations, the relevant experiments were carried out. The photographs of the experimental After simulations, the relevant experiments were carried out. The photographs of the platform are shown in Figure 12. parameters of the control board are shown in Table 2.with Thean control Table 2. The control board the The servo-drive by ourselves, is equipped NXP experimental platform areisshown in Figuredesigned 12. The parameters of which the control board are shown in board is the servo-drive designed by ourselves, which is equipped with an NXP DSP KV3125612 DSP KV3125612 microprocessor and an intelligent power module. With 220 V AC as the input, the microprocessor intelligentcan power module. With 220which V ACbelongs as the input, thepower outputservo power of the output power ofand thean servo-drive reach up to 0.75 Kw, to a low system. servo-drive can reach up to 0.75 Kw, which belongs to a low power servo system. The load motor The load motor and load driver are Fuji RYC102C3-VVT2 and the rated output torque is 5 Nm. and The load driver arebandwidth Fuji RYC102C3-VVT2 and the rated is 5 Nm. loopspeed bandwidth current loop of the experiment servooutput motortorque is 1 kHz, andThe thecurrent improved loop of the experiment 1 kHz, and the improved speedisloop bandwidth the servo system bandwidth of the servo servo motor systemiswith a low-resolution encoder 300 Hz, which isofgreatly improved with a low-resolution encoder is 300 Hz, which is greatly improved compared with the M/T method. compared with the M/T method.

Ptototype PMSM

Load PMSM

Figure 12. Photographs of the experimental platform. Table 2. Table 2. Parameters Parameters of of the the experimental experimental platform. platform. Parameters Parameters

Values Values

Rated Power (W) (W) Rated Power Rated Torque (Nm) Rated Torque (Nm) Rated Speed (Rpm) Rated Speed (Rpm) Rated Current (A)

750 750 2.42.4 3000 3000 4.5

Rated Current (A)

4.5

6.2. Step and Frequency Response Experiment The above algorithm was tested by a step test through the experimental platform. As shown in Figure 13, the speed estimated by the reduced-order EKF has a faster response and is more stable than the speed calculated by the incremental photoelectric encoder. Figure 14 shows the speed response at very low-speed excitation (10 rpm) and very high-speed excitation (1000 rpm),

Energies 2018, 11, 2886

13 of 16

6.2. Step and Frequency Response Experiment The above algorithm was tested by a step test through the experimental platform. As shown in Figure 13, the speed estimated by the reduced-order EKF has a faster response and is more stable than the speed calculated by the incremental photoelectric encoder. Figure 14 shows the speed response at very low-speed excitation (10 rpm) and very high-speed excitation (1000 rpm), respectively. It can be seen from the step response results under different speed excitations that the speed feedback estimated by EKF has a smaller feedback delay. The above results also show that the mentioned algorithm can Energies 2018, x FOR PEER REVIEW 14 of 17 run11, well at different speeds. Energies 2018, 11, x FOR PEER REVIEW

14 of 17 input speed EKF_speed input speed Encoder_Speed EKF_speed Encoder_Speed

120 120 100

80 80

Speed(rpm)

Speed(rpm)

100

103 103 102 102

60 60

101 101

40 40

100 100 99

20

99

20

98

98 0.050 0.050

0

0

0.000

0.025

0.000

0.025 time(sec)

0.055

0.060

0.055

0.060

0.050

0.050

time(sec) Figure 13. Step response and partial amplification at 100 rpm excitation (experiment). Figure 13. Step response and partial amplification at 100 rpm excitation (experiment).

Figure 13. Step response and partial amplification at 100 rpm excitation (experiment).

10 8 6

1200

12

1000 1200

10

800 1000

8

600

6 4 input speed Kalman_speed Encoder_speed

0 0.00

2

0.01

0.02

0.03

0.04 input 0.05speed 0.06

0.07

0 0.01

0.02

0.03

(a) 0.04

0.05

0.06

600 input speed Kalman_speed Encoder_speed

200

time(sec)Kalman_speed Encoder_speed 0.00

800

400

2

4

Speed(rpm) Speed(rpm)

Speed(rpm)

12

Speed(rpm)

14

14

400 0 0.00 200

0.01

0.02

0 0.00

0.07

0.03

0.04

0.05

time(sec) 0.01

(b) 0.03 0.02

0.06 input 0.07 speed

Kalman_speed Encoder_speed

0.04

0.05

0.06

0.07

time(sec) time(sec) Figure 14.14. Step response speedexcitation, excitation,respectively respectively (experiment). Figure Step responseatat(a) (a)10 10rpm rpmand and(b) (b) 1000 1000 rpm speed (experiment).

10 0 -10 -20 -30 -40

-10

-10

20

-20

-20

-3010

-30

-40 0

-40 0.010

Speed(rpm) Speed(rpm)

20

Speed(rpm)

Speed(rpm)

(a) the speed response at 100 Hz and 200 Hz velocity excitation. (b) It can be seen that Figure 15 shows input speed the velocity by EKF has a faster dynamic response and a smaller amplitude attenuation. input speed 40 Step estimated 40 Figure 14. response at (a) 10 rpm Kalman_speed and (b) 1000 rpm speed excitation, respectively (experiment). Kalman_speed Encoder_speed Therefore, the cutoff frequency of the speed loop is extended and the dynamic performance of the Encoder_speed 30 30 speed loop is significantly improved. Through step experiments and frequency response experiments, 20 20 it is verified that the control strategy input with speed the reduced-order EKF for speed feedback has a faster 10 10 speed 40dynamic Kalman_speed response and less steady-state fluctuation. It also proves the effectivity and feasibility input of the 40 Kalman_speed Encoder_speed 0 0 Encoder_speed 30above algorithm. 30

0.015

0.020

time(sec) (a)

0.025

0.030

0.0200

-10 -20

0.0225

0.0250

0.0275

0.0300

time(sec) (b)

-30

-40 excitation, respectively (experiment). Figure 15. Loop response at (a) 100 Hz and (b) 200 Hz speed 0.0200

0.0225

0.0250

0.0275

0.0300

Encoder_speed

0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

Encoder_speed

0 0.00

0.07

0.01

0.02

0.03

0.04

0.05

0.06

0.07

time(sec)

time(sec) (a)

(b)

EnergiesFigure 2018, 11, 14 of 16 14.2886 Step response at (a) 10 rpm and (b) 1000 rpm speed excitation, respectively (experiment).

input speed Kalman_speed Encoder_speed

30

30

20 10 0 -10 -20

20 10 0 -10 -20 -30

-30

-40

-40 0.010

input speed Kalman_speed Encoder_speed

40

Speed(rpm)

Speed(rpm)

40

0.015

0.020

0.025

0.0200

0.030

0.0225

0.0250

0.0275

0.0300

time(sec)

time(sec)

(b)

(a)

Figure 15. Loop response at (a) 100 Hz and (b) 200 Hz Hz speed speed excitation, excitation, respectively respectively (experiment). (experiment).

6.3. 6.3. Load Load Torque Torque Experiment Experiment Figure 50 Hz Hz excitation. excitation. Figure 16 16 shows shows the the results results of of the the speed speed feedback feedback and and observed observed load load torque torque at at 50 Since the observed composite load torque includes a fixed load torque and a friction torque Since the observed composite load torque includes a fixed load torque and a friction torque as as aa function function of of speed, speed, the the frequency frequency of of the the observed observed composite composite torque torque should should be be consistent consistent with with the the frequency of the speed feedback. The result shown in Figure 15b is consistent with the theoretical frequency of the speed feedback. The result shown in Figure 15b is consistent with the theoretical analysis. analysis. However, However, due due to to the the delay delay of of calculation, calculation, the the phase phase of of the the observed observed composite composite load load torque torque will hysteresis The actual actual load load torque torque is is 0.55 will have have a certain certain hysteresis compared to to the the speed speed feedback. feedback. The 0.55 Nm, Energies 2018, a11, x FOR PEER REVIEWcompared 15 ofNm, 17 which is similar to the observed value (0.545 Nm). which is similar to the observed value (0.545 Nm). input speed Kalman_speed

40

0.560

20

Torque(Nm)

Speed(rpm)

Observed Load Torque

0.565

30

10 0 -10 -20 -30

0.555 0.550 0.545 0.540 0.535 0.530

-40

0.525 0.01

0.02

0.03

time(sec) (a)

0.04

0.01

0.02

0.03

0.04

time(sec) (b)

Figure Figure16. 16. (a) (a)Loop Loopresponse responseatat5050Hz Hzspeed speedexcitation, excitation,and and(b) (b)observed observedload loadtorque torqueby byCLTO CLTO (experiment). (experiment).

6.4.Inertia InertiaExperiment Experiment 6.4. Figure17 17shows showsthe theresults resultsof ofthe thevelocity velocitystep stepexperiment experimentunder underdifferent differentinertia inertiaerror. error.ItItcan can Figure be seen that when the inertia is not matched, the speed estimated by the reduced-order EKF will be seen that when the inertia is not matched, the speed estimated by the reduced-order EKF will have have different time delays the true speed. However, work stablyand andthe thedynamic dynamic different time delays than than the true speed. However, theythey cancan all all work stably performanceisisnot notsignificantly significantlydegraded. degraded.Through Throughthe theEKF EKFequation, equation,we wecan cansee seethat thatthe theinertia inertiaand and performance torque are used to predict the change of speed and thus reduce the speed feedback delay. When the torque are used to predict the change of speed and thus reduce the speed feedback delay. When the inertia does not match, it will lead to inaccurate prediction models, which will lead to speed prediction inertia does not match, it will lead to inaccurate prediction models, which will lead to speed errors. However, the torquethe corrected by the PI controlled CLTO canCLTO be used suppress model prediction errors. However, torque corrected by the PI controlled cantobe used to the suppress error caused by the inertia error. This shows that the torque observer through PI convergence the model error caused by the inertia error. This shows that the torque observer through has PI a certain compensation effect on the inertia error, which improves the robustness of the algorithm. convergence has a certain compensation effect on the inertia error, which improves the robustness of the algorithm. At the same time, the speed has different degrees of overshoot under different moments of inertia error. This is because, in practical applications, the choice of the speed loop parameter is related to inertia. When the input inertia is too large, the set PI parameter will be too large, resulting in a certain overshoot.

different time delays than the true speed. However, they can all work stably and the dynamic performance is not significantly degraded. Through the EKF equation, we can see that the inertia and torque are used to predict the change of speed and thus reduce the speed feedback delay. When the inertia does not match, it will lead to inaccurate prediction models, which will lead to speed prediction errors. However, the torque corrected by the PI controlled CLTO can be used to suppress Energies 2018, 11, 2886 15 of 16 the model error caused by the inertia error. This shows that the torque observer through PI convergence has a certain compensation effect on the inertia error, which improves the robustness of the algorithm. At the the speed same has time,different the speed has different degrees ofdifferent overshoot under different At the same time, degrees of overshoot under moments of inertia moments of isinertia error. This is because, in practical applications, the choice of the speed loop error. This because, in practical applications, the choice of the speed loop parameter is related parameter is related to inertia. When the input inertia is too large, the set PI parameter will be tooa to inertia. When the input inertia is too large, the set PI parameter will be too large, resulting in large, resulting in a certain overshoot. certain overshoot. 600

Speed(rpm)

500 400 300 200

input speed J 0.7J 1.3J 2J

100 0 0.00

0.01

0.02

time(sec) Figure17. 17.Experimental Experimentalresults resultsof ofvelocity velocitysteps stepsunder underdifferent differentinertia inertiaerror error(experiment). (experiment). Figure

7. Conclusions 7. Conclusions The improved speed loop control strategy performs well in terms of both a dynamic and The improved speed loop control strategy performs well in terms of both a dynamic and steadysteady-state performance. The experimental results of the step and frequency response show that state performance. The experimental results of the step and frequency response show that it solves it solves the problem of the large speed delay of the traditional M/T method in a low speed range. the problem of the large speed delay of the traditional M/T method in a low speed range. Using a Using a CLTO to reduce the order of EKF can reduce the computational complexity of EKF and make it CLTO to reduce the order of EKF can reduce the computational complexity of EKF and make it easy easy to use in different occasions. Through theoretical analysis and experiments, it is verified that the torque observer using the PI convergence method can effectively improve the robustness of the system. At the same time, the CLTO observed the load torque and the friction torque together, avoiding the problem that the coefficient of friction is not easy to measure in practical applications. The feasibility and effectivity of the above algorithm are verified by the simulations and experiments. Due to limited capabilities, the above algorithm still has some limitations. For example, the dynamic response of the observed torque is not fast enough. More advanced parameter-insensitive torque observation methods are needed. More advanced control strategies need to be applied to the control of the speed loop. Further work will focus on more advanced control algorithms, more advanced torque observers, and inertia identification methods. Author Contributions: T.L., Q.L., and Q.Z. conceived and designed the experiments; T.L., Q.L., L.L., and Z.W. performed the experiments; T.L., Q.L., L.L., and Z.W. analyzed the data; Q.T. and Q.Z. contributed materials and the experimental platform; T.L., Q.T., Q.Z., Q.L., L.L., and Z.W. wrote the paper. Funding: This research received no external funding. Acknowledgments: This work is supported by School of Optical and Electronic Information, Huazhong University of Science and Technology. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3.

Ohmae, T.; Matsuda, T.; Kamiyama, K. A microprocessor-controlled high-accuracy wide-range speed regulator for motor drives. IEEE Trans. Ind. Electron. 1982, 29, 207–211. [CrossRef] Robert, D.L.; Keith, W.V. High-resoolution velocity estimation for all-digital, ac servo drives. IEEE Trans. Ind. Appl. 1991, 27, 701–705. Lee, S.H.; Lasky, T.A.; Velinsky, S.A. Improved velocity estimation for low-speed and transient regimes using low-resolution encoders. IEEE/ASME Trans. Mechatron. 2004, 9, 553–560. [CrossRef]

Energies 2018, 11, 2886

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

16 of 16

Saito, K.; Kamiyama, K.; Ohmae, T.; Matsuda, T. A microprocessor-controlled speed regulator with instantaneous speed estimation for motor drives. IEEE Trans. Ind. Electron. 1988, 35, 95–99. [CrossRef] Ovaska, S.J. Improving the velocity sensing resolution of pulse encoders by FIR prediction. IEEE Trans. Instrum. Meas. 1991, 40, 657–658. [CrossRef] Marc, B.; John, C.; Robert, T.N. Nonlinear speed observer for high-performance induction motor control. IEEE Trans. Ind. Electron. 1995, 42, 337–343. Lee, S.H.; Song, J.B. Acceleration Estimator for low-velocity and low-acceleration regions based on encoder position data. IEEE/ASME Trans. Mechatron. 2001, 6, 337–343. Su, Y.X.; Zheng, C.H.; Sun, D.; Duan, B.Y. A simple nonlinear velocity estimator for high-performance motion control. IEEE Trans. Ind. Electron. 2005, 52, 1161–1169. [CrossRef] Mercorelli, P. A Motion-Sensorless Control for Intake Valves in Combustion Engines. IEEE Trans. Ind. Electron. 2017, 64, 3402–3412. [CrossRef] Yang, S.M.; Ke, S.J. Performance evaluation of a velocity observer for accurate velocity estimation of servo motor drives. IEEE Trans. Ind. Appl. 2000, 36, 98–104. [CrossRef] Kim, H.W.; Sul, S.K. A new motor speed estimator using Kalman filter in low-speed range. IEEE Trans. Ind. Electron. 1996, 43, 498–504. Ali, W.H.; Gowda, M.; Cofie, P.; Fuller, J. Design of a speed controller using extended Kalman filter for PMSM. Trans. Int. Midwest Symp. Circuits Syst. 2014, 57, 1101–1104. Bolognani, S.; Tubiana, L.; Zigliotto, M. Extended Kalman filter tuning in sensorless PMSM drives. IEEE Trans. Ind. Appl. 2003, 39, 1741–1747. [CrossRef] Zheng, Z.D.; Fadel, M.; Li, Y.D. High performance PMSM sensorless control with load torque observation. Trans. China Electrotech. Soc. 2007, 22, 18–23. Dhaouadi, R.; Mohan, N.; Norum, L. Design and implementation of an extended kalman filter for the state estimation of a permanent magnet synchronous motor. IEEE Trans. Power Electron. 1991, 6, 491–497. [CrossRef] Bolognani, S.; Oboe, R.; Zigliotto, M. Sensorless full-digital PMSM drive with EKF estimation of speed and rotor position. IEEE Trans. Ind. Electron. 1999, 46, 184–191. [CrossRef] Bolognani, S.; Zigliotto, M.; Zordan, M. Extended-range PMSM sensorless speed drive based on stochastic filtering. IEEE Trans. Power Electron. 2001, 16, 110–117. [CrossRef] Park, J.B.; Wang, X. Sensorless Direct Torque Control of Surface-Mounted Permanent Magnet Synchronous Motors with Nonlinear Kalman Filtering. Energies 2018, 11, 969. [CrossRef] Yang, M.; Liu, Z.; Long, J.; Qu, W.; Xu, D. An Algorithm for Online Inertia Identification and Load Torque Observation via Adaptive Kalman Observer-Recursive Least Squares. Energies 2018, 11, 778. [CrossRef] Xu, B.; Mu, F.; Shi, G.; Ji, W.; Zhu, H. State Estimation of Permanent Magnet Synchronous Motor Using Improved Square Root UKF. Energies 2016, 9, 489. [CrossRef] Chen, L.; Liu, S. A Kalman estimator for detecting repetitive disturbances. In Proceedings of the 2005, American Control Conference, Portland, OR, USA, 8–10 June 2005; Volume 3, pp. 1631–1636. Masi, A.; Butcher, M.; Martino, M.; Picatos, R. An application of the extended Kalman filter for a sensorless stepper motor drive working with long cables. IEEE Trans. Ind. Electron. 2012, 59, 4217–4225. [CrossRef] Alonge, F.; D’Ippolito, F.; Sferlazza, A. Sensorless control of induction-motor drive based on robust Kalman filter and adaptive speed estimation. IEEE Trans. Ind. Electron. 2014, 61, 1444–1453. [CrossRef] Zarei, J.; Shokri, E. Convergence analysis of non-linear filtering based on cubature Kalman filter. IET Sci. Meas. Technol. 2015, 9, 294–305. [CrossRef] Wang, X.; Yaz, E.E. Stochastically resilient extended Kalman filtering for discrete-time nonlinear systems with sensor failures. Int. J. Syst. Sci. 2014, 45, 1393–1401. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).