Fractional order adaptive kalman filter for sensorless speed control of dc motor

Trang 1 Full Terms & Conditions of access and use can be found athttps://www.tandfonline.com/action/journalInformation?journalCode=tetn20ISSN: Print Online Journal homepage: www.tandfonl

Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=tetn20

ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/tetn20

Fractional order adaptive Kalman filter for

sensorless speed control of DC motor

Ravi Pratap Tripathi, Ashutosh Kumar Singh & Pavan Gangwar

To cite this article: Ravi Pratap Tripathi, Ashutosh Kumar Singh & Pavan Gangwar (2023)

Fractional order adaptive Kalman filter for sensorless speed control of DC motor, International Journal of Electronics, 110:2, 373-390, DOI: 10.1080/00207217.2021.2025452

To link to this article: https://doi.org/10.1080/00207217.2021.2025452

Published online: 05 Feb 2022.

Submit your article to this journal

Article views: 278

View related articles

View Crossmark data

Citing articles: 1 View citing articles

Fractional order adaptive Kalman filter for sensorless speed control of DC motor

Ravi Pratap Tripathi , Ashutosh Kumar Singh and Pavan Gangwar

Electronics and Communication Engineering Department, Indian Institute of Information Technology Allahabad, Prayagraj, India

ABSTRACT

State estimation is a challenging and most crucial issue in the

industry for proper monitoring and controlling of the plants

These kinds of control systems have the requirement of costly

measurement sensors/equipment for measurable and

unmeasur-able state variunmeasur-ables of the dynamical plants These drawbacks can

be overcome by designing a sensorless system to estimate the state

variables In the proposed work, sensorless speed control of DC

motor is implemented by using a fractional-order adaptive Kalman

filter (FOAKF) The FOAKF algorithm uses a fractional feedback loop

of the previous Kalman gain along with the current Kalman gain

The motor shaft speed is estimated by the FOAKF state estimator

Furthermore, a performance comparison of extended Kalman filter

(EKF) and FOAKF estimator under a similar condition is realised

using MATLAB/Simulink environment To validate the performance

of the FOAKF estimator, a hardware prototype model has been

presented with the help of the arduino board In the proposed

work, the root-mean-square error (RMSE) and Euclidean distance

error between reference speed and estimated speed have been

used for the performance metric The performance comparison

result shows that FOAKF is a more robust and accurate state

esti-mator in comparison with EKF.

ARTICLE HISTORY

Received 31 July 2021 Accepted 5 December 2021

KEYWORDS

DC motor; Kalman filter; fractional calculus; sensorless control

1 Introduction

In all industrial work, mechanical movement is carried out with the use of electric motors, hydraulics and pneumatic systems To drive these systems, motors are required For motion transmission systems generally, AC motors are used while DC motors are used in robotic manipulators and in industrial applications, where high load torque is needed To imple-ment the industrial task, various control approaches are used, in general emphasis is given

to controlling the speed of the machines To achieve effective speed control, a closed-loop control system is used, in which motor state variables like current, position and rotor speed are feedbacked A suitable controller can be implemented when all the state variables of the system are known To measure the entire state variable, it uses various sensors; thereby cost increases and makes the system more complex To reduce the number of sensors, hardware complexity and system cost, state observers/estimators are used in the control loops of

CONTACT Ravi Pratap Tripathi rse2018510@iiita.ac.in ; Indian Institute of Information Technology Allahabad, Prayagraj, Uttar Pradesh, India

https://doi.org/10.1080/00207217.2021.2025452

electrical motors (Jang et al., 2003; Mohan et al., 2020) The advantages of state observers over sensors are that they are not affected by any environmental causes and any machine distortions (Wang et al., 2019) Some sensorless speed control methods were performed (Lascu et al., 2006; Raca et al., 2008) An observer is a mathematical tool that estimates states with the help of system dynamical models and some measured state variables

Some works have been presented in previously published literature in which sensors are not mounted on the motor shaft and stochastic observer/estimator are used to estimate the speed of the DC motor In the previous works, the state estimation was performed by using the sliding mode observer (Lascu et al., 2009; Qiao et al., 2013) Kalman Filter (KF) (Rigatos, 2009), Extended Kalman Filter (EKF; Boizot & Busvelle, 2007; Terzic & Jadric, 2001; E Zerdali, 2019) and the particle filter (Ristic et al., 2004)

KF is an elementary concept and an extensively used mathematical tool for state estimation KF is applicable only for linear systems and gaussian noise In a nonlinear dynamic system generally, EKF is used (O Aydogmus & Talu, 2012; Z Aydogmus & Aydogmus, 2015; Habibullah & Lu, 2015; Tiwari et al., 2017; Yin et al., 2019; Zhou et al.,

2016) EKF is applicable only when the system is differentiable, well defined, noises are gaussian and their covariance is known; otherwise, it starts diverging In such cases, the other versions of KF like UKF (Jafarzadeh et al., 2013), CDKF (Zhao et al., 2020), Adaptive KF/EKF (Jiancheng & Sheng, 2011; Emrah Zerdali, 2020; ZhiWen et al., 2013) were pro-posed In the open literature, EKF and its derivative-based sensorless speed control algorithms are present but they could not able to provide sufficient contribution for sensorless speed control when the system is diverging and not well defined To incorpo-rate the above difficulties in state estimation FOAKF algorithm has been proposed FOAKF

is a robust adaptive state estimation technique, which is more adaptive in the field of signal and image processing, thermodynamics and tracking applications (Kaur & Sahambi,

2016) In the presented FOAKF algorithm, modified steady-state Kalman gain is proposed that is obtained by inserting a fractional feedback loop of previous Kalman gain with current steady-state Kalman gain The main advantage of FOAKF is that its gain will never

be diverse because of the fractional derivative of the previous Kalman gain The main contribution of this work is to develop a MATLAB/Simulink model for analysing the performance of the EKF and FOAKF estimator at nonlinear load in the closed-loop sensorless speed control Furthermore, a simulink model and its experimental validation have been presented of a FOAKF estimator-based sensorless speed control mechanism for

DC motor at nonlinear and sudden load conditions

The paper is organised in the following sections: In section 2, a DC motor mathematical model has been presented The state estimation algorithm is presented in section 3 In section 4, a stability analysis of the FOAKF estimator has been discussed The simulation and experimental results are described in section 5 and at last, the concluding remarks are presented in section 6

2 DC motor mathematical model

DC motor mathematical model in the time domain form is required for state estimation algorithm The DC motor model has two main equations, one is electrical and the other one is mechanical (Kothari & Nagrath, 2010) On applying Kirchhoff’s law in the armature circuit shown in Figure 1(a), the electrical equation is given as:

vðtÞ ¼ e aðtÞ þ R a i aðtÞ þ L a

d

where R a is armature resistance, i a is armature current, L a is the self-inductance produced

by the armature flux, v t is armature supply voltage, and ðe aðtÞ ¼ K e i f ω mÞis back emf

A separately excited DC motor is shown in Figure 1(a), which requires an extra DC voltage

v f to produce a magnetic field R f and L f represent the resistance and inductance of the

field winding respectively It is well known, when v f constant and steady-state exists on

the field circuit, then i f (field current) is constant, therefore Equation (1) can be written as:

vðtÞ ¼ K0e ω mðtÞ þ R a i aðtÞ þ L a

d

For motoring operation, the dynamic equation for the mechanical system is

TðtÞ ¼ K t i fðtÞi aðtÞ ¼ J d

dt ω mðtÞ þ T LðtÞ þ T fðtÞ þ Dω mðtÞ (3) where J is moment of inertia of motor and load, ω m is the motor speed, D viscous damping coefficient, T fðtÞ and T L is coulomb friction torque and load torque, respectively The load torque is shown in Figure 1(b) and can be mathematically written as:

T L¼mgL cos θ þ mL2 d

where L is arm length, m is the mass of load, g is gravitational constant and θ is angular displacement (integral of ω m)

As i f constant, then Equation (3) can be written as:

TðtÞ ¼ K0

t i aðtÞ ¼ J d

dt ω mðtÞ þ T LðtÞ þ T fðtÞ þ Dω mðtÞ (5)

Figure 1 DC motor circuit diagram and load model (a) Circuit diagram of DC motor (b) Nonlinear load model

With the help of Equations (2) and (5), the state-space model can be implemented The

model comprises two state variables armature current (i a ) and motor speed (ω m) The state-space model of DC motor in the time domain is given as:

_

ω m

_i a

¼

D

mL2 þJ

K0

t

mL2 þJ

K0

t

L a

R a

L a

ω m

i a

þ

mgL cos θ T f

mL2 þJ 0

L a

1

v t

(6)

From the state-space model, θ produces nonlinearity in the system For digital

implemen-tation, the above Equation (6) must be discretised The discrete-time model of the above system is represented by equations given below:

In the above equation, state variable x ¼ ω½ m i a�t , input vector u ¼ 1 v½ t�t, and output

vector y ¼ 0½ i a�t whereas v (mean 0, covariance Q) and w (mean 0, covariance R) are the

process noise, and measurement noise, respectively After putting the motor parameters

value given in Appendix in Equation (6), the discretised matrix coefficients A d ; B d and C d

at sampling interval ðT sÞcan be written as:

0:007 0:9828

(9)

(10)

With the help of controllability ðθ c¼½B d A d B d�Þand observability ðO ¼ C½ d A d C d�tÞtest,

we find that θ c and O are full rank (n = 2) matrix, which implies that we can design

a controller and observer The details of EKF and FOAKF observer and their performance comparison are present in the next sections

3 State estimation algorithm

The state-space model obtained in section 2 is used by the EKF and FOAKF algorithm to

estimate the motor speed ðω mÞand armature current ði aÞfrom the measurement i a

3.1 EKF algorithm

KF is a state estimator that uses an iterative mathematical equation to compute the best possible state from the noisy state The KF is not applicable for the nonlinear system, as in

the calculation of θ using ω m leads the DC motor model to nonlinear, so EKF is used to overcome this difficulty EKF is the extended version of KF that uses first-order Taylor series approximation of nonlinear dynamical function to estimate the state by noisy measured data The EKF has two main steps one is state prediction and the other one is state correction In the prediction step, a dynamical model of the system with process

noise covariance Q is used In the correction, the predicted state is corrected with the help

of continuously measured data and Kalman gain ðKÞ However in the application of EKF, the system dynamical model and initial values of all the covariance matrices, namely, Q; R and P must be known accurately If these are not defined correctly then the system may

become unstable and can diverge The standard EKF estimation has a two-step process, one is prediction and the other one is corrections, which are presented below

The prediction step is

^

x kþ1=k¼f ð^x k ; u kÞ (12)

P kþ1=k¼A d P k A t dþQ (13)

^

x kþ1=k is predicated state of x kþ1

The correction step is

K kþ1¼P kþ1=k C t dðC d P kþ1=k C d t þRÞ 1 (14)

^

x kþ1¼ ^x kþ1=kþK kþ1ðy kþ1 C d^x kþ1=kÞ (15)

P kþ1¼P kþ1=kðI K kþ1 C dÞ (16)

^

x0kþ1 is the updated states, I is the identity matrix and K kþ1 is Kalman gain Equations (12)- (16) are one complete cycle of the EKF algorithm

3.2 FOAKF algorithm

The mathematical model of the system is nonlinear, and generally, the system model and

their covariance Q; R and P are not normally well defined Therefore, the other adaptive

state estimation algorithm known as FOAKF has been presented in this section

The main concept behind FOAKF is to introduce a fractional derivative of previous Kalman gain as feedback in the Kalman gain loop Grunawald-Letnikov’s definition (Sierociuk & Dzieliński, 2006) has been used for the computation of fractional feedback Kalman gain in the proposed technique Based on Grunawald-Letnikov definition, the

fractional difference of signal x k can be given as:

Δ α x k¼ 1

h α

Xk

j¼0

1

α is fractional order ð0 < α < 1ÞÞ, k number of samples in signal x k , h (Sampling interval = 1)

In the proposed work, for all the simulation we assume α ¼ 0:5 because if we take α close

to zero then the time taken to stabilising x k will be more and for α close to one it takes less time to stabilise The γ j (gamma function) can be given as:

γ j ¼ α j

� �

¼

α α 1ð Þ ::ðα jþ1Þ

The FOAKF gain is calculated by using fractional calculus For calculating FOAKF gain, following step has been used as given below:

(1) Calculate standard Kalman gain ðK kþ1Þ.

(2) Calculate fractional difference gain ðf kÞof prior standard Kalman gain

(3) FOAKF gain is the algebraic sum of fractional difference gain and standard Kalman gain

It is well known that the working of EKF is a two-step process The first step of EKF is prediction while the second step is correction Similar to EKF, FOAKF is also a two-step process which is given below:

Predicted state of Equations (7) and (8) is:

^

x kþ1=k¼A d^x kþB d u k (18)

P kþ1=k¼A d P k A t dþQ (19)

In the correction step, first calculate standard Kalman gain ðK kþ1Þ using Equation (14) Applying the theory of fractional calculus presented in Equation (17) on prior Kalman gain

ðK kÞthe fractional difference gain ðf kÞcan be calculated as:

f k¼E X

k

j¼0

1

ð Þjþ1 γ j K k j

(20)

f k presented in Equation (20) is the mean of the fractional derivative of the prior Kalman

gain The proposed FOAKF gain ðKmodÞis given by adding Equations (14) and (20)

Now the corrected, states and error covariance of discrete time system can be calculated

similarly as Equations (15) and (16) using Kmod, which are given below:

^

x kþ1¼ ^x kþ1=kþKmodðy kþ1 C d^x kþ1=kÞ (22)

P kþ1¼P kþ1=kðI KmodC dÞ (23) Equations (18)-(23) are the one complete cycle of the FOAKF algorithm The FOAKF gain

ðK Þcan be calculated by minimising the error covariance P :

Figure 2 Control strategy of sensorless speed control of DC motor

P kþ1 ¼E ðx kþ1 ^x kþ1Þ2

(24)

P kþ1¼E x kþ1 ^x0kþ1=k ðK kþ1þf kÞ y kþ1 C d^x0kþ1=k

��2

(25)

For minimising the P kþ1 put @P kþ1

@K ¼0, and after solving we get FOAKF gain ðKmodÞ:

Kmod¼K kþE X

k

j¼0

1

ð Þjþ1 γ j K k j

(26)

Figure 3 Detailed simulink model of the system (a) EKF and FOAKF block (b) DC motor model

4 Stability analysis

For FOAKF estimator stability analysis, the error between the actual state ðx kÞand the

estimated state ð^x kÞshould be considered Because if error reaches zero, then ^x k will reach

to x k The dynamics of the x k and ^x k are given in Equations (7) and (23), respectively Now

the error ðeÞ and its derivatives ð_eÞ can be given as:

_e ¼ ðAx kþBuÞ A^x kþBu þ KmodðCx k C^x0kÞ

Equation (28) will be stable only if ðA KmodCÞ is hurwitz matrix If it is hurwitz then ^x k will

keep becoming a better estimation of x k as time goes on and the system become stable Therefore to prove stability, a Lyapunov function can be considered as:

_VðeÞ ¼ e0PðA KmodCÞe þ e0

_VðeÞ ¼ e0

ðPA0

According to Lyapunov stability concept for asymptotic stability, _VðeÞ < 0 and using the

riccati equation given in (Aminifar, 2016):

PA0

Therefore, Equation (32) can be written as

Now substitute the Equation (34) value into Equation (32), _VðeÞ will be:

_VðeÞ ¼ e0

From Equation (35), it is clear that VðeÞ will be a valid quadratic Lyapunov of Equation (29),

which implies that the corresponding matrix must be hurwitz

5 Simulation and experimental results

The block diagram of the sensorless speed control strategy is shown in Figure 2 In the block diagram, a PI controller, motor driver, DC motor and a FOAKF state estimator are present The FOAKF state estimator estimates the speed, which is feedbacked to perform the sensorless speed control of the DC motor

5.1 Simulation results

A sensorless PI-based speed control system is built using Simulink/MATLAB code generator to compare the EKF and FOAKF performance The detailed MATLAB simu-link structure of the system is shown in Figure 3 The Simulink structure has two

Figure 4 Performance comparison of EKF and FOAKF estimator at variable load

ðT L¼mgL cos θ þ mL 2 dω m

dtÞ (a) Speed estimations result and actual motor shaft speed (b) Current estimations result and measured current speed (c) Euclidean distance speed estimation error of EKF and FOAKF (d) Root mean square error of EKF and FOAKF (e) Load torque