Backstepping control using nonlinear state observer for switched reluctance motor

Vietnam Journal of Science and Technology 60 (3) (2022) 554 568 doi 10 15625/2525 2518/16047 BACKSTEPPING CONTROL USING NONLINEAR STATE OBSERVER FOR SWITCHED RELUCTANCE MOTOR P h i H o a n g N h a 1’2[.]

BACKSTEPPING CONTROL USING NONLINEAR STATE OBSERVER FOR SWITCHED RELUCTANCE MOTOR

P h i H o a n g N h a1’2, *, P h a m H u n g P h i1, D a o Q u a n g T h u y 3, P h a m X u a n D a t 1,

L e X u a n H a i4

1 Hanoi University o f Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Ha Noi, Viet Nam

2Hanoi University o f Industry, 298 Cau Dien, Bac Tu Liem, Ha Noi, Viet Nam

3 Ministry o f Science & Technology, 39 Tran Hung Dao, Ha Noi, Viet Nam

4International School, Vietnam National University, 114 Xuan Thuy, Ha Noi, Viet Nam

Email: phihoansnhata),smail.com: vhihoanenha(d),haui.edu.vn

Received: 6 May 2021; Accepted for publication: 24 August 2021

hence it is difficult to control This paper presents a new method to control the speed of switched reluctance motor based on backstepping technique and nonlinear state observation This controller is first applied for switched reluctance motor with nonlinear drive model This model

is a combination of both the commutator and the motor in the same model The combined model

of switched reluctance motor helps to reduce the influence of nonlinearity due to the switching lock, increasing the accuracy in controlling this motor The state variables of the controller are approximated by nonlinear state observer, including speed observer, flux observer, and rotor position observer The observer state variables are compared with directly measured state variables This nonlinear state observer improves the switched reluctance motor drive system by reducing actual measuring devices, such as incremental encoder and torque transducer The stability of the closed control loop was analyzed using Lyapunov stability criterion The simulation is carried out with both traditional backstepping controller and the backstepping controller combined with nonlinear state observation The quality of the nonlinear state observer and backstepping control system are analyzed theoretically and through numerical simulations

1 IN T R O D U C T IO N

The switched reluctance motor (SRM) is an electric motor with many outstanding advantages such as low manufacturing cost, simple structure, wireless rotor that allows high working temperature, large torque, etc [1 - 3], Due to the structure of the SRM and the operating principle of continuous switching between each phase, the SRM has strong nonlinearity In many works, SRM models have been presented as linear or nonlinear for independent phases [4

- 6] In the work [7], it was the first combined dynamic model of SRM with logical transitions in

one model Then, the author of this work has linearized the dynamic model to design the linear controller for SRM

In order to reduce error caused by linear modeling process, in this paper, we propose a new design method based on backstepping technique combined with nonlinear state observer The research results are verified through numerical simulations

After general introduction, this paper is organized as follows Section 2 presents the nonlinear model of SRM (the model including the phase shift switches and dynamics of SRM)

In section 3, a controller based on backstepping technique and nonlinear state observer are designed Section 4 illustrates the simulation results Finally, section 5 is conclusion

2 N O N L IN E A R M O D E L O F SR M SY ST E M

The mathematical model of SRM used to design the controller is represented in the form of differential equations based on the basic machine equations The dynamics of SRM includes voltage equations, torque equations and mechanical equations

Differential equation of SRM with m phases has the following form:

di//.

u, = RJ ,+-!-+

where: / = 1, 2, 3, ., m; u is voltage of phase j; R is resistor of phase j; iJ is current of phase

j; y/j is flux of phase j.

From equation (1), flux of any phase j is represented:

i

Flux y/j depends on both current r and the angle 6 , so it is represented in details as follows:

¥ j{ ij,d ).

The mechanical equation of SRM:

where: Te is torque of one phase; Tt is torque of load; J is moment of inertia.

(3)

According to the principle of energy conversion in SRM, the torque generated is equal to the energy variation of magnetic field in stator coil according to rotor angular position

where:

aw,.

r (£,/,) = —

dW'j(9,ij) = J’y/j (9,iJ)dij

(4)

(5)

The torque of the SRM is a nonlinear function in terms of current and rotor position Then, the total torque generated is equal to the total torque of phases:

Te (6>, , i2, , J Tj (9, ij) (6)

/=i

To control the SRM, we need to determine the flux y / j (/,,# ) as accurately as possible For convenience in the research and development of control algorithms, flux property can be approximated as a continuous function [7], as follows:

with j = 1, 2, m; y/s is magnetic flux saturation.

In general, due to special structure of SRM, the performance of this motor is not the same

as that of general electric motor Rotor of SRM rotates at discrete angles so the function fj{ 0 )

can be represented by a Fourier series as follows:

/,.(0) = a + f > , , s i n [ ^ - ( / - l )— \ + cncos[nNre - ( j - \ ) — ]} (8)

in which, N r is the number of rotor poles, and if higher order components in Fourier series is

omitted, the simpler function (8) is obtained:

9 7T

m

The torque of phase j is represented as follows:

The state space equations of SRM as follows:

dO

— = co

dt

^ = 7

dij

7=1

(11)

+

8iJ J dt

The state model of SRM is illustrated below based on [7], Considering SRM with m = 4 phases, the state vector is x = [#,r0,zj,z2,z3,z4]r =[jq,x2,x^,x4,x5,x6]r The state equation of motor is:

r2= i [ r 1(0,x3)+r2(0,x4) + r3(0,x5) + r 4(0,x6)-7 ;(x 1,x2)]

j

+vF ;- , % - 1- ^ { 1- [ 1+^ / 3 U )] ^ s/3(Ji)}+ , dff - ^ {i -[ 1 + x6./4(x,)]g- ^ ^ ' )}

- B x 2 - mgl sin(x,)

(1 3 )

x3 = [ ^ ^ rt)/ ( x i) J 1[ ^ + ( ^ ^ , ) ( ^ ^ ) x i ] + [ VrlC-^^ > > ;(JCi)]_Iii1

*4 = [ - ^ * 4/2 (Xl)/ 2 (*,)]"' [R x 4 + ( v se~XiMx' ))( x 4 ^ X) x 2] + \ y se x'h(x' >/2 (x,)]"' u 2

X 5 = \j~ysse~Xsfi (x' V3 (x{)] \te 5 + ( V s e~XsMx'))(x5^ ) x 2^ + \y/se~XiMx')f i (x S \ ' u 3

*6 =\_~¥se~xjA(x° f 4{x,)\ ' [ f e 6 + ( ^ e '" 6/4(j:i))(x6^ ) x 2] + [ ^ e ‘A:6/4(JC')/ 4(x1) J 1w4

Where:

%_

dxx

r

= bN, cos N rxx - ( / - 1 ) —

m ;

(14) (15) (16) (17)

(18)

Note that, in the upper state spatial model, Bx2 is an opposite component to the rotation while mgl is the torque of the load.

From (13), we denote:

1 /,(* )

ga(x) =

fb(x) =

gb(x) =

f:(x) =

ge(x) =

fd(x) =

U 2(x.) Sc,

L/i2(x,) Sr,

Vs 3/2(x,)

L / 22(*,) Sr,

U 2(x>) Sc,

Vs 3/3(x,) _/32(x,) Sc,

Vs S/, (*,)

Vs 3/4 (x,)

jVr { l - e ^ /lW }

N r |l - e Xsf'(x,)^

K { - M x ^ X5fM)}

Nr { - U x x) e ^ }

Equation (13) can be rewritten as follows:

X2 = [ fa (X) + Sa (X)X3 } + [ fh(x) + gb (x)X4 ] + [ fc (x) + gc (x)xs ] +

(19)

Derivative (19) with respect to time, we have:

* 2 = [ f a O ) + 8 a ( * ) * 3 + 8 a ( X ) X 3 ] + [ f b ( X ) + 8 b (XK + 8 b( X ) X 4 ] +

From equation (14) to equation (17), we denote:

Pa(x) = J' [Rx3 + (¥se~x^ )(x3 ^ ) x 2]

9a{ x ) - \w se xMx') f { x S \ 1

P b ( x ) = [ - y s e ^ f 2 ( x^l) J 1[ ^ 4 + ( ^ 4/2(Xl)) ( x4 ^ ) x2]

qb{x) = \ ¥ se~xj2ix')f2{xx) \ '

Pc(x) = * [ ^s + (v' e v'M ''')(x, ^ ) x 2]

qc(x) = [ v se~X5MXi)f3(x,)] 1

p d(x) = [~yfse ^ M a x S \ l [R x 6 + ) ( x 6 ^ ) x 2 ]

^ W = [ ^ se ^ /4("l)/ 4(x1)] 1

The equations from (14) to (17) are rewritten as follows:

*3 = p a ( x ) + q a ( x ) u ,

x4 = p b(x )+ q b(x)u2

<

X 5 = P c ( x ) + q c ( x ) u 3

X 6 = P d ( x ) + g d ( x ) u 4

Substituting (21) into (20), we have:

x2 =[fa (x) + 8 a {x)x3 + g a (x)pa (x) + ga (x)qa (x)«, ] + [fb (x) + gb (x)x4 + g b (x)ph (x) + g b (x)qb (x)u2 ] + [ f c (x) + gc (x)x5 + gc (x)pc (x) + gc (x)qc (x)w3 ] + [fd (x) + gd (x)x6 + gd (x)pd (x) + gd (x)qd (x)u4 ] - j x 2

(22)

mgl

J C O S (X j)x ,

The SRM operates based on the principle of supplying voltage to each phase If we consider the number of phases is 4, we have U j = k j U , with j = 1, 2, 3, 4, k j is phase shift switch that only allows the values of 0 and 1 Equation (22) can be represented as follows:

fa 00 + ga (x)x3 + ga (x)pa (x) + f b (x) + g h (x)x4 + gb (x)pb (x) + f c (x) +

g c (x)x5 + gc (x)pc (x) + f d (x) + gd (x)x6 + gd (x)pd (x)

(23)

[ga(x)qa(x)kl + gb(x)qb(x)k2 + g c(x)qc(x)k3+ g d(x)qd(x)k4\ - - j x 2 —cos(x,)x;

J

Then we denote:

, \ f a M + 8 a 0 ) * 3 + 8 a (X ) P a 0 0 + f b 0 0 + 8 b 0 0 * 4 + 8 b (X ) P b 0 0 + f c ( X ) +

F(x)=

8 c ( X ) X 5 + 8c (X)Pc0 0 + fd0 0 + 8d ( X ) X 6 + 8d(X)Pd ( x )

and

G(x)=[ga(x)qa(x)kx + gh(x)qb(x)k2 + gi (x)qi (x)k3 + gJ{x)qij{x)k4}

We have another form of equation (23) as follows:

X- =F(x) + G(x) - —x2 - cos(x, )x,

We denote:

<|/ ( x ) = F(x) ^ x 2 J cos(x,)x, (25)

g(x) = G(x)

we have:

To facilitate the design, we represent equation (26) as a state model Let x2 = zx we have a

state model of SRM:

f Zi = z2

With / (x ),g (x ) are defined in equation (25)

The model in equation (27) is perfectly suitable to use backstepping technique to design the controller for SRM

3 BACKSTEPPING CONTROL DESIGN USING NONLINEAR STATE OBSERVER

FORSRM 3.1 Backstepping control

In section 2 of this paper, the dynamic model of SRM is represented as state model (27):

Z i= z2

= f ( x ) + g ( x )u

(28)

This is a model of a second order tight feedback system According to the backstepping technique [8 -11], we need to design in 2 steps

Step 1: Let speed tracking error zd = cod is ex, we have:

Take derivative of the function ex over time, we have:

Let e2 = z2 - a l in which, a, is virtual control signal for the first sub-system.

Substituting to equation (30), we have:

^ = z i - z d =z2 - z d =e2 + al - z d (31)

To determine virtual control signal that ensures e, —>0, we choose member Lyapunov function:

K = - e 2

1 2 1

Take derivative of Vl over time we have:

(32)

K = eA = e,(e2+«, " i / ) = -CjC? + e,e2

To have equation (33), virtual control signal has following form:

(33)

=~cie< + A

in which, q is a positive constant If el -» 0 then e2 —>0

Step 2:

(34)

e2 = z2— a x

Take derivative of the function e2 over time, we have:

(35)

e2 — z2—

From (28), we have:

(36)

To determine the signal control u that ensures e2 —>0, we choose Lyapunov function for

the closed loop:

Take derivative over time we have:

Substituting equation (33) and (37) to equation (39), we have:

Select control signal for system in equation (40):

SKX)

with c2 is positive constant

Theorem: SRM has a state model (28) which is controlled by backstepping controller (41) with

c ,, c2 are positive constants to ensure closed loop Lyapunov stable.

Proof. Select Lyapunov function for closed loop as follows:

Take derivative of V over time we have:

V - -c,e,2 + exe2 + e2 [ / (x) + g (x )u - cex ] (43)

Substituting u from equation (41) to equation (43), we have:

V = - Cle f+ ele2+e2\_ f( x ) - c 2e2- e l - [ f ( x ) - d l] - d x~\

V = -c xef - c2e\ < 0

What needs to be proven has been proven

3.2 Nonlinear State Observer

The nonlinear state observer is intended to estimate the states: flux, position and rotor speed from observing directly the value of voltage, current and moment The structure of nonlinear state observer is illustrated in Figure 1

Figure 1 Structure of nonlinear state observer.

3.2.1 Nonlinear flux observer

The model of nonlinear flux observer is shown below We set a new state:

= - ln ' 1 - ^w >

V V ' J With y/j is flux of phase j.

The electro dynamic equation of the SRM:

^ j =( - rij +uj ) s M )

With

T S

Flux Observer is shown as follows:

^ = diag(-rit + « / ) g ( f Z i ) + yO(z)Or (i)(ai

-With y > 0 and g(<f) = [ g } (<ft) g 2( j ) g g 4(</>) J

(45)

(46)

(47)

(48) (49)

® (0=

o

h

0

3.2.2 Rotor’s position observer

A model of rotor’s position observer is presented in this part The matrix A 3(z) is calculated from [12]:

* s ( 0 =

z, 0

0 z2 -z3 0

0 — i

Matrix G(i,tj>) is defined as:

G{i,4) := [_Xl (0^3 (0]"1 (0-The rotor’s position observer is shown as follows:

-ai G,{b<t>)

G2{U4>\

6 = — arctan N

G2{i,<j>)

(50)

(51)

(52)

3.2.3 Rotor’s speed observer

The rotor’s speed observer in this part is designed based on equation (11) that illustrates SRM motor and position of rotor observed above The first step of this design observer is to approximate the torque of load After that, Luenberger’s observer is designed to observe the speed of rotor

First of all, we determine the filter as follows:

in which: p := — and T > 0 is filter time constant

dt

The torque of load is approximated with the following formula:

Based on equation (11) that illustrates SRM, the motor observer is designed as follows:

jtj = x2

-(55)

*2 = - t 2 (x 1 0 ) + -J [r* (i, 0) - T with x2 =d>, i x, i 2 > 0 , f L is calculated from equation (54), 0 is designed in angular observer

of rotor from equation (52)

The accuracy of flux observer, rotor’s position observer and rotor’s speed observer were verified in [12]

3.3 Backstepping control design using the state observer

The backstepping controller (41) proposed in section 3.1 is only available when the state variables of the SRM are directly observed In order to control the SRM without measuring the output, we proposed a method of combining the backstepping controller (41) with the observer (Figure 2) Thus, instead of having to measure the flux, position and speed of rotor directly, it is only necessary to measure current, voltage and torque

The structure of the backstepping controller combined with the state observer is shown in Figure 2

Figure 2 Structure of backstepping control system with nonlinear state observer.

The quality of the nonlinear state observer and system of SRM controller is verified in the next section

4 NUMERICAL SIMULATION

Figure 3 Schematic diagram of SRM backstepping control system.

Table 1 Parameters of SRM model, controller and observer.

b = 1.364xl0‘3( / / )

The simulation is performed on MATLAB/Simulink SRM parameters and selected force parameters of the controller and observer are shown in Table 1 The schematic diagram of SRM backstepping control system is shown in Figure 3

The results of the nonlinear state observer test are shown in Figures 4, 5, 6; the phase current and torque of the system in Figures 7 and 8, and the control results are shown in Figure 9

0.2 _Q

§ 0.1

- 0.2 -0.3

(b)

Time(s)

Figure 4 Flux observer and Flux.

(a)

O) 4

cf o

• i-2o Q.

1- -3 o S

H—

o

S _

2 LL1 -4

(b )

Time(s)

1.5

Figure 5 Rotor position observer.