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[.]
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, *, 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 t1, L e X u a n H a i4 1Hanoi University o f Science and Technology, 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 3Ministry 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: May 2021; Accepted for publication: 24 August 2021 A bstract Switched reluctance motor (SRM) has many advantages with very strong nonlinearity, 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 Keywords: Swiched Redutance Motor; Backstepping Technique; Nonlinear State Observer Classification numbers: 4.10.3; 4.10.4 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 Backstepping control using nonlinear state observer for switched reluctance motor 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 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 illustrates the simulation results Finally, section is conclusion N O N L IN E A R M O D EL O F SRM SY STEM 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,+-!-+ ( 1) dt 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 y/j = J(v; -R ij)dt ( 2) Flux y/j depends on both current r and the angle , so it is represented in details as follows: ¥ j{ ij,d ) The mechanical equation of SRM: J ^d = T - T dt2 e 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 aw, r (£,/,) = — JJ 89 (4) dW'j(9,ij) = J’y/j (9,iJ)dij (5) where: 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 555 Phi Hoang Nha, Pham Hung Phi, Dao Quang Thuy, Pham Xuan Pat, Le Xuan Hai 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: (7) y/j(0,ij) = y/sQ - e 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{ ) can be represented by a Fourier series as follows: /,.(0) = a + f > , , s i n [ ^ - ( / - l )— \ + cncos[nNre - ( j - \ ) — ]} m m (8) in which, Nr is the number of rotor poles, and if higher order components in Fourier series is omitted, the simpler function (8) is obtained: 97T f j (0) = a + b sinfAr.# - ( j -1 )— ] (9) m The torque of phase j is represented as follows: The state space equations of SRM as follows: dO — = co dt ( 11) ^ = 7=1 dij (d rA ' dt \ l diJ J dy/j + ) l 8iJ J The state model of SRM is illustrated below based on [7], Considering SRM with m = 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: ( 12) xl —x2 r2= i [ r 1(0,x3)+r2(0,x4) + r3(0,x5) + r 4(0,x6)-7 ;(x 1,x2)] j Z J\ - - - ~ X~ N r { ! - [ ! + x 3f l ( x l ) ] e - ^ M } + V^i) { l-[l + J + v F ;- , % - 1- ^ { 1- [ 1+^ / U )] ^ s/3(Ji)}+ W , dff - ^ {i -[1 + x6./4(x,)]g- ^ ^ ')} - Bx2 - mgl sin(x,) (13) 556 Backstepping control using nonlinear state observer for switched reluctance motor x3 = [ ^ ^ rt)/ ( x i) J 1[ ^ + ( ^ ^ , ) ( ^ ^ ) x i ] + [ VrlC-^^> > ;(JCi)]_Iii1 *4 = [ - ^ * 4/2(Xl)/ 2(*,)]"' [Rx4 + ( v se~XiMx' ))( x4 ^ X) x2] + \ y se x'h(x'>/2(x,)]"' u2 (14) (15) X5 = \j~ysse~Xsfi(x'V3 (x{)] \te + ( V s e~XsMx'))(x5^ ) x 2^ + \y/se~XiMx')f i (x S \ ' u3 (16) *6 =\_~¥se~xjA(x° f 4{x,)\ '[ f e + ( ^ e '" 6/4(j:i))(x6^ ) x 2] + [ ^ e ‘A:6/4(JC')/ 4(x1) J 1w4 (17) Where: %_ = bN, cos r Nrxx - ( / - ) — dxx m; (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: /,(* ) Qfi(.x,) jVr { l - e ^ /lW } U 2(x.) Sc, ¥s ga(x) = Vs d f(x t) L/i2(x,) Sr, fb(x) = Vs 3/2(x,) L/ 22(*,) Sr, gb(x) = Vs dfiix,) j H x , ) Sr, f:(x) = dfi(X\) Nr |l - e Xsf'(x,)^ U 2(x>) Sc, ge(x) = 3/3(x,) K { - M x ^ X5fM)} _/32(x,) Sc, fd(x) = S/, (*,) Sc, U 'w Vs Vs Vs Vs 3/4 (x,) Nr { - U x x) e ^ } [ f h x 1) Sr, Equation (13) can be rewritten as follows: X2 = [fa(X) + Sa (X)X3} + [fh(x) + gb(x)X4] + [fc(x) + gc(x)xs ] + (19) [fdM + Sd(x)x6h ~ x Sin(x,) 557 Phi Hoang Nha, Pham Hung Phi, Dao Quang Thuy, Pham Xuan Pat, Le Xuan Hai Derivative (19) with respect to time, we have: *2 = [ f a O ) + a (* )* +8 a ( X ) X3 ] + [ fb ( X) + b (XK +8 b ( X) X4 ] + [fc(x) + 8c(x)x5 + 8c(x)x5] + [/«,(*) + 8d(x)x6 + gd(x)x6] - -y x2 cos(x,)i, From equation (14) to equation (17), we denote: J' [Rx3+ (¥se~x^ Pa(x) = )(x3^ ) x 2] 9a{ x ) - \w se xMx')f { x S \ P b ( x ) = [ - y se ^ f (x ^l) J 1[ ^ + ( ^ 4/2(Xl)) ( x4 ^ ) x2] qb{x) = \ ¥ se~xj 2ix')f 2{xx)\ ' Pc(x) = * [ ^s + (v' e v' M''')(x, ^ ) x 2] qc(x) = [ v se~X5MXi)f3(x,)] p d(x ) = [~yfse ^ M a x S \ l [R x + )(x 6^ ) x 2] ^ W = [ ^ se ^ /4("l)/ 4(x1)] 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 X5 = P c ( x ) + q c( x ) u X6 = P d ( x ) + g d ( x ) u Substituting (21) into (20), we have: x2 =[fa (x) + a{x)x3 + g a(x)pa(x) + ga(x)qa(x)«, ] + [fb (x) + gb(x)x4 + gb(x)ph(x) + gb(x)qb(x)u2] + (22) [ f c(x) + gc(x)x5 + gc(x)pc(x) + gc(x)qc(x)w3] + mgl C O S (X j)x , J 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 [fd (x) + gd(x)x6 + gd(x)pd(x) + gd(x)qd(x)u4] - j x that only allows the values of and Equation (22) can be represented as follows: fa 00 + ga(x)x3 + ga(x)pa(x) + f b(x) + gh(x)x4 + gb(x)pb(x) + f c(x) + X ,= + gc(x)x5 + gc(x)pc(x) + f d(x) + gd(x)x6 + gd(x)pd(x) mgl B [ga(x)qa(x)kl + gb(x)qb(x)k2 + gc(x)qc(x)k3+ g d(x)qd(x)k4\ - - j x —cos(x,)x; J Then we denote: 558 (23) Backstepping control using nonlinear state observer for switched reluctance motor , F(x)= \ f aM + a0 )* c( X ) X + + a(X)Pa 0 8c (X)Pc 0 + fd 0 + fb 0 + + b0 * 8d ( X ) X + + b(X)Pb 0 + fc (X ) + 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, */ *J (24) We denote: |/( x ) = F(x) < ^x2 J cos(x,)x, (25) g(x) = G(x) we have: (26) = f(x ) + 8(x )u we have a To facilitate the design, we represent equation (26) as a state model Let x2 = zx state model of SRM: fZi = z2 (27) [z2 = / ( x ) + g(x)« X2 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 BACKSTEPPING CONTROL DESIGN USING NONLINEAR STATE OBSERVER FORSRM 3.1 Backstepping control In section of this paper, the dynamic model of SRM is represented as state model (27): Zi=z2 (28) = f ( x ) + g ( x )u This is a model of a second order tight feedback system According to the backstepping technique [8 -11], we need to design in steps Step 1: Let speed tracking error zd = cod is ex, we have: ^ = z i - z rf Take derivative of the function ex over time, we have: el =zx- z d =z2 - z d 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 (2 ) (30) (31) 559 Phi Hoang Nha, Pham Hung Phi, Dao Quang Thuy, Pham Xuan Pat, Le Xuan Hai To determine virtual control signal that ensures e, —>0, we choose member Lyapunov function: K = -e 2 (32) K = eA =e,(e2+«, " i / ) = -CjC? + e,e2 (33) Take derivative of Vl over time we have: To have equation (33), virtual control signal has following form: =~cie0 Step 2: e2 = z2—a x (35) Take derivative of the function e2 over time, we have: e2 —z2— (36) From (28), we have: e2 =z2- d \ = f { x ) + g {x )u -c c l (37) To determine the signal control u that ensures e2 —>0, we choose Lyapunov function for the closed loop: V2 =Vl+ ± e (38) V2 =Vl+ e2e2 (39) Take derivative over time we have: Substituting equation (33) and (37) to equation (39), we have: V2 = -qq2+ qe2+ e2[ / ( x ) + g (x )u - a x] (40) Select control signal for system in equation (40): U= ^ I - LV> L U SKX) (41) 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: V = ^ { e?+ e22) = Vl + ± e22 =V2 Take derivative of V over time we have: 560 (42) Backstepping control using nonlinear state observer for switched reluctance motor V - -c,e,2 + exe2 + e2[ / (x) + g (x )u - cex] (43) Substituting u from equation (41) to equation (43), we have: V = - Clef+ ele2+e2\_ f( x ) - c 2e2- e l - [ f ( x ) - d l] - d x~\ V = -c xef - c2e\ < 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 Figure Structure of nonlinear state observer 3.2.1 Nonlinearflux observer The model of nonlinear flux observer is shown below We set a new state: ' w> = -ln - ^ V (45) V 'J With y/j is flux of phase j The electro dynamic equation of the SRM: ^ j =( - rij +uj ) s M ) (46) With TS (47) yO(z)Or (i)(ai - (48) Flux Observer is shown as follows: ^ = diag(-rit + « / )g (fZ i) + (49) With y > and g(