Integral High Order Sliding Mode Control of Single-Phase Induction Motor

Transcription

1 213 1th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Integral High Order Sliding Mode Control of Single-Phase Induction Motor Guillermo J Rubio, Juan Diego Sánchez-Torres, José M Cañedo and Alexander G oukianov Department of Electrical Engineering CINVESTAV Unidad Guadalajara, 4515 México grubio, dsanchez, josec, louk]@gdlcinvestavmx Abstract An observer-based controller for the single-phase induction motor is proposed in this paper The scheme presented is formulated using block control feedback linearization technique and high order sliding mode algorithms with measurements of the rotor speed and stator currents A second order sliding mode observer is included into the controller design in order to obtain estimates of the rotor flux The stability of the complete closedloop system is analyzed in the presence of model uncertainty, namely, rotor resistance variation and bounded time-varying load torque I INTRODUCTION This paper is aimed to present an observer-based controller using high order sliding mode (HOSM) algorithms for capacitor-run single-phase induction motor (SPIM) It is well known the SPIM is widely used in many household applications as compressors, pumps, air conditioning systems, washer, refrigerators, and other equipment which require low power motors 1] Therefore, the design of control algorithms which improve the SPIM performance is a relevant task An important class of solutions for this problem is the observer-based controllers For the SPIM case, it consists of: (i) a feedback controller for speed profile tracking and flux magnitude regulation,(ii) a state observer to estimate the rotor flux, and (iii) stability analysis of the whole system closed by the designed observer-based feedback For the feedback controller case, several approaches have been proposed for the induction motor control For example, a classical vector control with field orientation technique, due to 2], the application of back-stepping 3], passivitybased control 4], 5], input-output feedback linearization 6], adaptive 7] and sliding mode (SM) 8] 11] including neural networks 12] and discrete controllers 13] However, most of the mentioned proposals are for the three phase motor (TPIM) The treatment of the SPIM control design problem is different from the TPIM controller, since the SPIM despite of symmetric TPIM has a basic control input which applies to the main winding, and the auxiliary winding is affected by the switched capacitor, it looks like a subactuated system Moreover, this control input that depends on switching parameter which can take just two values or 1 In addition, most of the proposed methods assume the rotor flux to be known Hence, it is necessary the development of a tool that allows the estimation of this variable The estimate is usually obtained from machine model and the measurement of speed and stator voltage and current 14], 15] Several flux observers have been proposed using adaptive 16], 17] and sliding mode (SM) 8], 18], 19] The proposed observers strategies guaranty robustness in the presence of plant model uncertainty Usually, the stability analysis of the complete observercontrol system is carried by using the separation principle proposed in 2] However, this principle was developed for a class of nonlinear minimum phase systems that can be presented in the observer canonical form The induction motor case covers a different scenario and the applicability of the observer-controller scheme described in 2] is questionable and, by far not trivial Thus, a more precise stability proof for the whole scheme is necessary In this paper, a robust observer-based controller design for the capacitor-run SPIM in the presence of uncertainty is considered The proposed control scheme is based on the motor dynamic model including the capacitor dynamics, described in a stationary reference frame (αβ) fixed in the stator First, a second-order SM observer based on equivalent control 21] and a generalization 22] of the super-twisting algorithm 23] is designed to estimate the rotor flux With the measured stator current and estimated rotor flux, the controller is proposed by using a combination of block control feedback linearization 24] and quasi-continuous SM algorithms 25] in order to design a nested integral structure as 26], 27] but with exact disturbance rejection, similar to the techniques presented in 28], 29] The super twisting algorithm for the basic control input and switching logic for the auxiliary input is proposed in order to ensure the design sliding manifold be a finite time attractive The closed-loop system exhibits the properties of exponential tracking and robustness, allowing to overcome the uncertainty due to the parameter variations and external disturbances as the load torque In the following, Section II provides the considered model of the SPIM Sections III and IV describe the proposed observer and controllers, including a detailed analysis of stability and robustness Simulation results which demonstrate the main characteristics of the proposed controller, are presented in Section VI Finally, in Section VII the conclusions are given II MATHEMATICA MODE FOR THE SPIM The dynamic model of the SPIM can be considered as the model of an unsymmetrical 2-phase (a, b) induction machine in the variables of circuit elements After the transformation to a fixed frame (αβ) 3], the single phase induction motor /13/$ IEEE 25

2 213 1th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) scheme with the stator current and the rotor flux as the state variables, is presented in Fig 1 i s i βs Using the relation between the voltages v αs and v βs in (1) of the form v αs = v s v βs = n 1 (2) v s v c ρ v s Main Winding R αs i αs ω r λ αr λ βr v c ρ αs Rotor 1 Auxiliary Winding βs R βs Fig 1: Single phase induction motor where the switching parameter ρ {,1}, the voltage v βs yields to { n 1 v s v c if ρ = 1 v βs = n 1 v s if ρ = being n 1 v s as a referred voltage of the main winding to the auxiliary winding with n = N A /N B, where N A is the number turns of main winding and N B is the number turns of an auxiliary winding and its dynamic equations are given by di αs di βs dλ αr dλ βr = c 1 a 1 i αs +c 1 c 4 λ αr c 1 c 3 n p ω r λ βr +c 1 v αs + αs = c 2 a 2 i βs +c 2 c 4 λ βr +c 2 c 3 n p ω r λ αr +c 2 v βs + βs = a 3 λ αr +n p ω r λ βr +a 4 i αs + αr = n p ω r λ αr a 3 λ βr +a 4 i βs + βr dω r =d 1 d 2 (λ βr i αs λ αr i βs ) d 2 T where λ αr and λ βr are the rotor magnetic-flux-linkage, i αs and i βs are the stator current, v αs and v βs are the voltage of the main and auxiliary stator windings, respectively, ω r is the rotor speed, n p is the number of pole pairs, T is the load torque This model considers variations on rotor resistance of the formr r (t) = R r + R r (t) where R r (t) is an unknown but bounded function of time, leading to a set of uncertain model parameters a 1 (t) = a 1 + a 1 (t), a 2 (t) = a 2 + a 2 (t), a 3 (t) = a 3 + a 3 (t), a 4 ((t) = a 4 + a 4 (t) and c 4 (t) ( = c 4 + c 4 (t) where a 1 = R αs +R 2 m r 2 ), a 2 = R βs +R 2 m r r 2 ), r a 3 = Rr,a 4 = Rr m andc 4 = Rr 2 m are the parameter r nominal values The parametric uncertainties are presented by a 1 (t) = a 2 (t) = 2 m 2 R r (t), a 3 (t) = 1 r R r (t), a 4 (t) = m R r (t), and c 4 (t) = m R 2 r (t) While r the model parameters which do not depend on the resistance variations are given by c 1 =, c 2 =, αs 2 m βs 2 m c 3 = m, d 1 = n p m r and d 2 = 1 J Thus, the unknown terms in (1) are defined by αs = c 4 (t)c 1 λ αr a 1 (t)c 1 i αs, βs = c 4 (t)c 2 λ βr a 2 (t)c 2 i βs, αr = a 3 (t)λ αr + a 4 (t)i αs, and βr = a 3 (t)λ βr + a 4 (t)i βs The dynamics of the capacitor (see Fig1) are given by dv c = ω X c i βs where X c is the capacitor reactance and ω = 2πf, with f being the fundamental frequency (1) III SECOND ORDER SIDING MODE OBSERVER FOR ROTOR FUXES Having the rotor speed ω r and stator current i αs, i βs measurements only, in this section a second order SM observer is designed to estimate the rotor flux Considering the transformation λ αr = i αs +c 1 c 3 λ αr, λ βr = i βs +c 2 c 3 λ βr (3) the flux and current dynamics (1) are represented in new variables of the form di αs di βs dλ αr dλ βr = ϑ 11 i αs +ϑ 12 λ αr ϕ 1ω r λ βr +c 1v αs + αs = ϑ 21 i βs +ϑ 22 λ βr +ϕ 2 ω r λ αr +c 2 v βs + βs = ς 11 i αs +ς 12 λ αr +c 1v αs + αr = ς 21 i βs +ς 22 λ βr +c 2 v βs + βr where ϑ 11 = c 1 a 1 + c1 c 4, ϑ 12 = ϑ 22 = c4 3, ϑ 21 = c 2 a 2 + c4 c 3, ϕ 1 = c 1 c 3 n p, ϕ 2 = c 2 c 3 n p, ς 11 = c 1 ( c 3 a 4 c 1 a 1 ) ς 12, ς 21 = c 2 c 3 a 4 c 2 a 1 ς 22, ς 12 = c1c 4 c 1c 3a 3 c 1c 3 and, ( ς 22 = c2c 4 c 2c 3a 3 c 2c 3 ) Here, the disturbances αs and βs are considered to be slow-varying, that is d αs = d βs = Based on (4), and defining ˆλ αr, ˆλ βr, î αs, and î βs as the estimates of λ αr, λ βr, i αs, and i βs, respectively, an observer based on the equivalent control method 21] is designed as dî αs dî βs dˆλ αr dˆλ βr = ϑ 11 i αs +ϑ 12ˆλ αr ϕ 1 ω rˆλ βr +c 1 v αs + ˆ αs +l 11 ρ 1 (ĩαs ) +V1 = ϑ 21 i βs +ϑ 22ˆλ βr +ϕ 2 ω rˆλ αr +c 2 v βs + ˆ βs +l 21 ρ 1 (ĩβs ) +V2 = ς 11 i αs +ς 12ˆλ αr +c 1 v αs +l 3 V 1 = ς 21 i βs +ς 22ˆλ βr +c 2 v βs +l 4 V 2 dˆ αs dv 1 ) = l 5 V 1, = l 12 ρ 2 (ĩαs dˆ βs dv 2 ) = l 6 V 2, = l 22 ρ 2 (ĩβs (4) (5) /13/$ IEEE 26

3 where ĩ αs = i αs î αs, and ĩ βs = i βs î βs are the estimation errors of i αs, and i βs, respectively With ρ 1 ( ) = µ sign( ) +µ 2 ( )+µ sign( ), ρ 2 ( ) = 1 2 µ2 1 sign( )+ 3 2 µ 1µ sign( ) + ( ) µ µ 1 µ 3 ( ) µ 2µ sign( ) µ2 3 2 sign( ), l j > for j = 1,,6 and, µ i > for i = 1,2,3 As a result, the rotor flux estimates ˆλ αr and ˆλ βr are obtained as ˆλ αr = ˆλ αr î αs c 1c 3 and ˆλ βr = ˆλ βr î βs c 2c 3 IV SIDING MODE CONTROER DESIGN Provided that the currents and speed are continuously measured and the rotor fluxes are estimated, the objective here is to design a SM controller which can effectively track the desired speed ω ref and the module to the square of the rotor flux φ ref reference signals by means of the continuous basic control v s and auxiliary control ρ as a discontinuous function A Sliding Manifold Design As first step, the state variables x 1 and x 2 are defined as x 1 = ω r φ and x 2 = i αs i βs, where φ = ψ 2 = λ 2 αr + λ 2 βr Thus, the system (1) can be represented in the nonlinear block controllable form with disturbance 24] dx 1 =f 1 (φ)+b 1 (λ r )x 2 +D 1 T + r (6) dx 2 =f 2 (ω r,λ r,i s )+B 2 u+ s where λ r = λ αr λ βr, u = v αs v βs, f 1 (φ) = f 11 f 12 = 2a 3 φ, D 1 = d 2, f 2 = f 21 f 22, r = 2 αr λ αr +2 βr λ βr, s = ] αs βs, d1 d 2 λ βr d 1 d 2 λ αr B 1 (λ r ) = c1 and, B 2 = ] 2a 4 λ αr 2a 4 λ βr, with f 21 = a 1 c 1 i αs +c 1 c 4 λ αr c 1 c 3 ω r λ βr c 2 and f 22 = a 2 c 2 i βs +c 2 c 3 ω r λ αr +c 2 c 4 λ βr Only the estimates of the rotor fluxes are available for the control design Hence, the estimated variables ˆφ = ˆλ 2 αr+ˆλ 2 βr, ˆλ r = (ˆλ αr,ˆλ βr ) and its errors φ = φ ˆφ, λ r = λ r ˆλ r, are defined Setting the controller-used error ẑ 1 = z 11 ẑ 12 and real tracking errors z 1 = z 11 z 12, with z 11 = ω r ω ref (t), ẑ 12 = ˆφ φ ref (t) and, z 12 = φ φ ref (t) = ˆφ+ φ φ ref (t) = ẑ 12 + φ, the dynamics of the first transformed block (6) become = f 1(ˆφ)+B 1 (ˆλ r )x 2 + Φ+ 1 (7) where Φ = d φ and 1 = D 1 T + r + dωref (t) 213 1th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) dφ ref (t) To stabilize the dynamics for z 1, x 2 can be selected as a stabilizing term in form of a virtual controller Therefore, the /13/$ IEEE 27 desired value for x 2 is defined as x 2des = i des αs it is proposed of the form i des βs, and x 2des = x 2des +x1 2des (8) where x 1 2des will be designed to reject the disturbance 1 in finite time by using the integral sliding mode technique 31] The term x 2des is such that z 1 converges exponentially to zero To establish the control x 2des in (7), the error variable z 2 = z 21 z 22 is defined as and (7) is rewritten as z 2 = x 2 x 2des (9) = f 1(ˆφ)+B 1 (ˆλ r )x 2des +B 1 (ˆλ r )x 1 2des +B 1 (ˆλ r )z 2 + Φ+ 1 (1) In order to calculate x 1 2des, the variable σ = σ 1 is proposed as σ 2 σ = ẑ 1 +ξ (11) where ξ = ξ 1 below ξ 2 is an integral variable to be defined From (11), the dynamics of σ are given by = f 1(ˆφ)+B 1 (ˆλ r )x 2des +B 1(ˆλ r )x 1 2des +B 1 (ˆλ r )z 2 + Φ+ 1 + dξ (12) With the selection of dξ as dξ = f 1(ˆφ) B 1 (ˆλ r )x 2des (13) where ξ() = z 1 (), the system (12) reduces to = B 1(ˆλ r )x 1 2des +B 1(ˆλ r )z 2 + Φ+ 1 (14) To enforce sliding motion on the manifold σ = despite of the disturbance 1, the term x 1 2des in (14) is chosen as x 1 2des = B1 1 (ˆλ r )ν, with ν = ν 1 ν 2 defined as the solution to dν 1 1 = k σ 1 dν 2 2 = k σ 2 Here, the derivatives 1 differentiator 32] +k δ1 σ sign(σ 1 ) +k δ1 σ k δ2 σ sign(σ 2 ) +kδ2 σ and 2 (15) are obtained using a SM When the motion on the manifold σ = is reached, the solution to = in (14) {B 1 (ˆλ r )x 1 2des } eq = B 1 (ˆλ r )z 2 + Φ+ 1 (16) shows that the disturbance Φ+ 1 is rejected by the equivalent control {B 1 (ˆλ r )x 1 2des } eq 15] Therefore, the dynamics on σ = are given by = f 1(ˆφ)+B 1 (ˆλ r )x 2des (17)

4 213 1th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Thus, the desired dynamics K 1 ẑ 1 for dz1 in (17) are introduced by means of x 2des = B 1 1 (ˆλ r ) f 1 (ˆφ) K 1 ẑ 1 ] (18) where K 1 = diag(k 1,k 2 ) with k 1 >, k 2 > Hence, with (18) in (13), dξ reduces to dξ = K 1ẑ 1 (19) B Inducing Sliding Modes From (9), it follows that dz 2 = f 2(ω r,ˆλ s,i s )+B 2 u+ 2 (2) where the term 2 = r dx2des is a bounded disturbance To induce a SM motion on the manifold on z 21 = or i αs = i des αs in the current loop, taking into account (2), the basic control v s is formulated as 23] v s = α 1 z 21 1/2 sign(z 21 ) α 3 z 21 +u 1 (21) du 1 = α 2 sign(z 21 ) with α 1 >, α 2 >, and α 3 > And to induce a quasisliding mode motion on the manifold z 22 = or i βs = i des βs, the auxiliary control ρ for the capacitor is designed by means of the following switching logic: ρ = 1 2 sign(z 22v c )+ 1 2 (22) V STABIITY ANAYSIS OF THE OBSERVER-BASED CONTROER The extended closed loop system is presented as { = K 1 ẑ 1 +ν +B 1 (ˆλ r )z 2 + Φ+ 1 (23) = ν +B 1 (ˆλ r )z 2 + Φ+ 1 1 dν 1 = k +k δ 1 σ sign(σ 1) σ1 1 +k δ1 σ (24) 2 dν 2 = k +k δ 2 σ sign(σ 2) σ2 2 +k δ2 σ { ( dz2 = f 2 ω r,ˆλ s,i s )+B 2 u+ 2 (25) dĩ αs = ϑ 12 λ αr ϕ 1 ω r λ βr + αs ) l 11 ρ 1 (ĩαs V1 dĩ βs = ϑ 22 λ βr +ϕ 2 ω r λ αr + (26) βs ) l 21 ρ 1 (ĩβs V2 d λ αr = ς 12 λ αr + αr l 3 V 1 d λ βr = ς 22 λ βr + βr l 4 V 2 (27) d αs = l 5 V 1 dˆ βs = l 6 V 2 It is possible to demonstrate for the block (24) that there is a SM on the manifold σ = in finite time by using the results exposed in 25] Similarly, it can be shown the finite time convergence of the system (25) to the manifold z 2 = 11] Finally, the uniform finite time convergence to zero of the estimation errors ĩ αs and ĩ βs in the system (26) can be proved with the results presented by 22] The SM motion is given by the systems (23) and (27) constrained to the set σ 1 σ 2 z 21 z 22 ĩ αs ĩ βs = as follows: = K 1(z 1 + z 1 ) d λ αr = (ς 12 l 3 ϑ 12 ) λ αr l 3 αs + αr +l 3 ϕ 1 ω r λ βr d λ βr = (ς 22 l 4 ϑ 22 ) λ βr l 4 βs + βr l 4 ϕ 2 ω r λ αr d αs = l 5 ϑ 12 λ αr l 5 αs +l 5 ϕ 1 ω r λ βr dˆ βs = l 6 ϑ 22 λ βr l 6 βs l 6 ϕ 2 ω r λ αr (28) To analyze the stability of the system (28), it can be written as a linear system with non-vanishing disturbance of the form ė = Me+ (29) wheree = z1 T λ αr λ βr αs βs, M is the block matrix K 1 ς 12 l 3 ϑ 12 l 3 M = ς 22 l 4 ϑ 22 l 4 l 5 ϑ 12 l 5 l 6 ϑ 22 l 6 and = z 1 T αr +l 3 ϕ 1 ω r λ βr βr l 4 ϕ 2 ω r λ αr l 5 ϕ 1 ω r λ βr l 6 ϕ 2 ω r λ αr For (29), the yapunov candidate function V = 1 2 et Pe is proposed, with P > With the adequate choice of l i when i = 3,,6 and K 1, the matrix M is Hurwitz Hence, there exists one unique solution P to the yapunov equation M T P+PM = Q, where Q = Q T and Q > For the system (29), it is satisfied λ min (P) e 2 2 et PQ λ max (P) e 2 2 (3) V e Me = et Qe λ min (Q) e 2 2 and the perturbation term is considered to be bounded by α 2 e 2 +β 2, with α 2 > and β 2 > The derivative of V, yields to V = e T Qe 2e T P (31) /13/$ IEEE 28

5 213 1th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) and, substituting the bounds (3) in (31), results V ( λ min (Q)+2α 2 λ max (P)) e β 2λ max (P) e 2 = α(1 θ) e 2 2 αθ e 2 2 +β e 2 where α = λ min (Q) 2α 2 λ max (P), β = 2β 2 λ max (P) and < θ < 1 Finally, V α(1 θ) e 2 2, e 2 > δ, with δ = β αθ Thus, the nominal system ė = Me has an exponentially stable equilibrium point e = and, the solution e(t) of (29) is ultimately bounded The ultimate bound is given by λmax(p) e 2 δ λmin(p) VI NUMERICA SIMUATION RESUTS To verify the effectiveness and efficiency of the proposed observer-based controller, numerical simulations are conducted using the Euler integration method with a time step t s = Parameters and data of the SPIM are in the Table 1 3]: 2) At 2 seconds, a 3% increase in the value of the rotor resistance is presented 3) The SPIM starts whit a increase in the value of inductances at 15% ωr rad/sec] φ Fig 2: Rotor speed ω r and module to the square of rotor flux φ 1 φ ω r ω rref φ ref Single-Phase HP 25 Vs 11 (V ) f 6 (Hz) np 2 n = N A 118 Rαs 22 (Ω) N B Rβs 513 (Ω) Rr 412 (Ω) αs 1846(H) βs 1833 (H) r 1828 (H) m 1772 (H) J 146 (Kgm 2 ) kd (kgm 2 /s) Imax 15 (A) Crun 35 µf λαrwb] λβr wb] TimeSec] TABE I: Parameters of SPIM Fig 3: Error of rotor flux in axis frame α β The controller gains are adjusted to k 1 = 5, k 2 = 75, k σ1 = 3, k σ2 = 1, k δ1 = 1, k δ2 = 15, α 1 = 36, and α 3 = 1 And, the gains for the observer are l 11 = 15, l 21 = 17, l 12 = 1, l 22 = 1, l 5 = 5 and, l 6 = 5 For the simulation purposes, the initial conditions of the state variables are selected to zero Tracking performance is verified for the two plant outputs: driving the square of rotor flux φ to a constant reference φ ref = 15, and a speed profile ω ref for ω r, proposed as follows: 1) The SPIM starts on repose with the reference speed on 1 rad/sec 2) At the first second, a change of the speed reference from 1 rad/sec to 12 rad/sec, is presented 3) Finally, at 3 seconds, a change of the speed reference from 12 rad/sec to 14 rad/sec, is presented In addition, the system is subject to disturbances which are introduced as follows: 1) The SPIM starts on repose with a load torque of 5 N-m, then at 1 sec a change of load torque from 5 N-m to 8 N-m After that at 3 sec another change of load torque from 8 N-m to 1 N-m And finally at 4 sec one more change of load torque from 1 N-m to 5 N-m iαsa] i βs A] Fig 4: Stator currents in axis frame α β The rotor speed tracking response is depicted in Fig 2 which shows a satisfactory performance under the change of the speed reference at t = 1,4 sec and t = 34 sec, where the speed tracking effect is achieved almost totally after 82 sec Fig 2 shows the module to the square of the rotor flux φ response too; it is possible to see that the module is maintained over the given reference The errors responses of rotor fluxes are shown in Fig /13/$ IEEE 29

6 213 1th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) On the other hand, the stator currents (see Fig 4) are in the appropriate range during the start ( < t < 2) that corresponds to the proposed control algorithm Finally, in Fig 5, the responses of the voltages are presented, where v αs is the super-twisting SM control and, v βs is the discontinuous SM control vαsv ] v βs V ] Fig 5: Stator control voltages of axis α β VII CONCUSIONS An observer-based control scheme was proposed to track the rotor angular speed ω r and module to the square of rotor flux φ It is based on SM algorithms allowing a robust design The stability conditions of the closed-loop system was derived The simulation results have shown a robust performance of the designed controller with respect to the perturbations caused by the load torque, fulfilling the stator current constraints REFERENCES 1-H iu, A maximum torque control with a controlled capacitor for a single-phase induction motor, Industrial Electronics, IEEE Transactions on, vol 42, no 1, pp 17 24, ] F Blaschke, The principle of field orientation applied to the new transvector closed-loop control system for rotating field machines, Siemens Review, vol 39, pp , ] I Kanellakopoulos, P T Krein, and F Disilvestro, Nonlinear fluxobserver-based control of induction motors, in American Control Conference, 1992, june 1992, pp ] Gokdere and M Simaan, A passivity-based method for induction motor control, Industrial Electronics, IEEE Transactions on, vol 44, no 5, pp , oct ] R Ortega, P J Nicklasson, and G Espinosa-Pérez, On speed control of induction motors, Automatica, vol 32, no 3, pp , ] R Marino, S Peresada, and P Valigi, Adaptive input-output linearizing control of induction motors, Automatic Control, IEEE Transactions on, vol 38, no 2, pp , feb ] R J Evans, B J Cook, and R E Betz, Nonlinear adaptive control of an inverter-fed induction motor linear load case, Industry Applications, IEEE Transactions on, vol IA-19, no 1, pp 74 83, jan ] V I Utkin, Sliding mode control design principles and applications to electric drives, Industrial Electronics, IEEE Transactions on, vol 4, no 1, pp 23 36, feb ] V A Utkin, AC drives control problems, Avtomatica i Telemekhanika, vol 12, pp 53 65, ] G J Rubio, J M Cañedo, A G oukianov, and J M Reyes, Second order sliding mode block control of single-phase induction motors, in Proceedings of the 18th IFAC World Congress, Milan, Italy, ] G J Rubio, J D Sanchez-Torres, J M Canedo, and A G oukianov, HOSM block control of SPIM, in Electrical Engineering, Computing Science and Automatic Control (CCE), 212 9th International Conference on, 212, pp ] A Y Alanis, E N Sanchez, A G oukianov, and M Perez-Cisneros, Real-time discrete neural block control using sliding modes for electric induction motors, Control Systems Technology, IEEE Transactions on, vol 18, no 1, pp 11 21, 21 13] B Castillo-Toledo, S Di Gennaro, A G oukianov, and J Rivera, Hybrid control of induction motors via sampled closed representations, Industrial Electronics, IEEE Transactions on, vol 55, no 1, pp , 28 14] G C Verghese and S Sanders, Observers for flux estimation in induction machines, Industrial Electronics, IEEE Transactions on, vol 35, no 1, pp 85 94, ] V Utkin, Sliding modes in control and optimization, ser Communications and control engineering series Springer-Verlag, ] R Marino, S Peresada, and P Tomei, Exponentially convergent rotor resistance estimation for induction motors, Industrial Electronics, IEEE Transactions on, vol 42, no 5, pp , oct ] A G oukianov, J M Canedo, O Serrano, V I Utkin, and S Celikovsky, Adaptive sliding mode block control of induction motors, in American Control Conference, 21 Proceedings of the 21, vol 1 IEEE, 21, pp ] C Aurora, A Ferrara, and A evant, Speed regulation of induction motors: a sliding mode observer-differentiator based control scheme, in Decision and Control, 21 Proceedings of the 4th IEEE Conference on, vol 3, 21, pp vol3 19] G Bartolini, A Damiano, G Gatto, I Marongiu, A Pisano, and E Usai, Robust speed and torque estimation in electrical drives by secondorder sliding modes, Control Systems Technology, IEEE Transactions on, vol 11, no 1, pp 84 9, 23 2] A N Atassi and H K Khalil, A separation principle for the stabilization of a class of nonlinear systems, Automatic Control, IEEE Transactions on, vol 44, no 9, pp , ] S V Drakunov, Sliding-mode observers based on equivalent control method, in Decision and Control, 1992, Proceedings of the 31st IEEE Conference on, vol 2, 1992, pp ] E Cruz-Zavala, J Moreno, and Fridman, Uniform second-order sliding mode observer for mechanical systems, in Variable Structure Systems (VSS), 21 11th International Workshop on, 21, pp ] A evant, Sliding order and sliding accuracy in sliding mode control, International Journal of Control, vol 58, no 6, pp , ] A oukianov, Robust block decomposition sliding mode control design, Mathematical Problems in Engineering, vol 8, no 4-5, pp , 22 25] A evant, Quasi-continuous high-order sliding-mode controllers, Automatic Control, IEEE Transactions on, vol 5, no 11, pp , 25 26] J Rivera and A oukianov, Integral nested sliding mode control: Application to the induction motor, in Variable Structure Systems, 26 VSS 6 International Workshop on, 26, pp ] H Huerta-Avila, A oukianov, and J Canedo, Nested integral sliding modes of large scale power system in Decision and Control, 27 46th IEEE Conference on, 27, pp ] A Estrada and Fridman, Quasi-continuous HOSM control for systems with unmatched perturbations, Automatica, vol 46, no 11, pp , 21 29] J Dávila, Exact tracking using backstepping control design and highorder sliding modes, IEEE Transactions on Automatic Control, 213, to be publised - early access 3] P Krause, O Wasynczuk, and S Sudhoff, Analysis of electric machinery and drive systems, ser IEEE Press series on power engineering, I Press, Ed IEEE Press, 22 31] V Utkin, J Guldner, and J Shi, Sliding mode control in electromechanical systems, ser Automation and control engineering CRC Press, 29 32] A evant, Robust exact differentiation via sliding mode technique, Automatica, vol 34, no 3, pp , /13/$ IEEE 3