Electrical Engineering (2007) 89:193–203 DOI 10.1007/s00202-005-0331-1 O R I G I N A L PA P E R A Mezouar · M K Fellah · S Hadjeri Robust sliding mode control and flux observer for induction motor using singular perturbation Received: 18 July 2005 / Accepted: 20 September 2005 / Published online: 24 February 2006 © Springer-Verlag 2006 Abstract This paper proposes a sequential methodology for designing a robust adaptive sliding mode observer for an induction motor drive using a two-time-scale approach This approach is based on the singular perturbation theory The two-time-scale decomposition of the original system of the observer error dynamics into separate slow and fast subsystems permits a simple design and sequential determination of the observer gains In the proposed method, the stator currents and rotor flux are observed on the stationary reference frame using the sliding mode concept The control algorithm is based on the indirect field oriented sliding mode control with an on-line adaptation of the rotor resistance to keep the machine field oriented The control–observer scheme seeks to provide an asymptotic tracking of speed and rotor flux in spite of the presence of an uncertain load torque and an unknown value of rotor resistance The validity for practical implementation has been verified through computer simulations Keywords Induction motor drives · Singular perturbation theory · Sliding mode control · Sliding mode observer List of Symbols Electrical rotor and reference speeds ω, ω∗ vsd , vsq Stator voltages in the synchronously rotating reference frame Stator currents in the synchronously isd , isq rotating reference frame φrd , φrq Rotor fluxes in the synchronously rotating reference frame A Mezouar (B) · M K Fellah · S Hadjeri Intelligent Control and Electrical Power Systems Laboratory, Department of Electrical Engineering, Faculty of Sciences Engineering, Djillali Liabes University, 22 000 Sidi Bel Abb`es, Algeria Tel.: +213-72-320609 Fax: +213-48-474262 E-mail: a mezouar@yahoo.com E-mail: mkfellah@yahoo.fr E-mail: shadjeri@yahoo.fr f T e , TL Sc S Stator voltages in the stationary reference frame Stator currents in the stationary reference frame Rotor fluxes in the stationary reference frame Synchronous frequency, slip frequency ωsl = ωs − ω Stator and rotor inductances Stator and rotor resistances Stator and rotor time-constants Rotor inverse time-constant αr = Rr /Lr Mutual inductance and leakage factor Moment of inertia of the rotor and numbers of pole pairs Coefficient of viscous friction Electromagnetic and load torques Control sliding surface Observer sliding surface (.), (.)∗ Estimated and reference value of (.) (.) t time-derivative of (.) (d(.)/dt) vsα , vsβ isα , isβ φrα , φrβ ωs , ωsl L s , Lr R s , Rr Ts , Tr αr M, σ J, p ∧ • Introduction The control of the induction motor has attracted much attention in the past few decades One of the most significant developments in this area was the field oriented control The orientation of the flux made it possible to act independently on the rotor flux and the electromagnetic torque through the intermediary of the components of the stator voltage [1, 2] Unfortunately, this control approach suffers from its sensitivity to the motor parameter variations When the motor parameters change with temperature and magnetic saturation, the performance of the system will deteriorate There are two ways to solve this problem The first is to perform an online identification of the motor parameters and accordingly update the values used in the controller The other solution 194 is to use a robust control algorithm (i.e., insensitive to the motor parameter variations) In the last years, the sliding mode technique has been widely studied and developed for the control and state estimation problems since the studies of Utkin [3] This control technique allows a good steady state and good dynamic behavior in the presence of system parameters’ variation and disturbances [4–6] Several methods of applying the sliding mode control to induction motor drives have been presented [4–7] All of these methods have a common feature: the analysis and design of the sliding mode controller are based on the mathematical model of the induction motor as used in indirect vector control Most control schemes for the induction motor require accurate information on state variables and parameters Therefore, fluxes are usually estimated with measured stator currents, stator voltages, and motor speed by using the current model for rotor flux or voltage model for stator flux [8,9] However, since the estimated fluxes are basically dependent on the motor model, a variation in the parameters inevitably propagates to the flux estimated error Among these parameters, rotor resistance varies mainly depending on the temperature One of the solutions to reduce the estimation error is to design an adaptive observer compensating for the variation To overcome this problem, numerous on-line identification schemes dealing with rotor resistance have been proposed Some researches have proposed various induction motor drives with rotor resistance or rotor time-constant identification [10–16] to produce better control performance, such as a model reference adaptive system, Luenberger observer, and the extended kalman filter On the other hand, the singular perturbation theory provides the mean to decompose two-time-scale systems into slow and fast subsystems of lower order described in separate time-scales, which greatly simplify their structural analysis and any subsequent control design Then, the control (and/or observer) design may be carried out for each lower order subsystem, and the combined results yield a composite control (and/or observer) for the original system [17–19] So, the idea of combining the singular perturbation theory and the sliding mode technique constitutes a good possibility to achieve classical control objectives for systems having unmodeled or parasitic dynamics and parametric uncertainties [20–22] In this paper, an adaptive sliding mode observer is developed for the simultaneous estimation of the rotor flux components and of the rotor resistance for an induction motor under the assumption that only the stator currents and the motor speed are available for measurement Singular perturbation theory is used for having a sequential and simple design of this observer In addition, an adaptive law based on the Lyapunov stability theory estimates the rotor resistance This paper is organized as follows: the main results of the two-time-scales approach and the design of the general twotime-scales sliding mode observer are presented in Sects and 3, respectively In Sect 4, we briefly review the indirect field oriented sliding mode control of the induction motors A Mezouar et al The design of a two-time-scale sliding mode observer for the induction motor model is presented in Sect In that section, a study of the stability analysis of this observer is made via the singular perturbation method with the sliding mode concept and the Lyapunov stability theory In Sect 6, and through simulation, the studied observer is associated to the indirect field oriented sliding mode control where rotor fluxes and rotor resistance are replaced by those delivered by the observer Finally, in Sect 7, we provide some comments and conclusions Two-time-scale approach The two-time-scale approach, based on the singular perturbation theory, can be applied to systems where the state variables can be split into two sets, one having “fast” dynamics, and the other having “slow” dynamics The difference between the two sets of dynamics can be distinguished by the use of a small multiplying scalar ε Generally, the scaling parameter ε is the speed ratio of the slow versus fast phenomena If the slow states are expressed in the t time-scale, then, the fast ones will be in the τ time-scale defined by (1) τ = (t − t0 )/ε, where t0 is the initial time The reader is referred to [17] and [18] for the general theory on singular perturbation 2.1 Nonlinear singularly perturbed systems Let us consider the following class of nonlinear singularly perturbed systems described by the so-called standard singularly perturbed form: d x = fx (x, z, u, t, ε), x(t0 ) = x0 , dt d (2) ε z = fz (x, z, u, t, ε), z(t0 ) = z0 , dt n m where x ∈ is the slow state, z ∈ is the fast state, u ∈ p is the control input and ε is a small positive parameter such that ε ∈ [0, 1] fx and fz are assumed to be bounded and analytic real vector fields, and we consider a vector of measurement that is linearly related to the fast state vector as (3) y = z, y ∈ m 2.2 Slow reduced subsystem In the limiting case, as ε → in (2), the asymptotically stable fast transient decays ‘instantaneously’, leaving the reducedorder model in the t time-scale defined by the quasi-steady states xs (t) and zs (t): d xs = fx (xs , zs , us , t, 0), (4) dt (5) = fz (xs , zs , us , t, 0), and the substitution of a root of (5) Robust sliding mode control and flux observer for induction motor using singular perturbation zs = h(xs , us , t), (6) into (4) yields a reduced model d xs = fx (xs , h(xs , us , t), us , t, 0), dt xs (t0 ) = x0 , (7) where the index (s) indicates that the associated quantity belongs to the system without ε 3.1 Sliding mode observer design By structure, an observer based on the sliding mode approach is very similar to the standard full order observer with replacement of the linear corrective terms by a discontinuous function [5–7] The corresponding sliding mode observer for the system of (11) can be written as a replica of the system with an additional nonlinear auxiliary input term as follows: x˙ˆ = fx (x, ˆ z, u, ε) + Gx ε x˙ˆ = fz (x, ˆ z, u, ε) + Gz 2.3 Fast reduced subsystem 195 s s, (12) The fast dynamic subsystem (also known as the boundary layer system) denoted by zf , which represents the derivation of zs from z, is obtained by transforming the slow time-scale t to the fast time-scale τ = (t − t0 )/ε System of Eq (2) becomes where s = sign S(y, y) ˆ is the switching function, and Gx and Gz are the observer gains with (n × m) and (m × m) dimensions, respectively, to be determined The sliding surface function S can be chosen as a linear function of (y − y) ˆ as given in [7], so d x = ε fx (x, z, u, ετ + t0 ), dτ d z = fz (x, z, u, ετ + t0 ) dτ S(y, y) ˆ = (8) (9) e˙x ε e˙z e˙x ε e˙z zf (0) = z0 − zs (0), where uf = u − us is the fast part of the input control = fx (x, z, u, ε) − fx (x, ˆ z, u, ε) − Gx s = fz (x, z, u, ε) − fz (x, ˆ z, u, ε) − Gz s , (14) = f x − Gx s = fz − Gz s , (15) where 2.4 Two-time-scale states approximation Fast and slow variables given by (7) and (9) can be combined into a composite structure in order to approximate the original states of (2) as given in [17]: (10) Two-time-scale sliding mode observer Consider the above continuous nonlinear singularly perturbed system of Eq (2) which is given by x˙ = fx (x, z, u, ε) ε z˙ = fz (x, z, u, ε) where (y − y) ˆ = (y1 − yˆ1 ) (y2 − yˆ2 ) · · · (ym − yˆm ) and is a (n × m) gain matrix to be specified The error dynamics is calculated by subtracting (12) from (11): or with x = xs (t) + O(ε), z = zs (t) + zf (τ ) + O(ε) (13) T Introducing the derivation of zs from z, i.e., zf = z − zs , and again examining the limit as ε → 0, it yields d zf = fz x0 , zs (0) + zf (τ ), uf (τ ), t0 , dτ (y − y), ˆ (11) In addition, it is assumed that the above system is observable Consequently, the observer design may be considered for the state observation of slow variables from the measurement of fast variables [20–22] ex = x − x, ˆ ez = z − zˆ , fx = fx (x, z, u, ε) − fx (x, ˆ z, u, ε), fz = fz (x, z, u, ε) − fz (x, ˆ z, u, ε) The observer gains design can be based on the sequential application of the resulted subsystems of (15) by applying the singular perturbation methodology We first need to analyze the fast variables tracking properly using the so-called reaching condition (based on measurable state variables), and, thereafter, the slow variables’ asymptotic-convergence (for inaccessible state variables) 3.2 Stability analysis in the fast time-scale For the fast error dynamic subsystem, the associated timescale is defined by τ = (t − t0 )/ε; then (15) can be transformed into de x = ε( fx − Gx s ) dτ (16) dez = fz − Gz s dτ 196 A Mezouar et al Setting ε = in (16), it yields dez = fz − Gz s , (17) dτ with dex = dτ In this time-scale, the stability analysis involves the determination of the observer gain Gz so that in this time-scale (τ ), the surface S(τ ) = is attractive It can be shown that when the sliding mode occurs on S(τ ), the equivalent value of the discontinuous observer auxiliary input is found by solving Eq (17) for Gz s after insuring a value of zero for dez dτ such as G z ˜ s = fz , and the equivalent switching vector is obtained as ˜ s = G−1 fz z (18) 3.3 Stability analysis in the slow time-scale The slow error dynamic subsystem can be found by considering ε = in (15); so dex = fx − Gx s , dt = fz − Gz s (19) phase induction motor expressed in the synchronously rotating reference frame (d − q) is Rλ µ µ d isd = − isd + ωs isq + φrd + ωφrq + vsd dt σ L σ L T σ L σ Ls s s r s Rλ µ µ 1 d isq = −ωs isd − isq − ωφrd + φrq + vsq dt σ L σ L σ L T σ Ls s s s r M d (21) isd − φrd + ωsl φrq φrd = dt Tr Tr M d dt φrq = T isq − ωsl φrd − T φrq r r p f dω = (Te − TL ) − ω, dt J J with the constants defined as follows: M2 M2 , Rλ = R s + Rr , σ = − Lr Ls Lr M , µ= Lr where the state variables are the stator currents (isd , isq ), the rotor fluxes (φrd , φrq ) and the rotor speed ω, and the stator voltages (vsd , vsq ) and slip frequency ωsl are the control variables The electromagnetic torque expressed in terms of the state variables is pM (φrd isq − φrq isd ) (22) Te = Lr 4.2 Rotor field oriented induction motor model Among the various sliding mode control solutions for the induction motor proposed in the literature, the one based on indirect field orientation can be regarded as the simplest one From (20), the equivalent switching vector can be re- Its purpose is to directly control the inverter switching by the found as use of two switching surfaces The induction motor equations in the synchronously rotat˜ s = G−1 fz z ing reference frame (d − q), oriented in such a way that the Therefore, by an appropriate choice of Gx , the desired rotor flux vector points into d-axis direction, are the followrate of convergence ex → can be obtained ing: d ω = f1 dt d Sliding mode control review of the induction motor φrd = f2 dt (23) d Assuming that the induction motor model system is control- i = f + v sd sd lable and observable, the sliding mode control consists of two dt σ Ls d phases [4,5]: isq = f4 + vsq dt σ Ls • designing an equilibrium surface, called the sliding surface, such that any state trajectory of the plant restricted to the with M isq sliding surface is characterized by the desired behavior; , (24) ωsl = and Tr φrd • designing a discontinuous control law to force the system where to move on the sliding surface in a finite time f1 = kc φrd isq − Jp TL − fJ ω f2 = M isd − φrd Tr Tr (25) 4.1 Dynamic model of induction machine Rλ f3 = − σ Ls isd + ωs isq + σµLs T1r φrd Under the assumptions of linearity of the magnetic circuit f4 = −ωs isd − σRLλs isq − σµLs ωφrd , and neglecting iron losses, the state space model of the three(20) Robust sliding mode control and flux observer for induction motor using singular perturbation Remark and kc = 197 p M J Lr 4.3 Speed and flux sliding mode controller Using the reduced nonlinear induction motor model of Eq (23), it is possible to design both a speed and a flux sliding mode controller Let us define the sliding surfaces d Sc1 = Sc1 (ω) = λω (ω∗ − ω) + (ω∗ − ω) dt (26) Sc2 = Sc2 (φr ) = λφ (φ ∗ − φdr ) + d (φ ∗ − φdr ), r dt r where λω > 0, λφ > 0, ω∗ and φr∗ are the reference speed and the reference rotor flux, respectively To determine the control law that leads the sliding functions (26) to zero in finite time, one has to consider the dynamics of Sc = (Sc1 , Sc2 )T , described by S˙c = F + D VS , (27) where F = ă + + fJ TL + + (ă r + r ) + + kφ c rd D= M/Tr , σ Ls Tr VS = f J f2 − vsq vsd f1 −kc (isq f2 + φrd f4 ) M f Tr , If the Lyapunov theory of stability is used to ensure that Sc is attractive and invariant, the following condition has to be satisfied ScT · S˙c < (28) So, it is possible to choose the switching control law for stator voltages as follows: vsq Kω sign(Sc1 ) = −D −1 F − D −1 , (29) vsd Kφ sign(Sc2 ) where Kω > 0, Kφ > (30) Proof see Appendix The sliding mode causes drastic changes in the control variables introducing high frequency disturbances To reduce the chattering phenomenon, a saturation function sat(Sc ) instead of the switching one sign(Sc ) has been introduced sat(Sc i ) = Sc i δi if |(Sc i )| ≤ δi sign(Sc i ) if |(Sc i )| > δi , where δi > and i = 1, • From the above control law of Eq (29), it can be seen that the implementation of these algorithms requires load torque and rotor flux estimations since stator currents, stator voltages and rotor speed are available by measures In the next section, we focus on by a robust estimation of rotor flux The estimated load torque can be easily obtained by using the mechanical equation of the motor model with estimated rotor fluxes and measured stator currents • In the following, we assume to operate with constant reference speed, constant reference rotor flux and constant load torque, so that ω˙ ∗ = 0, φ˙ r∗ = and T˙L = Two-time-scale sliding mode observer design for the induction motor Consider only the first four equations of the induction motor model of Eq (21) in which the speed motor will be considered as a time-varying parameter The objective of the studied observer is to estimate the unmeasured rotor fluxes The sliding mode observer design procedure comprises of the following two steps [7]: • designing an equilibrium surface such that the estimation error trajectories restricted to this surface have the desired stable dynamics; and • determining the observer gains to drive the estimation error trajectories to the sliding surface and maintain it on the set 5.1 Dynamic model of the induction motor Using the model of Eq (21), the state space model of the induction motor, without mechanical equation, expressed in the fixed stator reference frame (α, β) is Rλ µ µ d isα = − isα + φrα + ωφrβ + vsα dt σ L σ L T σ L σ Ls s s r s d Rλ µ µ 1 isβ − ωφrα + φrβ + vsβ isβ = − dt σ Ls σ Ls σ L s Tr σ Ls (32) d M φrα = isα − φrα − ωφrβ dt T T r r d M φrβ = isβ + ωφrα − φrβ dt Tr Tr Voltage, current and flux transformation from the synchronous to the stationary reference frame and vice versa is made as [1,2]: xα xβ (31) = cos(θs ) − sin(θs ) xd sin(θs ) cos(θs ) xq cos(θs ) sin(θs ) − sin(θs ) cos(θs ) xα xβ , (33) , (34) and xd xq = 198 A Mezouar et al where x = v, i, φ, and θs is the angular displacement of the synchronously rotating reference frame with S= s1 s2 = z1 − zˆ z2 − zˆ = ez1 e z2 (39) Setting exi = xi − xˆi and ezj = zj − zˆ j for i ∈ {1, 2} and j ∈ {1, 2}, and using Eqs (35) and (36), the estimation error Based on the well-known of the induction machine model dynamics are dynamics [20,21], the slow variables are (φrα , φrβ ) and the ε e˙z1 = µ (+αr ex1 +ωex2 )+ αr (Mz1 − xˆ1 ) −Gz1 s fast variables are (isα , isβ ) Therefore, the corresponding stan- T dard singularly perturbed form with ε = σ Ls , x = (φrα , φrβ ) ε e˙ = µ (−ωe +α e )+ α (Mz − xˆ ) −G z2 x1 r x2 r 2 z2 s and z = (isα , isβ )T is (40) e˙x1 = − (+αr ex1 +ωex2 )+ αr (Mz1 − xˆ1 ) − Gx2 s ε z˙ = −Rλ z1 + µαr x1 + µωx2 + vsα e˙x2 = − (−ωex1 + αr ex2 ) + αr (Mz2 − xˆ2 ) − Gx2 s ε z˙ = −Rλ z2 − µωx1 + µαr x2 + vsβ (35) Equation (40) can be expressed in a matrix form as x˙1 = Mαr z1 − αr x1 − ωx2 ε e˙z = µ (αr I − ωJ )ex + αr (Mz − x) ˆ − Gz s x˙2 = Mαr z2 + ωx1 − αr x2 , (41) ˆ − Gx s , e˙x = − (αr I − ωJ )ex + αr (Mz − x) where Rr where I is the (2 × 2) identity matrix and J is the (2 × 2) = , R λ = R s + M µ αr αr = skew symmetric matrix defined by Tr Lr 5.2 Singularly perturbed induction motor model Remark • All parameters of the induction motor will be considered as constant except for the rotor resistance The rotor resistance Rr will be treated as an uncertain parameter with Rrn as its nominal value An additional assumption is that Rr varies slowly (practical assumption), so that R˙ r ≈ • The motor speed will be treated as a bounded time-varying variable J = −1 Exploiting the time-properties of the multi-time-scales system of Eqs (35) and (36), ez = (ez1 , ez2 )T are the fast variables and ex = (ex1 , ex2 )T are the slow variables Therefore, the stability analysis of the above system involves the determination of Gz1 and Gz2 to ensure the attractiveness of the sliding surface S(τ ) = in the fast time-scale Thereafter Gx1 and Gx2 are determined, such that the reduced-order ˙ )∼ system obtained when S(τ ) ∼ = S(τ = is locally stable 5.3 Singularly perturbed sliding mode observer From Sect 3, the observer equations of the above model based on the sliding mode concept are the following ˙ ˆ ε zˆ = −Rλ z1 + µαˆ r xˆ1 + µωxˆ2 + vsα + Gz1 s ε z˙ˆ = −Rˆ λ z2 − µωxˆ1 + µαˆ r xˆ2 + vsβ + Gz2 s (36) x˙ˆ = M αˆ r z1 − αˆ r xˆ1 − ωxˆ2 + Gx2 s x˙ˆ = M αˆ z + ωxˆ − αˆ xˆ + G r r x s, where αˆ r = αr + αˆ r = αr and Rˆ λ = Rs + M µ αˆ r , in which Rˆ r Rr Rr = + , Lr Lr Lr (37) where xˆi and zˆ j are the estimation of xi and zj for i ∈ {1, 2} and j ∈ {1, 2} and Gx1 , Gx2 , Gz1 and Gz2 are the observer gains The switching vector s is chosen as s = sign (s1 ) , sign (s2 ) (38) 5.4 Fast reduced-order error dynamics From the singular perturbation theory, the fast reduced-order system of the observation errors can be obtained by introducing the fast time-scale τ = (t − t0 )/ε The system of Eq (41) gives d ez = µ (αr I − ωJ )ex + αr (Mz − x) ˆ − Gz s dτ (42) d ex = −ε (αr I −ωJ )ex + αr (Mz − x) ˆ −ε Gx s dτ Considering ε = in the above system, it yields d ez = µ (αr I − ωJ )ex + dτ αr (Mz − x) ˆ − Gz s, (43) d ex = (44) dτ By an appropriate choice of the observer gain terms Gz1 and Gz2 , sliding mode occurs in (43) along the manifold ez = Robust sliding mode control and flux observer for induction motor using singular perturbation Proposition Assume that ex1 and ex2 are bounded in this time (practical assumption) and that ω varies slowly, and consider the system of (43) with the following observer gains matrix Gz = η1 0 η2 (45) The attractivity condition of the sliding surface S(τ ) = is given by ST dS dτ < (46) In this time-scale dx/dτ = and dex /dτ = So, d ˆ − Gz S T S = S T µ (αr I −ωJ )ex + αr (Mz − x) dτ s , (47) or ST d S = −s1 η1 sign(s1 ) dτ −µ αr ex1 + ωex2 + αr (Mz1 − xˆ1 ) αr (Mz2 − xˆ2 ) (48) Taking into account that all states and parameters of the induction motor are bounded, there exists sufficiently large positive numbers η1 and η1 such that ST dS dτ < Thus, (46) is verified with the set defined by the following inequalities η1 > µ αr ex1 + ωex2 + αr (Mz1 − xˆ1 ) η2 > µ αr ex2 − ωex1 + αr (Mz2 − xˆ2 ) (49) Once the trajectory reaches the sliding surface S = ez = 0, the system in the sliding mode behaves as if Gz s is replaced by its equivalent value (Gz s )eq , which can be calculated from the subsystem (43) assuming ez = and e˙z = 5.5 Slow reduced-order error dynamics For slow error dynamics (when S ≡ 0), we use the system (41) and set ε = So, we can write = µ (αr I − ωJ )ex + e˙x = − (αr I − ωJ )ex + αr (Mz − x) ˆ − Gz αr (Mz − x) ˆ − Gx s, αr (Mz − x) ˆ 5.6 Stability analysis of the slow reduced-order error dynamics We chose the positive-definite candidate Lyapunov function as follows 1 (ex )T ex + ( αr )2 , (56) W = q where q > The t time-derivative of W can be expressed as W˙ = −kαr (ex )T ex d αr + k(ex )T (Mz − x) ˆ (57) q dt The condition for (57) to be negative-definite will be satisfied if k>0 (58) and d αr + k(ex )T (Mz − x) ˆ = (59) q dt With the assumption of Eq (59), it yields d αr = −q k (ex )T (Mz − x) ˆ (60) dt Equation (60) provides an adaptive law to estimate the value of the rotor resistance Unfortunately, the flux errors (ex ) are not available So, by defining the function (61) E = ε ez + µ ex , and by using Eq (41), we obtain (62) E˙ = −(Gz + µ Gx ) s It is possible to reconstruct the estimated fluxes error as (63) ex = (E − ε ez ) µ Using the fluxes error estimation of Eq (63), the adaptive law of Eq (60) becomes feasible: d q αr = − k (E − ε ez )T (Mz − x) ˆ (64) dt µ s (50) (51) From Eq (50), we can obtain the equivalent switching vector ˜ s as ˜ s = µ G−1 (αr I − ωJ )ex + z In this time-scale, we can replace s by ˜ s in Eq (51) Hence, subsystem (51) can be written as the following system ˆ , (53) e˙x = −K (αr I − ωJ )ex + αr (Mz − x) with (54) K = (I + µ Gx G−1 z ) in which we assume that Gx is a diagonal matrix such that K = diag(k, k) (55) + αr −s2 η2 sign(s2 ) −µ αr ex2 − ωex1 + 199 (52) Table Nominal parameters of the induction motor 1.5 kW 220/380 V 3.68/6.31 A N = 1420 rpm Ls = 0.274 H p=2 Rs = 4.85 Lr = 0.274 H J = 0.031 Kg m2 Rr = 3.805 M = 0.258 H f = 0.00113 N m s/rd 200 A Mezouar et al ∗ Fig Sensitivity of the system performance to changes on the rotor resistance: first by 50% and next by 100% with φrd = 1.0 Wb a Reference signals of rotor resistance (solid) and load torque (dotted), b reference (dotted) and actual (solid) speed, c rotor fluxes estimation: φˆ rd (solid) and φˆ rq (dotted), d rotor fluxes error: eφrd (solid) and eφrq (dotted), e reference (dotted) and estimated (solid) rotor resistance, f estimated load torque Simulation results The proposed estimation algorithm has been simulated for the induction motor whose data are given in Table As a controller, the indirect field oriented sliding mode control is used It is assumed that the load torque is unknown and that all the parameters are known and constant except for the rotor resis- tance which will change during the operating motor For this closed loop system, the rotor flux feedback signal and rotor resistance are replaced with the estimated corresponding values of Eqs (36) and (64), respectively With the assumption that all states including rotor flux and all parameters are known, the rotor flux and rotor resistance estimated by the proposed method are compared to their actual values Robust sliding mode control and flux observer for induction motor using singular perturbation 201 ∗ Fig Sensitivity of the system performance to change in the external load with Rr = 1.25Rrn and φrd = 1.0 Wb a Real (dotted) and estimated (solid) load torque, b reference (dotted) and actual (solid) speed, c load torque (dotted) and motor torque (solid), d reference (dotted) and estimated (solid) rotor resistance, e rotor fluxes estimation: φˆ rd (solid) and φˆ rq (dotted), f rotor fluxes error: eφrd (solid) and eφrq (dotted) The sliding mode control and observer parameters were chosen as λω = 120, λφ = 120, Kω = 80, Kφ = 80, δ1 = δω = 0.5, δ2 = δφ = 0.5, Gz = diag(50, 50), Gx = diag(5, 5) and q = 700 The results are summarized in this section 6.1 Rotor resistance variation effect This test involves in increasing the rotor resistance As shown in Fig 1a, the motor is started with its nominal rotor resistance value Rrn = 3.805 Then, the rotor resistance of the motor model is suddenly set to 1.5Rrn at t = s, and to 2Rrn at t = s The reference speed and reference rotor flux are maintained at 1400 rpm and 1.0 Wb, respectively 202 A Mezouar et al ∗ Fig Sensitivity of the system performance to change in the reference speed with Rr = 1.25Rrn and φrd = 1.0 Wb a Reference (dotted) and ˆ actual (solid) speed, b reference (dotted) and estimated (solid) rotor resistance, c estimated rotor fluxes: φrd (solid) and φˆ rq (dotted), d rotor fluxes error: eφrd (solid) and eφrq (dotted) Figure 1b shows the speed response of the motor; a very good speed regulation is obtained Fig 1c, d shows the estimated rotor fluxes and the error between them and the actual values High flux tracking and good rotor flux orientation can be observed Figure 1e compares the estimated and actual rotor resistance.After a short convergence time, the estimated rotor resistance reaches the actual value Figure 1f shows the estimated load torque These results show that the sliding mode control with the proposed observer can track the reference command accurately and quickly It is important to notice that the q-axis rotor flux error is greater than the d-axis rotor flux error in the transient state This report is very clear since the rotor resistance estimation error (in transient state) propagates on the slip frequency which directly affects the rotor field orientation [see Eq (24)] 6.2 Performance under external load disturbances The sensitivity of the observer to external load disturbances is also investigated in this study The objective is to follow the speed and rotor flux references in spite of disturbances in the load torque with a constant error (of +25%) in the rotor resistance value This practical error is made to test the efficacy of the adaptive law of Eq (64) Figure 2a shows the actual and the estimated applied load torque Due to the rotor inertia, the estimated load torque presents negative values in the start-up motor and later follows exactly the actual signal Figure 2b presents a very good performance for speed regulation Figure 2c shows the motor and the real load torque Figure 2d presents the actual and estimated rotor resistance Figure 2e, 2f show that the completely decoupled control of rotor flux and torque is obtained and that the observer is very robust to external load disturbances 6.3 Performance over wide speed range In this case, we consider the speed tracking performances for a wide variation range of reference speed The rotor flux reference is kept at its rated value of 1.0 Wb and the motor operates without external load disturbances The observer performance for speed tracking is presented in Fig 3a The actual and the estimated rotor resistances are shown in Fig 3b At very low speed, the rotor resistance error is in the order of 0.7% On the other hand, the rotor resistance estimation is very good at a high speed Figure 3d, 3c show the estimation of the rotor fluxes and the error between the estimated rotor fluxes and the actual rotor fluxes, respectively These results prove that the speed tracking is quite good and that the rotor field is always well-oriented Robust sliding mode control and flux observer for induction motor using singular perturbation Conclusion A robust sliding mode rotor flux observer with an on-line adaptation of the rotor resistance for a 1.5 kW induction motor has been derived using the singular perturbation theory Following the procedure presented in this paper, the conditions for convergence are established through Lyapunov’s theory The accurate rotor flux obtained by this algorithm has been applied to indirect field oriented sliding mode control The efficiency of the control–observer structure scheme with the rotor resistance estimation has been successfully verified by simulation The proposed sliding mode observer-based control demonstrated very good performance, especially; it is robust under rotor resistance variation, external load disturbances and speed tracking In this paper, it has been shown that the use of a sliding mode observer with slow and fast parts, based on the singular perturbation approach, and sliding mode control, is effective for solving the control problem of induction motors Future work is oriented at experimental validation Appendix (Proof 1) Let consider the Lyapunov function T S Sc c Its time-derivative is V˙ = ScT S˙c V = (65) (66) Using equations (27) and (29), we can rewrite (66) as Kω sign(Sc ) V˙ = ScT F + D −D −1 F − D −1 Kφ or V˙ = −ScT Kω 0 Kφ sign(Sc ) (67) Then, the manifold Sc is attractive if V˙ < 0, i e., Kω > Kφ > References Leonhard W (1996) Control of electrical drives Springer, Berlin Heidelberg Newyork 203 Vas P (1990) Vector control of AC machines Oxford University Press, London, UK Utkin VI (1977) Variable structure systems with sliding modes IEEE Trans Automat Contr AC-22:212–222 Sabanovic A, Bilalovic F (1989) Sliding-mode control of ac drives IEEE Trans Ind Appl IA-25:70–75 Utkin VI (1993) Sliding mode control design: principles and applications to electric drives IEEE Trans Ind Electron 40:23–36 Chan CC, Wang HQ (1996) New scheme of sliding mode control for high performance induction motor drives IEE Proc Electr Power Appl 43:177–185 Benchaib A, Rachid A, Audrezet E, Tadjine M (1999) Real-time sliding-mode observer and control of an induction 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Castro R (1995) Sliding mode control and state estimation for non-linear singularly perturbed systems: application to an induction electric machine In: proceedings of the 34th IEEE conf Control App, New York, pp 998–1003 22 De-Leon J, Castro R, Alvares JM (1996) Two-time sliding mode control and state estimation for non-linear systems In: proceedings of the 13th triennial world cong, IFAC, San Francisco, USA, pp 265–270 ... good and that the rotor field is always well-oriented Robust sliding mode control and flux observer for induction motor using singular perturbation Conclusion A robust sliding mode rotor flux observer... Speed and flux sliding mode controller Using the reduced nonlinear induction motor model of Eq (23), it is possible to design both a speed and a flux sliding mode controller Let us define the sliding. .. dτ By an appropriate choice of the observer gain terms Gz1 and Gz2 , sliding mode occurs in (43) along the manifold ez = Robust sliding mode control and flux observer for induction motor using