International Journal of Automation an Computing 11(6), December 214, 598-64 DOI: 1.17/s11633-14-842-1 Circle-criterion Base Nonlinear Observer Design for Sensorless Inuction Motor Control Wafa Bourbia 1 Fari Berrezzek 2 Bachir Bensaker 3 1 University Baji Mokhtar, Department of Electrotechnic, BP. 12, Annaba 23, Algeria 2 University Me Cherif Messaia, Souk Ahras 41, Algeria 3 Laboratory of Electromechanical Systems, University Baji Mokhtar, BP.12, Annaba 23 Algeria Abstract: This paper eals with the esign of a nonlinear observer for sensorless inuction motor control. Base upon the circle criterion approach, a nonlinear observer is esigne to estimate pertinent but unmeasurable state variables of the consiere inuction machine for sensorless control purpose. The observer gain matrices are compute as a solution of linear matrix inequalities (LMI) that ensure the stability conitions of the state observer error ynamics in the sense of Lyapunov concepts. Measure an estimate state variables can be exploite to perform a state feeback control of the machine system. The simulation results are presente to illustrate the effectiveness of the propose approach for nonlinear observer esign. Keywors: Nonlinear observer, circle-criterion, Lyapunov stability, linear matrix inequalties (LMI), inuction motor. 1 Introuction It is well known that inuction motor is one of the most wiely use machines in inustrial applications. This is ue to its high reliability, relatively low cost, an moest maintenance requirements. However, inuction motor is also known as a complex nonlinear system, in which timevarying parameters entail aitional ifficulties for machine control, conitions monitoring an fault iagnostic purposes 1. Furthermore, only a few state variables of the machine are available for on line measurement because of technical an/or economical constraints of the consiere application. In orer to perform avance control techniques, there is a great nee of a reliable an accurate estimation of the unmeasurable key state variables of the machine. A state estimator, also calle state observer, is a ynamic system that is riven by the inputs an outputs of the consiere system, an estimates asymptotically its unmeasurable state variables. It is a soft sensor that plays an important role not only in sensorless control techniques but also in conitions monitoring, fault iagnosis, preictive maintenance an fault tolerant control techniques 1 3. A control literature review shows that nonlinear state observer esign approaches can be roughly ivie into three classes 4 9. The first class of approaches attempts to eliminate the system nonlinearities by a technique of linearization 3. Its rawback is a set of extremely restrictive conitions that can harly be met by a physical system. The secon class of approaches attempts to ominate the system nonlinearities by using a high gain output correction term 4, 5. Its rawbacks are the block triangular structure, the estabilizing effect of the peaking phenomenon, an the Regular paper Special Issue on Recent Avances on Complex Systems Control, Moelling an Preiction II Manuscript receive February 13, 213; accepte December 2, 213 sensitivity against measurement noises. The thir class of approaches to esign nonlinear observers exploits the system nonlinearity. Lipschitz an sector properties are the main nonlinearity properties that are exploite 6, 1 12. In orer to benefit from some recent avances in nonlinear observer base control evelope by the control community, in this paper we focus our attention on the application of one metho of the thir class, the so-calle circlecriterion approach, to esign a nonlinear observer for inuction motor sensorless control. The circle criterion approach is a new line of research introuce for continuous-time systems by 1. It permits to hanle irectly the system nonlinearities with less restriction than linearization an high gain approaches 1, 11. A etaile proof of the main theorem is given in this paper. The paper is organize as follows: In the secon section, we present the ingreient of nonlinearity satisfying the sector properties an the esign of the nonlinear observer. The consiere nonlinear inuction motor moel is presente in the thir section. In the fourth section, we present simulation results an comments. A conclusion ens the paper. 2 Circle criterion base nonlinear observer esign In contrast to the linearization base an high-gain approaches which attempt to eliminate the system nonlinearities using a nonlinear state transformation or to ominate them by a high gain term of correction, circle-criterion approach exploits the type of system nonlinearities to esign nonlinear observer. In its basic form introuce by Arcak an Kokotovic 1, the approach is applicable to a class of nonlinear systems that can be ecompose in linear an nonlinear parts with a conition that the nonlinearities satisfy the sector property.
W. Bourbia et al. / Circle-criterion Base Nonlinear Observer Design for Sensorless Inuction Motor Control 599 2.1 Basic sector properties A nonlinear function f(z, t) such as f(z, t) : + R p R p is sai to belong to the sector + if zf(z, t). This relation is known as the sector property of a nonlinear function. It is equivalent to the following relation: (v 1 v 2)f(v 1,t) f(v 2,t), v 1,v 2 R + (1) here v 1 v 2 = z an f(v 1,t) f(v 2,t) = f(z, t), where v 1 an v 2 are two real positive numbers. Relation (1) states that the nonlinear function f(z, t) is a nonecreasing function. On the other han, if f(z, t) is a continuously ifferentiable function the above relation is also equivalent to the following 1, 11 : f(z, t), z R. (2) z If the function f(z, t) oes not satisfy the positivity conition (2), we can introuce a new function g(z,t) such that g(z,t) =f(z, t)+ρz, ρ One can see that g(z,t) = z f(z, t) z f(z, t) z, z R. (3) + ρ, z R. (4) In the multivariable case, the sector property can be written as: z T f(z, t), here z an f(z, t) are the vectors of appropriate imension, respectively. 2.2 Nonlinear observer esign The circle criterion base nonlinear observer esign can be performe for a class of nonlinear systems that can be ecompose into linear part an nonlinear part as the following 1 13 : ẋ(t) =Ax(t)+φ u(t),y(t) + GfH x(t) (5) y(t) =Cx(t) (6) where A, C an G are known constant matrices with appropriate imensions. The pair (A, C) is assume to be observable. The term φ u(t),y(t) is an arbitrary real-value vector that epens only on the system inputs u(t) anoutputs y(t). The nonlinear part of the system is moele by the term fhx(t) which is a time-varying vector function verifying the sector property. In the following, we recall the main theorem an conitions that are use in this work to stuy the feasibility of nonlinear observer esign for inuction motor sensorless control with respect to sector property. Theorem 1 1, 13. Consier a nonlinear system of the form (5) an (6) with the nonlinear part satisfying the circle criterion relations (1) (4). If there exists a symmetric an positive efinite matrix P R n n an a set of row vectors K R p such that the following linear matrix inequalities (LMI) hol: (A LC) T P + P (A LC)+Q (7) PG+(H KC) T =. (8) Then a nonlinear observer can be esigne as ˆx(t) =Aˆx(t) +φ u(t),y(t)+ly(t) ŷ(t)+ GfHˆx(t) +K(y(t) ŷ(t)) (9) ŷ(t) =C ˆx(t). (1) An the limit of the state estimation error, e(t) =x(t) ˆx(t), tens to zero when the time instant t tens to infinity. ˆx(t) is the estimate of state vector x(t) of the nonlinear system. Q = εi n is a known positive efine matrix, I n is an n-th orer unity matrix, an ε is a small positive real number. The nonlinear observer esign refers to the selection of the gain matrices L an K satisfying the LMI conitions (7) an (8). One can see that the structure of the nonlinear observer is compose of a linear part, that is similar to linear Luenberger observer, an a nonlinear part that is an aitional term that represents the time-varying nonlinearities satisfying the sector property. The circle criterion base nonlinear observer esign takes avantage of the sector property by introucing a nonlinear term in the structure of the observer. In the light of the summary proof presente in 1, we present in the following a etaile proof of the theorem. Proof. The state estimation error is given as: e(t) = x(t) ˆx(t), where ˆx(t) is the estimate of the state vector x(t) of the nonlinear system (5) an (6). The ynamics of the state estimation error are then ė(t) = (A LC)e(t)+ Gf(H x(t)) f(h ˆx(t)+K(y(t) ŷ(t)). (11) Let v 1 = H x(t), an v 2 = H ˆx(t)+K(y(t) ŷ(t)). By setting z = v 1 v 2 =(H KC)e(t), the term between brackets in (11) can be seen as a function of the variable z, an then f(v 1) f(v 2) = f(z, t). Taking into account the above result, the error ynamics in (11) can be rewritten as ė(t) =(A LC)e(t)+Gf(z, t) (12) z =(H KC)e(t). (13) Note that the error ynamics, relations (12) an (13), once again, can be consiere as a linear system controlle by a time-varying nonlinear function f(z, t) satisfying the sector property. Circle criterion establishes that a feeback interconnection of a linear system with a time-varying nonlinearity satisfying the sector property is globally uniformly asymptotically stable 1, 13. Base upon the error ynamics, i.e., relations (12) an (13), the nonlinear observer esign problem is then equivalent to stabilization of the error ynamics problem. To this en, a caniate Lyapunov function V = e T Pe is consiere. With the help of relation (12) an (13), the erivative of the Lyapunov function becomes V = e T (A LC) T P + P (A LC) e+ f T (z, t)g T Pe+ e T PGf(z, t). (14)
6 International Journal of Automation an Computing 11(6), December 214 By setting an (A LC) T P + P (A LC) Q (15) PG = (H KC) T. (16) With Q = εi n an ε, the erivative of the Lyapunov function can be rewritten as V e T Qe 2z T f(z, t). (17) The esign of nonlinear observer base on circle-criterion approach presents the avantage of removing the global Lipschitz restrictions. However, it introuces linear matrix inequality (LMI) conitions. An extension to multivariable iscrete-time case is given in 13 for systems with multiple nonlinearities. In 13, the author has investigate globally Lipschitz systems an boune-state nonlinear systems. A robust version of the circle-criterion is evelope taking into account inputs uncertainties an also measurement noises 14. Boune-state nonlinear systems constitute a large class of systems that inclues electric machine systems. Electric machine moels involve the magnetic flux as a key an boune state variable that is when combine with other state variables of the machine, such as rotor angular velocity, leas to the existence of nonlinear part of the machine moel. This is ue to the effect of the magnetic material saturation property that is similar to the sector nonlinearity. 3 Inuction motor nonlinear moel Inuction motor, as various other electric machines, constitutes a theoretically interesting an practically important class of nonlinear systems. Inuction motor is known as a complex nonlinear system in which time-varying parameters entail aitional ifficulty for system control an conitions monitoring. Base on the fact that the nonlinear moel of the inuction motor system can be significantly simplifie, if one applies the -q Park transformation, ifferent structures of the nonlinear moel can be investigate aniscusseasin1. In this paper, the consiere nonlinear moel of the inuction motor is escribe in stator fixe -q Park reference frame by the following nonlinear ifferential equations with the stator current, rotor flux an rotor angular velocity as selecte state variables of the machine. t i s = γi s + β ϕ r + βω rϕ rq + 1 u s (18) T r σl s t isq = γisq βωrϕ r + β ϕ rq + 1 u sq (19) T r σl s t ϕ r = m i s 1 ϕ r ω rϕ rq (2) T r T r t ϕrq = m i sq + ω rϕ r 1 ϕ rq (21) T r T r t ωr = α(ϕ ri sq ϕ rqi s ) k f ω r k l T l (22) where α = ( 1 σ 1 T s ) + 1 σ T r n2 p m Jl r, β = 1 m ( 1 σ ) σ, σ = 1 m 2 l sl r, γ =, k f = fr, k J l = np,anωr = npωr. J The inices s an r refer to the stator an the rotor components, respectively. The inexes an q refer to the irect an quarature of the fixe stator reference frame components, respectively (Park svectorcomponents). i an u are the current an the voltage vector, respectively, ϕ is the flux vector, r is the resistance, l is the inuctance, an m is the mutual inuctance. T s an T r are the stator an the rotor time constant, respectively. ω r is the rotor angular velocity, f r is the friction coefficient, J is the moment of inertia coefficient, n p is the number of pair poles, Ω r is the mechanical spee of the rotor, an finally T l is the mechanical loa torque. The consiere inuction motor system moel has three inputs an two outputs. Only two state variables are available for measurements which are the stator current components. The nonlinearity of the moel is mainly introuce by the prouct of the rotor angular velocity an the rotor flux components, i.e., relations (18) (21), an the torque, i.e., relation (22), as the prouct of two state variables namely the stator current components an the rotor flux components. In orer to take into account the effect of the time-varying parameters, such as stator (rotor) resistance, one has to introuce an aitional equation relating to the consiere parameter variation. In this paper, we consier only the nonlinearity introuce by the variation of the rotor angular velocity. This type of nonlinear moel is generally use for performing nonlinear control, conitions monitoring an faults iagnosis of electric machine systems. Performing these techniques requires the estimation of unmeasure rotor flux linkage an rotor angular velocity base on the stator current an voltage measurements. In this context, the circle criterion approach application is investigate to esign a nonlinear observer for inuction motor sensorless control. To satisfy sector conitions (1) (4), nonlinearities of the machine moel (18) (22) are function of the flux state variable that is a boune state variable. The nonlinearities of the moel are of the form ω rϕ r that can be expresse as ω rϕ r =(ω rϕ r + ρω r) ρω r. (23) One can verify that (ω rϕ r + ρω r)=ϕ r + ρ. (24) ω r With ϕ r 2, one can choose ρ =2. Once again the system nonlinearity is ecompose into a nonlinearity satisfying the sector property an a linear part to be ae to the linear part of the inuction motor moel. 4 Simulation results an comments Characteristics of the consiere inuction machine are liste in Table 1. In orer to implement the circle criterion approach, the nonlinear inuction motor moel, relations (18) (22), is
W. Bourbia et al. / Circle-criterion Base Nonlinear Observer Design for Sensorless Inuction Motor Control 61 written in the form of moel (5) an (6) taking into account properties (3) an (4), with the following notation for the nonlinear term: Gf(H x) = 4 G if i(h ix(t)) (25) i=1 here the nonlinear functions f i are efine as f 1(H 1x(t)) = ω r(ϕ rq + ρ),f 2(H 2x(t)) = ω r(ϕ r + ρ) f 3(H 3x(t)) = i sq(ϕ r + ρ),f 4(H 4x(t)) = i s (ϕ rq + ρ). G 3 =, G 4 = 121.497 1 H 1 = H 2 = H 3 = 1 H 4 = 1. 121.497 With x(t) =x 1,x 2,x 3,x 4,x 5 T =i s,i sq,ϕ r,ϕ rq,ω r T as the state vector of the inuction machine. The input-output value function is efine as φu(t), y(t)) = Bu(t), where u(t) =u s,u sq,t l T is control input of the machine, B is a constant matrix an y(t) =i s i sq T is the measure output vector. Table 1 Characteristics of the inuction motor Symbol Quantity Numerical value P Power 1.5kW f Supply frequency 5 Hz U Supply voltage 22 V n p Number of pair poles 2 R s Stator resistance 4.85 Ω R r Rotor resistance 3.85 Ω l s Stator inuctance.274 H l r Rotor inuctance.274 H m Mutual inuctance.258 H ω r Rotor angular spee 297.25 ra/s J Inertia coefficient.31 kg 2 /s f r Fiction coefficient.114 N s/ra T l Loa torque 5 N m Taking into account the numerical values of the ifferent parameters of the machine liste in Table 1, one can easily obtain the following numerical moel matrices: 264.7163 42.9129 6.624 264.7163 42.9129 6.624 A = 3.5828 13.8869 2 3.5828 13.8869 2 242.994 242.994.366 B = C = G 1 = 32.1898 32.1898 64.5161 1 1 3.312 1, G 2 = 3.312 1 The first step of the simulation consists of resolving the LMI conitions, relation (7) an (8), using an aequate LMI tool such as the LMI tool-box of the Matlab software. The obtaine nonlinear observer gain matrices L an K i are the following: L = 1.6749.1188.1188 1.6749.7172.175.175.7172 1.621 1.621 K 1 = 1.637.7381 K 2 =.7381 1.637 K 3 =.3948.9193 K 4 =.9193.3948. The corresponing Lyapunov matrix for this LMI feasibility test with ε =.4 is P =.155.71.514.1486.274.71.155.1486.514.274.514.1486 5.61.4659.55.1486.514.4659 5.61.55.274.274.55.55.173 The secon step of simulation consists of injecting the obtaine numerical values of the gain matrix L an the vectors K i in the observer expression, relation (9) an (1), in which the nonlinear term takes the following form: Gf(H ˆx(t)+K(y(t) ŷ(t))) = 4 G if ih i ˆx(t)+K i(y(t) ŷ(t)) i=1 where the vector parameters G i an H i are efine as above as well as the matrices A, B an C. With the help of Matlab S-function an Matlab Simulink, the nonlinear machine system an the nonlinear observer are simulate as shown in Fig. 1. Taking into account the following starting conitions X =4 for the initial state vector of the machine, the measure an unmeasure state variables of the inuction machine are generate..
62 International Journal of Automation an Computing 11(6), December 214 Fig. 1 Matlab-Simulink simulation scheme The simulation results of the esigne nonlinear observer are presente in the following. Figs. 2 an 3 show the measure (re line) an estimate (blue line) stator current an rotor flux for the -axis an the corresponing error. The q-axis components contain the same information as the -axis but they are shifte by π 2 from the -axis components. (The color figures in this paper can be foun in the electronic version.) Figs. 4 an 5 show the measure (re line) an estimate (blue line) rotor angular velocity an the electromechanical torque, respectively, with the corresponing estimation error. One can see that the estimate state variables of the machine follow the esire trajectories. Fig. 2 Measure (re line) an observe (blue line) -stator current components an the corresponing estimation error Fig. 3 Measure (re line) an observe (blue line) -rotor flux component an the corresponing estimation error For further clarity, a zoome part is ae in each figure to show the ifference between the measure an the estimate state variables of the inuction machine system. One can see that the estimation error is null after a transient state. Thus it confirms that the esigne nonlinear observer, base on the circle criterion, estimates effectively the unmeasure state variables of the consiere inuction machine. Measure an estimate state variables of the consiere inuction machine can be use to control the machine system via an aequate state feeback control technique. Evaluation of the performance of the esigne observer in the low spee region an even at zero spee, with an without loa torque, is a research area that effectively opens
W. Bourbia et al. / Circle-criterion Base Nonlinear Observer Design for Sensorless Inuction Motor Control 63 a large variety of applications of sensorless inuction motor rive, which will be investigate in the forthcoming paper. chine for possible sensorless control. References 1 B. Bensaker, H. Kherfane, A. Maouche, R. Wamkeue. Nonlinear moeling of inuction motor rives for nonlinear sensorless control purposes. In Proceeings of the 6th IFAC Symposium on Nonlinear Control Systems, IFAC, Stuttgart, Germany, vol. 3, pp. 1475 148, 24. 2 M. G. Campbell, J. Chiasson, M. Boson, L. M. Tolbert. Spee sensorless ientification of the rotor time constant in inuction machines. IEEE Transactions on Automatic Control, vol. 52, no. 4, pp. 758 763, 27. 3 R. Yazanpanah, J. Soltani, G. R. A. Markaeh. Nonlinear torque an stator flux controller for inuction motor rive base on aaptive input-output feeback linearization an sliing moe control. Energy Conversion an Management, vol. 49, pp. 541 55, 28. Fig. 4 Measure (re line) an observe (blue line) rotor angular velocity an the corresponing estimation error 4 A. N. Atassi, H. K. Khalil. Separation results for the stabilization of nonlinear systems using ifferent high-gain observer esigns. Systems & Control Letters, vol. 39, no. 3, pp. 183 191, 2. 5 N. Boizot, E. Busvelle, J. P. Gauthier. An aaptive highgain observer for nonlinear systems. Automatica, vol. 46, no. 9, pp. 1483 1488, 21. 6 A. Zemmouche, M. Boutaeib. On LMI conitions to esign observers for Lipschitz nonlinear systems. Automatica, vol. 49, no. 2, pp. 585 591, 213. 7 H. Beikzaeh, H. D. Taghira. Exponential nonlinear observer base on the ifferential state-epenent Riccati equation. International Journal of Automation an Computing, vol. 9, no. 4, pp. 358 368, 212. 8 L. P. Liu, Z. M. Fu, X. N. Song. Sliing moe control with isturbance observer for a class of nonlinear systems. International Journal of Automation an Computing, vol.9, no. 5, pp. 487 491, 212. 9 J. Chen, E. Prempain, Q. H. Wu. Observer-base nonlinear control of a torque motor with perturbation estimation. International Journal of Automation an Computing, vol.3, no. 1, pp. 84 9, 26. Fig. 5 Measure (re line) an estimate (blue) electromechanical torque of the machine an the corresponing estimation error 5 Conclusions A circle criterion base nonlinear observer esign for inuction motor sensorless control has been presente. The main avantage of the circle criterion approach is that it permits to exploit irectly the nonlinearities of the system without attempting to eliminate them. However, it introuces linear matrix inequalities as conitions for the convergence of the observer an the error of state estimation. Resolving the LMI etermines the gain matrices of the nonlinear observer. Simulation results show that the circle criterion base nonlinear observer esign can effectively be performe to estimate unmeasurable state variables of the inuction ma- 1 M. Arcak, P. Kokotovic. Nonlinear observers: A circle criterion esign an robustness analysis. Automatica, vol. 37, no. 12, pp. 1923 193, 21. 11 M. Arcak. Certainty-equivalence output-feeback esign with circle-criterion observers. IEEE Transactions on Automatic Control, vol. 5, no. 6, pp. 95 99, 25. 12 W. Bourbia, F. Berrezzek, B. Bensaker. A Circle-criterion base nonlinear observer esign for inuction motor conitions monitoring. In Proceeings of the 7th European Nonlinear Dynamics Conference, Roma, Italy, 211. 13 S. Ibrir. Circle-criterion approach to iscrete-time nonlinear observer esign. Automatica, vol. 43, no. 8, pp. 1432 1441, 27. 14 M. Chong, R. Postoyan, D. Nesic, L. Kuhlmann, A. Varsavsky. A robust circle criterion observer with application to neural mass moels. Automatica, vol. 48, no. 11, pp. 2986 2989, 212.
64 International Journal of Automation an Computing 11(6), December 214 Wafa Bourbia grauate from University of Baji Mokhtar Annaba (UBMA), Algeria. She receive her B. Sc. an M. Sc. egrees in electrotechnical engineering from the UBMA in 23 an 26, respectively. Currently, she is involve in octorate stuies on nonlinear sensorless control of electric machine systems. She has publishe a few referee conference papers. Her research interests inclue system control techniques an their application to electric machines. E-mail: bourbia wafa@yahoo.fr Fari Berrezzek grauate from University Baji Mokhtar Annaba (UBMA), Algeria. He receive his B. Sc. an M. Sc. egrees in electrotechnical engineering from the UBMA in 1994 an 26, respectively. Currently, he is involve in octorate stuies on nonlinear control of electric machine systems. He has publishe a few referee conference papers. His research interests inclue system control techniques an their application to electric machines. E-mail: berrezzek fari@yahoo.fr Bachir Bensaker grauate from University of Science an Technology of Oran (USTO), Algeria. He receive his B. Sc. egree in electronics engineering in 1979. He receive his M. S. an Ph. D. egrees in instrumentation an control from the Universities of Rouen an Le Havre, France, in 1985 an in 1988, respectively. Since 1988, he has been with the Department of Electronics, University of Annaba (UBMA), Algeria, where, since 24, he has been a full professor. He has been an IFAC affiliate since 1991. He has publishe about 5 referee journal an conference papers. His research interests inclue system moelling, control, ientification, estimation, an system reliability an their applications in nonlinear control, conition monitoring, fault etection an iagnostics of electrical machines. E-mail: bensaker bachir@yahoo.fr (Corresponing author)