Two simple and novel SISO controllers for induction motors based on adaptive passivity

Transcription

1 ISA Tansactions 47 (2008) Two simple and novel SISO contolles fo induction motos based on adaptive passivity Juan C. Tavieso-Toes a, Manuel A. Duate-Memoud b, a Univesity of Santiago of Chile, Depatment of Electical Engineeing, Av. Ecuado 359, Santiago, Chile b Univesity of Chile, Depatment of Electical Engineeing, Av. Tuppe 2007, Casilla 42-3, Santiago, Chile Received 3 Mach 2006; accepted 29 May 2007 Available online 2 August 2007 Abstact The design of two single-input single-output (SISO) contolles fo induction motos based on adaptive passivity is pesented in this pape. The two contolles wok togethe with a field oientation block. Because of the adaptive natue of the poposed contolles, the knowledge of the set moto load paametes is not needed and obustness unde vaiations of such paametes is guaanteed. Simple popotional contolles fo the toque, oto flux and stato cuent contol loops ae used, due to the contol simplification given by the use of feedback passive equivalence. A new pinciple called the Toque Flux Contol Pinciple is also stated in this aticle, which consideably simplifies the contolle design, diminishing the contol effots and avoiding also oto flux estimation. c 2007, ISA. Published by Elsevie Ltd. All ights eseved. Keywods: Contol of induction motos; Adaptive passivity; Adaptive contol; Electic dive contol; Nonlinea contol of single-input single-output systems. Intoduction The design of suitable contol algoithms fo induction motos has been widely investigated duing the last decade and at the beginning of this. Although the field oientation pinciple has been imposed as an obligatoy pat of the contol schemes that guaantees high pefomance, the speed, toque and oto flux contolles continue evolving. Since the beginning of the vecto contol of AC dives, seen as a viable eplacement of the taditional DC dives, seveal techniques of linea contol have been used, such as P, PI and PID egulatos, and exact feedback lineaization,2]. Due to thei chaacteistics these techniques do not guaantee a suitable machine opeation fo all opeation anges and they do not conside the vaiations of the set moto load paametes. In this sense, thee ae some applications whee it is equied to opeate at seveal speeds between zeo and the nominal, the moto can wam up diminishing the stato and oto esistances, o pehaps the inetia of the load can vay (e.g. a eel being filled in Coesponding autho. Tel.: ; fax: addesses: jctavieso@jctavieso.cl (J.C. Tavieso-Toes), mduatem@ing.uchile.cl (M.A. Duate-Memoud). the textile industy). Seveal nonlinea contol techniques have been poposed in ode to ovecome the above poblems. Sliding mode techniques 3 5] guaantee obustness unde vaiations of some paametes such as load toque and oto esistance. Nevetheless this contol technique pesents the chatteing effect and acoustic noise as main disadvantages. Atificial intelligence techniques using fuzzy logic, neuonal netwoks o thei combinations, which can even include optimization algoithms based on genetic algoithms, have also been poposed by some authos 6,7]. The fuzzy logic can guaantee obustness unde paamete vaiations and load distubances, but it uses membeship functions which equie a lage pactical expeience fo thei choice. On the othe hand, the neuonal netwoks guaantee good esults but they equie a taining pocess that can be off-line, on-line o a combination of these. Contol techniques based on passivity have also been suggested 5,8] but they ae not obust unde vaiations of the set moto load paametes. All the pevious techniques ae based on athe complex contol schemes that might involve a geat deal of wok in: the choice of the ight paametes and functions (in the case of fuzzy contol), the off-line taining o the on-line paamete estimation at evey instant of time (in the case of ANN), the /$ - see font matte c 2007, ISA. Published by Elsevie Ltd. All ights eseved. doi:0.06/j.isata

2 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) state vaiable estimation (oto flux estimation) and the design of the contol stategy based on the estimations. Fo induction moto contol it becomes necessay to use obust adaptive techniques, to avoid the paamete and oto flux estimations to end up with simple contolles consideing the model nonlineaities. Recently, in 9,0,4,5] two novel adaptive feedback passive equivalence techniques wee poposed fo SISO and MIMO systems. They conside the nonlinea model chaacteistics and they ae adaptive in natue, guaanteeing obustness unde all model paamete vaiations. Based on the contol techniques pesented in 9,0] fo SISO systems, two contol stategies fo induction motos ae poposed in this pape, one with fixed adaptive gains and the othe with time-vaying adaptive gains. These stategies ae suitably simplified by using the new pinciple called the Toque Flux Contol Pinciple which is also poposed in the pape. This pinciple simplifies the design of the toque and flux contolle that wok togethe with a field oientation block. By applying this pinciple to the design of the two contolle given by 9,0], two contol schemes without equiing paamete and vaiable estimations and using simple popotional gains fo the speed, oto flux and stato cuents contol loops ae obtained. A MIMO vesion of the techniques poposed in this pape was developed in 6], having a lage numbe of adjustable paametes. The esults poposed hee ae much simple since both contolles pesent only two adjustable paametes using simple adaptive laws, guaanteeing esults simila to those pesented in 6] fom the obustness viewpoint, unde a wide ange of moto load paamete vaiations as well as unde a wide ange of popotional gain vaiations. The Contolle with Fixed Adaptive Gains (CFAG) and the Contolle with Time-Vaying Adaptive Gains (CTVAG) ae peliminay esults of a eseach poject aimed to develop a beta poduct that will impact the induction moto contol industy. The fist stategy (CFAG) is simple but a bette tansient behavio is expected when using the second one (CTVAG). Futhe expeimental studies will detemine which technique is bette, depending pehaps on technical equiements, such as pecision, tansient esponse, opeation ange o on load chaacteistics. If the alpha poduct justifies maketing this appoach, then movement onto beta site development should poceed. 2. Moto model and passive decomposition scheme 2.. Moto model An induction moto model obtained fom the genealized electical machine equations will be used fo the contolle design. Magnetic field linealy distibuted though the ai gap is assumed and the esulting model is applicable to a p-poles machine. Ion losses, satuation and hysteesis phenomena ae neglected. The efeence axes ae fixed in abitay x y coodinates that otate at a geneic speed ω g. A geneic coodinate system is adopted to allow appeciation of the advantages of the contolle if it is expessed on divese axes. The esulting equations ae ] ( Rs i sx = + L2 m R ) L 2 i sx + ω g i sy + L 2 R ψ x + ( Rs i sy = ω g i sx + L2 m R L 2 ω ψ y + ) i sy u sx ω ψ x + R L 2 ψ y + u sy ψ x = R i sx R ψ x + (ω g ω )ψ y ψ y = R i sy (ω g ω )ψ x R ψ y ω = (T em T c ) J B p J ω whee i sx, i sy ae the stato cuents, ψ x, ψ y ae the oto fluxes, ω is the oto speed and u sx, u sy ae the stato voltages, consideed as contol inputs., L s, ae the mutual, stato and oto inductances espectively, R s, R ae the stato and oto esistances, J is the oto inetia, T em is the electomagnetic toque poduced by the moto, T c is the load toque and B p is the mechanical viscous damping coefficient. We also define σ = L2 m L s, R s = R s + L2 m R,, L 2 leakage o coupling facto. stato tansient inductance. stato tansient esistance. The electomagnetic toque is given by T em = 3 p (ψ 2 2 g i sg ) = 3 p ψg i sg sin α 2 2 = 3 p (ψ x i sy ψ y i sx ). (2) 2 2 The epesentation of the vectos, togethe with the nomenclatue used in the study, is shown in Fig Passive decomposition A passive decomposition is used in 5,8], valid also fo the genealized machine. It consides the induction moto as the inteconnection of the electical and the mechanical systems by means of a negative feedback, as seen in Fig. 2. Stating fom the pevious statement the induction moto contol can be caied out by contolling the electical system, consideing the mechanical system as a passive distubance of (T em T c ) on the vaiable ω. Stating fom the pevious idea the following assumption is stated. Assumption. Thee exists an input vecto voltage u s = u sx u sy ]T which guaantees that the desied cuents i s = ()

3 62 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) In ode to apply the techniques poposed in 9,0] it is necessay to expess the electical system as two SISO systems. Then the MIMO system (3) will be divided into two SISO subsystems defined as follows: ẏ = A (y i, z) + B (y } i, z)u ż = f 0 (z) + P (y fo i =, 2 (4) i, z)y ẏ 2 = A 2 (y i, z) + B 2 (y i, z)u 2 ż = f 0 (z) + P 2 (y i, z)y 2 } fo i =, 2 (5) Fig.. Angle between stato cuent and oto flux vectos. with y, y 2 scala outputs, u, u 2 scala inputs, A, A 2, B, B 2, scala functions, f 0 R2, P, P 2 R2 and z R 2. Fo ou paticula case we will have: Subsystem : Fig. 2. Electical system to be contolled with a passive distubance. i sx i sy ]T and fluxes ψ = ψ x ψ y ]T can be established in the induction moto. Then we can define the vaiable eos as the diffeence between the desied vaiable and the eal one. Fo example, e isx = isx i sx, whee isx is the efeence cuent component and i sx the cuent component obtained though measuements. Then we can wite the electical system in tems of the vaiable eos (deviation vaiables) as follows: ] ẏ = ėisx ė isy R s e isx + ω g e isy + R = L 2 e ψ x + L m ω e ψ L y ω g e isx R s e isy ω e ψ L x + L m R L 2 e ψ y (3) 0 + u 0 L ] m R e isx R e ψ ėψ ż = x L = L x + (ω g ω )e ψy ė ψ y R e isy (ω g ω )e ψ L x R e ψy whee y = i s i s = (i sx i sx) (i sy i sy)] T = e isx e isy ] T = y y 2 ] T R 2, z = ψ ψ = (ψ x ψ x) (ψ y ψ y)] T = e ψ x e ψ y ] T = z z 2 ] T R 2 and u = u s u s = (u sx u sx) (u sy u sy)] T = e usx e usy ] T = u u 2 ] T R 2. ẏ = ė isx = R s e isx + ω g e isy + R L 2 e ψ x + L ] m ω e ψ L y ] + e usx ] R e ψ ėψ ż = x = L x (ω g ω )e ψ y (6) ė ψ y (ω g ω )e ψ x R e ψ L y + T Subsystem 2: T ė isy ė isx e i sx. ẏ 2 = ė isy = ω g e isx R s e isy ω e ψ L x + L ] m R L 2 e ψ y ] + e usy ] R e ψ ėψ ż = x L x (ω g ω )e ψ y = (7) ė ψ y (ω g ω )e ψ x R e ψ L y + ė isx T ė isy T with T = /R. e i sy 3. Contolle design though passive equivalence 3.. Model adjustments In ode to apply the theoems of 9,0], both subsystems must be paameteized in the following nomal fom 3] with linea explicit paametic dependence:

4 ẏ i = Λ T ai A i(y i, z) + Λ bi B i (y i, z)µ i ż = Λ 0 f 0 (z) + Λ pi P i (y i, z)y i J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) } i =, 2 (8) with z R 2, y i R, u i R, A i (y i, z) R 4, B i (y i, z) R, f 0 R 2, P i (y i, z) R 2. The paametes Λ ai R 4, Λ bi R, Λ 0 R 2 2, Λ pi R 2 2. The tem function ż = f 0 (z), in both subsystems, is the so called zeo dynamics 7,8]. Now it is necessay to check whethe both subsystems ae locally weakly minimum phase by finding a positive definite W0 (z) z ] T diffeentiable function W 0 (z) satisfying Λ0 f 0 (z) 0, Λ 0 3]. In both cases we define the function W 0 (z) = 2 (e2 + ψ e2 ); then W 0(z) x ψ y z = e ψ x e ψy ] T. Since Λ 0 f 0 (z) = ] R e ψ x (ω g ω )e ψ y (ω g ω )e ψ x R e ψ y it can be checked that ] W0 (z) T Λ 0 f 0 (z) z = ] R e ψ e ψ x e L x (ω g ω )e ψy ψ y (ω g ω )e ψ x R e ψy = R e 2 ψ x (ω g ω )e ψ x e ψy + (ω g ω )e ψ x e ψy R e 2 ψ y = R ( e 2 ψ x + e 2 ψ y ) 0 (9) leading us to conclude that both subsystems ae locally weakly minimum phase. Now we poceed to expess them in the nomal fom defined by (8). Fo the Subsystem we have Λ a = R s e isx R L 2 A (y i, z) = ω g e isy e ψ x, ω e ψ y Λ b =, B (y i, z) =, Λ p = T 0 0 P (y i, z) = T ] P = P 2, y = e isx, ė isy ė isx ] T,, u = e usx. (0) Fo the Subsystem 2 we can wite Λ a2 = R s A 2 (y i, z) = Λ p2 = R L 2 e isy ω g e isx e ψ y ω e ψ x, Λ b2 =, B 2 (y i, z) =, L m 0 T 0 P 2 (y i, z) = T ] P2 = P 22, y 2 = e isy, ė isx ė isy L ] T m,, u 2 = e usy. Fo both subsystems (0) and () we have R e ψ Λ 0 f 0 (z) = L x (ω g ω )e ψ y (ω g ω )e ψ x R. e ψ L y 3.2. Poposed contolle with fixed adaptive gains (CFAG) () Fom 9] the subsystems (0) and () must be locally weakly minimum phase and matices B i (y i, z) fo i =, 2 should be invetible. The fist condition was aleady demonstated in (9). Fom Eqs. (0) and () it can be eadily checked that matices B i (y i, z) = fo i =, 2, so they ae in fact invetible. Then, accoding to 9] thee exists an adaptive contolle of the fom u i (y i, z, θ hi ) = B i (y i, z) θi T (t)a i(y i, z) θ 2i (t)p i (y i, z) W 0(z) z θ 3i (t)p 2i (y i, z) W 0(z) z 2 + θ 4i (t)µ i ] fo i =, 2 (2) with z R 2, y i R, u i R, µ i R, A i (y i, z) R, B i (y i, z) R, f 0 R 2, P i (y i, z) R 2, P 2i (y i, z) R 2 and θ hi (t) fo i =, 2 and h =,..., 4 adjustable paametes with θ i (t) R 4, and θ 2i (t), θ 3i (t), θ 4i (t) R updated with the adaptive laws θ i (t) = sign(λ bi )A i (y i, z)y i ( ) W0 (z) θ 2i (t) = sign(λ bi )y i P i (y i, z) ( z ) W0 (z) θ 3i (t) = sign(λ bi )y i P 2i (y i, z) z 2 θ 4i (t) = sign(λ bi )y i µ i i =, 2 (3)

5 64 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) that applied sepaately to each subsystem (8) make them locally feedback equivalent to a C 2 -passive subsystem fom the new input µ i to the output y i 9]. The paametes Λ ai R, Λ bi R, Λ 0 R, Λ pi R 2 2 fo i =, 2 epesent constant but unknown paametes fom a bounded compact set Ω. µ, µ 2 signals will be chosen as in (27) Poposed contolle with time-vaying adaptive gains (CTVAG) Anothe adaptive contolle with time-vaying gains 0] can also be poposed fo this case. This contolle denoted as CTVAG has the following fom: u i (y i, z, θ hi ) = θi T B i (y i, z) (t)a i(y i, z) θ 2i (t)p i (y i, z) W 0(z) z θ 3i (t)p 2i (y i, z) W 0(z) z 2 + θ 4i (t)µ i ] fo i =, 2 (4) with z R 2, y i R, u i R, µ i R, A i (y i, z) R, B i (y i, z) R, f 0 R 2, P i (y i, z) R 2, P 2i (y i, z) R 2, fo i =, 2 and θ hi (t) fo i =, 2 and h =,..., 4, ae adjustable paametes with θ i (t) R 4 and θ 2i (t), θ 3i (t), θ 4i (t) R, updated with the adaptive laws θ i (t) = sign(λ bi ) γ i + (t) γ i (t) T γ i (t) A i (y i, z)y i γ θ 2i (t) 2i (t) = sign(λ bi ) + ( W0 (z) P i (y i, z) z γ θ 3i (t) 3i (t) = sign(λ bi ) + ( W0 (z) P 2i (y i, z) z 2 γ θ 4i (t) 4i (t) = sign(λ bi ) + γ i (t) T γ i (t) ) y i γ i (t) T γ i (t) ) y i γ i (t) T γ i (t) µ i y i i =, 2 (5) whee γ i (t) R 4 4 and γ 2i (t), γ 3i (t), γ 4i (t) R ae timevaying adaptive gains defined by ] γ i (t) = γ i A i (y i, z)ai T (y i, z)γ i ( )] W0 (z) 2 γ 2i (t) = γ 2i (t)p i (y i, z) z ( )] i =, 2 (6) W0 (z) 2 γ 3i (t) = γ 3i (t)p 2i (y i, z) z 2 γ 4i (t) = γ 4i (t)µ i ] 2 with γ i (t) = Tace{γ i (t)} γ 2i (t) γ 3i (t) γ 4i (t) ] R 4. This contolle can be applied to each sepaate subsystem (8) and accoding to 0] this will convet each of them into C 2 - passive equivalent systems. The paametes Λ ai R, Λ bi R, Λ 0i R, Λ pi R 2 2 epesent constant but unknown paametes fom a bounded compact set Ω. µ, µ 2 signals will be chosen as in (27). By applying the CFAG given by Eqs. (2) and (3) and CTVAG given by Eqs. (4) (6), we will obtain two contol schemes that do not need the knowledge of any moto load paametes. Howeve it is necessay to know the eo of the oto flux components (z and z 2 ), i.e. oto flux estimation is needed. We will show in Section 4 that this will be not necessay. 4. Pinciple of Toque Flux Contol 4.. Basis and statement Fom now on we will wok in a contol scheme consisting of two kinds of blocks: the adaptive passive contolles, to contol the stato cuent components and oto flow (Fig. 3(a) and (b)), and the field oientation block (Fig. 3(c)). As can be seen in Fig. 3(a), the block will have an extenal closed-loop speed contol feeding the intenal closed-loop i sy cuent contol. In Fig. 3(b) it can be seen thee is a block with an extenal open-loop ψ x contol feeding the intenal closedloop i sx cuent contol. The second block is e jρ g, epesented in Fig. 3(c), which tansfoms fom a stationay to a otating coodinate system. This is a vey impotant contol block employed to guaantee an adequate field oientation such that the maximum output toque is obtained. Once this point is made clea and since the contolle woks togethe with the field oientation block the following assumption can be made. Assumption 2. It is assumed that the coodinate tansfomation block is equal to e jρ g, guaanteeing an adequate field oientation. Then we can wite ρ g = ω g = ω + T i sy ψ x. (7) The pevious assumption implies seveal consequences that ae mentioned next. Consequences:. Fom the fouth pat of Eq. () we can wite ψ y = R i sy (ω g ω )ψ x R ψ y, with ω g = ω + ψ y = R i sy T i sy ψ x ( ω + T i sy ψ x ψ y + R ψ y = 0 lim t ψ y = 0. ) ] ω ψ x R (8) ψ y In pactice the quadatue flux component will be pactically zeo at t = 5T, because ψ y is given by a fist-ode diffeential equation.

6 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 3. Contol blocks involved in the contol scheme. 2. Fom Eq. () and consideing the pevious consequence given by (8) the following model fo the electical system is obtained: ) i sx + ω g i sy + ( Rs i sx = + L2 m R L 2 + ω ψ y + u sx i sy = ω g i sx ( Rs + L2 m R L 2 + R L 2 ψ y + u sy ψ x = R i sx R ψ x. ) i sy L 2 R ψ x ω ψ x (9) 3. Fom thid pat of equation (9) it can be concluded that ψ x is a function of de i sx. In ode to emulate an independent field DC moto, the objective will be to maintain the oto flux constant. 4. Fom Eq. (2) we have T em = 3 p 2 2 L (ψ x i sy ψ y i sx ) and due to Consequence given by Eq. (8), the electomagnetic toque can be expessed as T em = 3 p 2 2 L ψ x i sy whee it can be seen that T em is a function of i sy. 5. The contol of the oto flux ψ = ψ x is caied out though the stato cuent diect component; it is desiable to keep it fixed. The contol of toque T em is caied out though the stato cuent quadatue component. All the above is obtained by using the coodinate tansfomation block and the coesponding field oientation. Based on above assumption and the five consequences deived fom it, the following contol pinciple is established. The Pinciple of Toque Flux Contol When designing contolles fo an altenating cuent moto based on a model of the Genealized Electical Machine, woking in a contol scheme with the coodinate tansfomation block e jρ g (the Field Oiented Scheme) to tansfom fom a stationay to a otating coodinate system, the contol of the toque and flux can be done by contolling the stato cuents. So, it is useless to make effots to contol the oto flux o oto cuent components. The contolle still guaantees a suitable toque and flux contol and it is possible to discad all the tems elated to the oto cuent o oto flux components. By applying this pinciple not only is the contolle design consideably simplified but also the oto cuent o oto flux estimations ae avoided. It is impotant to emphasize that sometimes, even if the field oientation effect is used, the consequences mentioned above ae not fulfilled due to a bad oientation, e.g. due to a bad estimation of paamete /T. Nevetheless, expeimental esults show that the validity of the Pinciple still holds in those cases. The contolle will take cae of an effective contol of the components of the stato cuent whethe the field oientation guaantees o not that the oto flux component eos tend to zeo Application to the contolle CFAG Consideing the Pinciple of Toque Flux Contol, we will use the theoem stated in 9]. Fo the CFAG we will choose adaptive laws such that the stoage functions V i fo i =, 2 satisfy the following condition: ( ) { Wo (z) n V i = Λ o f o (z) + Φ θ( z j+)i P ji (y i, z) j= ( ) ] Wo (z) y i Λ bi z + Φ θ ( j+) i Φ } θ( j+)i Λ bi j + { Φ θi Ai T(y i, z)y i Λ + Φ bi θi Φ θi Λ } { bi + Φ θ(n+)i µ i y i Λ + Φ bi θ(n+)i Φ θ(n+)i Λ } 0 bi fo i =, 2. (20)

7 66 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) ( If the) oto flux components ae eliminated thee esults Wo (z) n ( ) z = z = 0, and the coesponding factos will vanish V 2i = Wo (z) Φ θ( j+)i P ji (y i, z) y i Λ without needing adaptive laws. Then θ ( j+)i = θ ( j+)i = j= + bi z j γi T γ i 0, j =, 2, can be imposed. So it is veified fom (20) that the contolle will still make the system () equivalent to a C 2 - passive system. + γ ( j+)i Φ θ( j+)i Φ θ( Λ j+)i bi Fom Eqs. (0) and () it can be seen that A (y, z) = ] T y ω g y 2 z ω z 2 ( ) W0 (z) = z = ] T z z 2 z B(y, z) = and A 2 (y 2, z) = ] T y 2 ω g y z 2 ω z ( ) W0 (z) = z z B(y 2, z) =. (2) (22) Now if we do not conside (discad) z = z z 2 ] T vaiables, by making z = 0 in Eqs. (2) and (3) the following simplified contolles ae obtained fo Subsystems and 2: Subsystem : ] u (y, z, θ h ) = θ T y + θ ω g y 4 µ 2 ] y h =, 4. (23) θ = y ω g y 2 θ 4 = y µ Subsystem 2: u 2 (y 2, z, θ h ) = θ T ] y2 θ = y 2 ω g y θ 4 = y 2 µ 2 y2 ] + θ ω g y 4 µ 2 h =, 4. (24) On the othe hand, a still moe simplified contolle is obtained by setting the contolle diectly feeding the moto in the stato coodinate system. This means that ω g = 0, so the following SISO contolle applicable to each subsystem (8) is finally obtained: u i (y i, z, θ hi ) = θ i y i + θ 4i µ i θ i = y 2 i θ 4i = y i µ i i =, 2 h =, 4. (25) + Φ θi A T + i (y i, z)y i Λ bi γi T γ i + γ i Φ θi Φ θi Λ bi + Φ θ(n+)i µ i y i Λ + bi γi T γ i + γ (n+)i Φ θ(n+)i Φ θ(n+)i Λ bi ( ) Wo (z) + Λ o f o (z) 0. + z γi T γ i Now, if the oto flux components ) ae eliminated fom Eqs. (4) and (5), i.e. = z = 0, fo i =, 2, and ( Wo (z) z the coesponding factos depending on z will vanish without employing any adaptive law, so θ ( j+) = θ ( j+) = γ ( j+) = γ ( j+) = 0, j =, 2 can be imposed. Besides, fom (8) and (9), by setting the contolle in the stato coodinate system, whee ω g = 0, a moe simplified contolle applicable to each subsystem (8) is obtained fom (4) (6) as follows: u i (y i, z, θ hi ) = θ i y i + θ 4i µ i θ i = sign(λ bi ) θ 4i = sign(λ bi ) γ i = (γ i y i ) 2 γ 4i = (γ 4i µ i ) 2 γ + γ T i γ i γ 4 + γ T i γ i y 2 i µ i y i i =, 2 h =, 4 (26) 4.3. Application to the contolle CTVAG In the same way as in Section 4.2 it can be veified that adaptive laws designed fo CTVAG ae such that the stoage functions V 2i fo i =, 2 satisfy the following condition: 4.4. Pactical consideations in the design The designs obtained up to hee ae based on Assumption and use both desied and eal vaiables. Next we will explain how to apply these contolles in pactice. In the contol scheme

8 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) we will use a typical cascade contol fo induction motos as shown in Fig. 3, obtaining the contolle output vaiables. Anothe assumption on these vaiables is made. Assumption 3. The desied cuent and flux vaiables, isx, i sy and ψ x, ψ y, obtained by applying a desied input voltage u sx, u sy (Assumption ) ae the same vaiables as ae obtained fom the speed contolle output (Fig. 3(a)) and the flux contolle output (Fig. 3(b)) espectively. On the basis of Assumption 3 we can wite u = u sx u sx, with u sx = µ u 2 = u sy u sy, with u sy = µ 2. (27) Fom (27) we obtain the eal voltages to be applied: u sx = µ u u sy = µ 2 u 2. Besides, it is known that a popotional contolle of the fom µ i = ky i fo i =, 2 tuns a passive system asymptotically stable 9,0]. This is exactly the kind of feedback that will be used hee (P contolle). 5. Simulation esults In ode to veify the advantages of the poposed contolles a compaison between the poposed contolles and a taditional cuent egulated PWM induction moto dive with PI loop contolles was caied out. In the simulations thee was consideed a squiel-cage induction moto whose nominal paametes ae: 5 kw] (20 HP]), 220 V], f p = 0.853, 4 poles, 60 Hz], R s = ], X ls = X l = ], x m = ], R = ], J = 2.8 kg m 2 ] and B p = 0. These paametes wee taken fom 2]. The efeence speed will incease in a amp fashion up to 0.5 s. and afte that time it will emain constant. On the othe hand the load toque will be fixed equal to the nominal value. All the simulations wee made using the softwae package MATLAB SIMULINK with the ODE 5s (stiff/ndf) integation method and a vaiable step size. In Fig. 4 some steady state moto chaacteistics ae shown, when a nominal voltage is supplied. It can be seen fom Fig. 4 that even though the nominal toque is equal to 69.5 N m], this moto can supply a toque of 230 N m]. Besides, in spite the fact that the nominal cuent is 49.7 A] it can each values of 280 A] at each phase. This is vey impotant because in some cases the moto should be able to develop such high toque values in tansient peiods, fo example to satisfy high pefomance. The contol schemes used in this study ae shown in Figs. 5 and 6, whee in the poposed contol scheme simple popotional contolles ae employed. The contol scheme shown in Fig. 5 was taken fom 2], whee it is impotant to note that the Toque Flux Contol block depends on the exact values o the estimates of paametes X, X m, R. Besides, the Field oient block depends on the exact values o the estimates of X m and T. The Fig. 4. Steady state chaacteistics of the 20 HP moto employed in this study. PI speed contolle is tuned with P = 30 and I = 0 (see Ref. 2]). The othe contol loops (flux and cuent) ae not PI contolles. The poposed contol scheme is shown in Fig. 6. The two poposed contolles CFAG and CTVAG wee developed and tested in the Poposed Contolle block. It is impotant to obseve that the speed, oto flux, and stato cuent contolles ae simple popotional ( P) gains. These popotional gains wee tuned such that the cuent, flux, speed and voltage wee in suitable anges. This contol scheme just needs the exact values o the estimates of paametes X m and T fo the Field oient block. No othe paamete o state estimations ae needed. Fig. 7 shows the infomation used to compae the two contol schemes. Thee ae shown the vaiations of the efeence speed ω (Fig. 7(a)), the vaiations in load toque (Fig. 7(b)), the vaiation of about 30% in the stato and oto esistance (Fig. 7(c) and Fig. 7(d)), the linea inceasing up to the double the load inetia duing the moto opeation (Fig. 7(e)) and the vaiations in the viscous fiction coefficient (Fig. 7(f)). Fo both poposed contol stategies and the selected taditional contol, five compaative tests consideing the vaiations shown in Fig. 7 wee caied out. These tests will allow us to study the behavio of the schemes in the following situations: Situations:. Initially all the schemes ae simulated with the moto nominal data. The efeence of speed is inceased as a amp fom 0 to 90 RPM] in 0.5 s] and the load toque is fixed at the nominal value 69.5 N m]. 2. Vaiations of the load toque, as indicated in Fig. 7(b). 3. Vaiations of the speed efeence, as shown in Fig. 7(a). 4. Vaiation of the moto esistances, as shown in Fig. 7(c) and 7(d). 5. Vaiation of the load paametes, as indicated in Fig. 7(e) and 7(f). 6. Changes in the contolle paametes (P and I ) of the contol loops.

9 68 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 5. Taditional cuent egulated PWM induction moto dive with PI contolles. Fig. 6. Poposed contol scheme with field oiented block. In all the simulation esults fo the poposed contolles shown in what follows, the initial conditions of all the contolle paametes and adaptive gains wee set equal to zeo, that is to say, θ ik (0) = γ ih (0) = 0, fo i =, 2 and h =, 4. Figs. 8(a), 8(b) and 8(c) show the esults obtained fo the thee contolles unde nomal conditions (i.e. accoding to Situation ). In Fig. 8(a) it can be seen how the taditional contol based on linea contolles exhibits a steady state velocity eo tending to 0.75%. The tansient esponse is slow and pesents initial oscillations. We can obseve how the machine is magnetized, with the flow expessed in volts. The toque duing the amp is lage to compensate fo the inetia. Something simila happens with the phase cuent. The othe phase voltages and cuents pesent a wavefom simila to that shown in Fig. 8(a), although shifted by 20. It is obseved fom Fig. 8(b) that the contolle CFAG pesents a bette tansient esponse afte 0.5 s] and with a moe accuate stationay state (with a velocity eo less than 0.5%) than the taditional case. In Fig. 8(c) we can see that the CTVAG is as accuate as the CFAG. Nevetheless, a bette tansient esponse is obseved, being as fast as the CFAG case but less oscillatoy. We see in Figs. 9(a), 9(b) and 9(c) how the diffeent schemes behave unde vaiations of the load toque, as descibed in Fig. 7(b). Fom Fig. 9(a) it can be seen that fo lowe load toque geate speed accuacy is obtained, tending to zeo eo fo the case when thee is no load toque. The eo in steady state is about % fo a nominal load and 0.45 % fo half a nominal load. In the case of the CFAGshown in Fig. 9(b), the eo values ae smalle than fo the taditional case, being 0.5% fo a nominal load and 0.22% fo half a nominal load. We obseve how this contolle is less affected unde abupt vaiations of load toque. The contolle CTVAG pesents a simila esponse to CFAG, but in all the cases the tansient esponse is of bette quality. This can be concluded fom Fig. 9(c). We see now in Figs. 0(a), 0(b) and 0(c) the behavio unde vaiations of the speed efeence at nominal load, accoding to the vaiations indicated in Fig. 7(a). In the case of the taditional contolle (Fig. 0(a)) the eo in stationay state

10 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 7. Paamete and efeence vaiations used in the set of compaative tests. Fig. 8(a). Results of the taditional contol scheme without consideing paamete vaiations, unde a speed amp. inceases fo lowe speed. The eo is about % at nominal speed and appoximately double fo half of nominal speed. The esults fo poposed contolles CFAG and CTVAG ae much bette than the taditional ones, getting smalle eos, wheeas the est of the vaiables have a suitable behavio, simila to the one found fo the taditional contolle (see Figs. 0(b) and 0(c)). In these cases we have an eo of about 0.5% fo nominal speed and appoximately.% at half of the nominal speed. When analyzing Situation 4 (Fig. 7(c) and (d)) all of the contolles pesent a good behavio unde changes of the stato esistance (see Figs. (a) (c)). Nevetheless, unde changes

11 70 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 8(b). Results using CFAG, without consideing paamete vaiations, unde a speed amp. Fig. 8(c). Results using CTVAG, without consideing vaiations of paametes, unde a speed amp. of the oto esistance the coect field oientation is lost and the speed esponse is consideably affected. Notice how the flow of the machine diminishes consideably. Fo the poposed contolles shown in Figs. (b) and (c), the esponses in all the cases ae much moe obust than the taditional contolle (Fig. (a)). Both poposed contolles pesent lesse speed eos in the steady state than the classical scheme. Consideing now the vaiations of the load paametes accoding to Situation 5 (Fig. 7(e) and (f)), none of the thee contolles unde study was substantially affected, as shown in Figs. 2(a) 2(c).

12 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 9(a). Results of the taditional contolle unde vaiations of load toque. Fig. 9(b). Results using CFAG, unde vaiations of the load toque. Fo the poposed contolles, diffeences found in the geneal behavio still emain. CFAG pesents a lesse eo in the steady state than the taditional contolle. CTVAG exhibits a simila esponse to that of the CFAG but with a bette tansient behavio. In Fig. 3(a) the esults coesponding to the taditional contolle ae shown fo vaiation of the popotional constant (P) in the speed loop, fom a value of 30 accoding to 2] to a value of P = 20. That is, fo a vaiation of 33.3%. The est of the flux and cuent contolle paametes and the

13 72 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 9(c). Results using CTVAG, unde vaiations of the load toque. Fig. 0(a). Results of the taditional contol unde vaiations in the speed efeence. value of the integal constant (I ) of the speed loop wee not modified. It impotant to point out that with the vaiation of only one contolle paamete the contol scheme does not pefom popely. That is why it was not necessay to exploe the influence of othe paamete vaiations. In the case of the poposed contolles, the popotional gains of all the contol loops wee changed while the integal

14 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 0(b). Results using CFAG unde vaiations in the speed efeence. Fig. 0(c). Results using CTVAG unde vaiations in the speed efeence. constant was kept fixed at the value 0. Fo both contolles CFAG and CTVAG, vaiations fo the speed contol paametes of 37.5% wee applied (P fom 80 to 50). The flux contol paamete was vaied by 3% ( P fom 69 to 60). The popotional gains in the cuent loops wee vaied by 33.3% (P fom 30 to 20). Fom Figs. 3(b) and 3(c) it can be seen that in spite of these simultaneous vaiations the speed eo continues to be less than % and the tansient esponse afte 0.5 s] was

15 74 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. (a). Results of the taditional contol scheme unde vaiations of moto paametes. Fig. (b). Results using CFAG, fo vaiations of moto paametes. not affected. CFAG as well as CTVAG guaantee good esults fo wide anges of vaiations of the popotional gains. With the poposed contol schemes wide paamete vaiations can be handled and a lage numbe of paametes can be changed without an impotant deteioation of the oveall system behavio. 6. Conclusions A pinciple called the Pinciple of Toque Flux Contol has been poposed in this pape; it is applied to the adaptive contolles based on passivity given in 9,0]. As a esult, two simplified SISO contol schemes have been obtained without state estimation (components of the oto flux) and without

16 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. (c). Results using CTVAG, fo vaiations of moto paametes. Fig. 2(a). Results of the taditional contol, fo vaiations of the load paametes. paamete estimation (paametes of the moto load set) except those used fo the field oientation block. Thei behavios wee studied and compaed with that of a classical contol scheme 2], and they wee found suitable and obust. Compaed with othe contol schemes poposed in the liteatue such as those based on sliding modes 3 5], atificial intelligence 6,7] and non-adaptive passivity 5,8], we have pesented two simple and novel SISO contolles. They have adaptive chaacteistics, they ae obust in the pesence of load paamete vaiations and they use simple popotional

17 76 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 2(b). Results of CFAG unde vaiations of load paametes. Fig. 2(c). Results of CTVAG unde vaiations of load paametes. contolles in the oto speed, oto flux and stato cuent contol loops. They ae also obust fo a lage ange of popotional gain vaiations. All of the above mentioned chaacteistics fo the poposed contol schemes guaantee high pefomance contol, such as high stating toque at low speed and duing the tansient, accuacy in steady state, wide ange of speed contol and good esponse unde speed changes. It was shown that the contolle CTVAG pesents bette tansient esponse than the contolle CFAG, due to the time-

18 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 3(a). Results fo the taditional contolle, unde changes in the tuning of the popotional gain of the speed loop contol. Fig. 3(b). Results using CFAG unde changes in the tuning of the popotional gains. vaying adaptive gains included. Besides, it has some additional adjustment possibilities such as adjustment of the initial values of the contolle paametes plus the initial values of timevaying adaptive gains. Acknowledgements The esults epoted in this pape have been obtained with the suppot of CONICYT-CHILE though gant FONDECYT N

19 78 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Fig. 3(c). Results using CTVAG unde changes in the tuning of the popotional gains. The suppot of Viceectoía de Investigación y Desaollo, Univesidad de Santiago de Chile (USACH), is geatly appeciated by the fist autho. Refeences ] Loenz RD, Lipo TA, Novotny DW. Motion contol with induction motos. Poceedings of the IEEE 994;82(8): ] De Luca A, Ulivi G. Design of an exact nonlinea contolle fo induction moto. IEEE Tansactions on Automatic Contol 989;34: ] Chan CC, Wang HQ. New scheme of sliding-mode contol fo high pefomance induction moto dives. IEE Poceedings on Electical Powe Applications 996;43: ] Dunnigan MW, Wade S, Williams BW, Yu X. Position contol of a vecto contolled induction machine using Slotine s sliding mode contol appoach. IEE Poceedings on Electical Powe Applications 998;45: ] Taoutaou D, Casto-Linaes R. A contolle obseve scheme fo induction motos based on passivity feedback equivalence and sliding modes. Intenational Jounal of Adaptive Contol and Signal Pocessing 2000;4(2 3): ] Bose BK, edito. Powe electonics and vaiable fequency dives, Technology and applications. IEEE PRESS; ] Vas P. Atificial-intelligence-based electical machines and dives. Oxfod Univesity Pess; ] Espinosa G, Otega R. An output feedback globally stable contolle fo induction motos. IEEE Tansaction on Automatic Contol 995;40: ] Casto-Linaes R, Duate-Memoud MA. Passivity equivalence of a class of SISO nonlinea systems via adaptive feedback. In: Poceedings of VIII Latinameican congess on automatic contol, vol p ] Duate-Memoud MA, Casto-Linaes R, Castillo-Facuse A. Adaptive passivity of nonlinea systems using time-vaying gains. Dynamics and Contol 200;(4): ] Vas P. Sensoless vecto and diect toque contol. Oxfod Univesity Pess; ] Chee-Mun O. Dynamic simulation of electic machiney, using Matlab/Simulink. Pentice Hall PTR; ] Bynes CI, Isidoi A, Willems JC. Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinea systems. IEEE Tansaction on Automatic Contol 99;36(): ] Duate-Memoud MA, Méndez-Miquel JM, Casto-Linaes R, Castillo- Facuse A. Adaptive passivation with time-vaying gains of MIMO nonlinea systems. Kybenetes 2003;32(9 0): ] Duate-Memoud MA, Casto-Linaes R, Castillo-Facuse A. Diect passivity of a class of MIMO nonlinea systems using adaptive feedback. Intenational Jounal of Contol 2002;75(): ] Duate-Memoud MA, Tavieso JC. Contol of induction motos: An adaptive passivity MIMO pespective. Intenational Jounal of Adaptive Contol and Signal Pocessing 2003;7(4): ] Isidoi A. Nonlinea contol systems. 3d ed. NY: Spinge-Velag; ] Nijmeije H, Van de Schaft A. Nonlinea dynamical contol systems. NY: Spinge-Velag; 996. Juan C. Tavieso-Toes was bon in 973, eceiving the degees of Electical Enginee and M.Sc. fom the Supeio Polytechnic Institute José Antonio Echeveía at Havana, Cuba, in 995 and 2000 espectively; and the Ph.D. degee at the Univesity of Santiago of Chile in He has moe than eleven yeas of pofessional expeience woking fo diffeent companies in Cuba and Chile, and moe than seven yeas of teaching and eseach expeience at the Electical Engineeing Depatments of the Univesity of Santiago of Chile, and at the Univesity of Chile. Since he woked at Fluo Copoation in Chile, whee he was consideed a global expet in the subjects of Digital Contol Systems (DCS/PLC), Pocess Contol (Contol Stategies), and Vaiable Speed Dives, which ae his main eseach aeas. Today he acts as Consultant fo the Mining Company Doña Inés de Collahuasi.

20 J.C. Tavieso-Toes, M.A. Duate-Memoud / ISA Tansactions 47 (2008) Manuel A. Duate-Memoud eceived the degee of Civil Electical Enginee fom the Univesity of Chile in 977 and the M.Sc., M.Phil. and the Ph.D. degees, all in electical engineeing, fom Yale Univesity in 985, 986 and 988 espectively. Fom 977 to 979, he woked as Field Enginee at Santiago Subway. In 979 he joined the Electical Engineeing Depatment of Univesity of Chile, whee he is cuently Pofesso. His main eseach inteests ae in obust adaptive contol (linea and nonlinea systems) and system identification. D. Duate is membe of the IEEE and IFAC. He is past Teasue and past Pesident of ACCA, the Chilean National Membe Oganization of IFAC, and past Vice-Pesident of the IEEE-Chile.