Nonlinear H-infinity Feedback Control for Asynchronous Motors of Electric Trains

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1 Intell In Syst (25) :85 98 DOI.7/s y ORIGINAL PAPER Nonlinear H-infinity Feeback Control for Asynchronous Motors of Electric Trains G. Rigatos P. Siano 2 P. Wira 3 F. Profumo 4 Receive: December 24 / Revise: 3 March 25 / Accepte: 3 July 25 / Publishe online: August 25 Springer Science+Business Meia Singapore 25 Abstract A new metho for feeback control of asynchronous electrical machines is introuce, with application example the problem of the traction system of electric trains. The control metho consists of a repetitive solution of an H-infinity control problem for the asynchronous motor, that makes use of a locally linearize moel of the motor an takes place at each iteration of the control algorithm. The asynchronous motor s moel is locally linearize roun its current operating point through the computation of the associate Jacobian matrices. Using the linearize moel of the electrical machine an H-infinity feeback control law is compute. The known robustness features of H-infinity control enable to compensate for the errors of the approximative linearization, as well as to eliminate the effects of external perturbations. The efficiency of the propose control scheme is shown analytically an is confirme through simulation experiments. B G. Rigatos grigat@ieee.org; mechatronicsanai@gmail.com P. Siano psiano@unisa.it P. Wira pwira@uha.fr F. Profumo francesco.profumo@polito.it Unit of Inustrial Automation, Inustrial Systems Institute, 2654 Rion Patras, Greece 2 Department of Inustrial Engineering, University of Salerno, 8484 Fisciano, Italy 3 Laboratoire MIPS, Université e Haute Alsace, 6893 Mulhouse Ceex, France 4 DENERG - Dipartimento Energia, Politecnico i Torino, 29 Turin, Italy Keywors Nonlinear H-infinity feeback control Robust control Asynchronous electric motors Electrical trains Introuction Efficient control of the traction system of electric trains is important for improving their performance inexes (e.g. acceleration, maximum spee, motor s torque) as well as their safety features [ 4]. To this en, in the recent years several research results have been prouce on control of inuction motors, which are frequently use for the traction of electric trains (an particularly of high-spee trains) [5 8]. Inuction motors (IM) have been the most wiely use machines in fixe-spee applications for reasons of cost, size, weight, reliability, ruggeness, simplicity, efficiency, an ease of manufacture. The inuction motor moel is a highly nonlinear one an is characterize by the ifficulty in measuring certain of its state vector elements (e.g. magnetic flux) [9]. With the fiel-oriente metho, the ynamic behavior of the inuction motor is rather similar to that of a separately excite DC motor [,]. A ecouple relationship is obtaine by means of a proper selection of state coorinates an thus, the rotor spee is asympotically ecouple from the rotor flux, while the spee can be controlle only by varying the stator s currents [2,3]. On the other han certain approaches try to evelop control for inuction motors base on feeback linearization of its ynamics [4 8]. The control performance of the inuction motor is influence by the uncertainties of motor s ynamic moel, such as mechanical parameter uncertainty, external loa isturbance, an unmoelle ynamics in practical applications. Thus other approaches try to provie inuction motor control with improve robustness features [9 2]. 23

2 86 Intell In Syst (25) :85 98 In this research article a new control metho for inuction motors is evelope, base on nonlinear H-infinity control theory. The application of an approximate linearization scheme for the ynamic moel of the inuction motor is propose, base on Taylor series expansion roun the motor s present operating point. To perform this linearization the computation of Jacobian matrices is neee while the inuce linearization error terms are treate as isturbances. For the linearize equivalent of the asynchronous motor s moel an H feeback control scheme is evelope. The formulation of the H control problem is base on the minimization of a quaratic cost function that comprises both the isturbance an the control input effects. The isturbance tries to maximize the cost function while the control signal tries to minimize it, within a mini-max ifferential game. The efficiency of the propose nonlinear H control scheme has been teste through simulation experiments, which have shown a satisfactory performance. Comparing to nonlinear feeback control approaches which are base on exact feeback linearization of the inuction motor (as the ones base on ifferential flatness theory an analyze in Ref. [22 24]) the propose H control scheme is assesse as follows: (i) it uses an approximate linearization approach of the system s ynamic moel which oes not follow the elaborate transformations (iffeomorphisms) of the exact linearization methos, (ii) it introuces aitional isturbance error which is ue to the approximative linearization of the system ynamics coming from the application of Taylor series expansion [25 27], (iii) it requires the computation of Jacobian matrices, which in the case of the sixth-orer asynchronous motor moel can be also a cumbersome proceure, (iv) unlike exact feeback linearization, the H control term has to compensate not only for moelling uncertainties an external isturbances but nees also to annihilate the effects of the cumulative linearization error, (v) the H control approach follows an optimal control metho for the computation of the control signal, however unlike exact feeback linearization control it requires the solution of Riccati equations which for the sixth-orer inuction motor s moel can be again a cumbersome proceure. The structure of the paper is as follows: in Mathematical Moel of the Inuction Motor section, the ynamic moel of the inuction motor is analyze. In Fiel Oriente Control section, an overview of fiel-oriente control of inuction motors an the associate ynamic moel in the q reference frame are given. In Linearization of the Inuction Motor s Dynamic Moel section, linearization of the inuction motor s moel is performe roun local operating points an through the computation of Jacobian matrices. In The Nonlinear H-infinity Control section, the nonlinear H feeback control law is formulate. In Lyapunov Stability Analysis section, Lyapunov stability analysis is provie for the control loop of the asynchronous motor. In Robust state estimation with the use of the H Kalman Filter section the problem of robust estimation for the inuction motor is treate with the use of the H-infinity Kalman Filter. In Simulation Tests section, the performance of the propose control scheme is teste through simulation experiments. Finally, in Conclusions section, concluing remarks are state. Mathematical Moel of the Inuction Motor To erive the ynamic moel of an inuction motor the threephase variables are first transforme to two-phase ones [4 7]. This two-phase system can be escribe in the statorcoorinates frame α b, an the associate voltages are enote as v s α an v s b, while the currents of the stator are i s α an i s b, respectively (see Fig. ). Then, the rotation angle of the rotor with respect to the stator is enote by δ. Next, the rotating reference frame q on rotor, is efine. The currents of the rotor are ecompose into q coorinates, thus resulting into i r an i rq. Since the frame q of the rotor aligns with the frame α b of the stator after rotation by an angle δ it hols that ( irα i rb ) ( cos(δ) sin(δ) = sin(δ) cos(δ) )( ir The voltage evelope along frame α of the stator is given by R s i s α + ψ s α t i rq ) () = v s α (2) Fig. AC motor circuit, with the a b stator reference frame an the q rotor reference frame 23

3 Intell In Syst (25) : where the magnetic flux ψ s α is the result of the magnetic flux that is generate by current i s α of the stator (self-inuctance) an of the magnetic flux which is generate by current i r α of the rotor (mutual inuctance), i.e. ψ s α = L s i s α + Mi r α (3) The voltage evelope along frame b of the stator is R s i s b + t ψ s b = u s b (4) where the magnetic flux ψ s b is the result of the magnetic flux that is generate by currrent i s b of the stator (self-inuctance) an of the magnetic flux which is generate by current i r b of the rotor (mutual inuctance), i.e. ψ s b = L s i s b + Mi r b (5) Similarly the voltage along frames an q of the rotor is calculate as follows R r i r + t ψ r = (6) R r i rq + t ψ r q = (7) After intermeiate computations the equations of the inuction motor are foun to be: R s i s α + M ) L r t ψ r α + (L s M2 L r t i s α = v s α (8) R s i s b + M ) L r t ψ r b + (L s M2 t i s b = v s b (9) R r ψ r L α R r M i s α + r L r t ψ r α + n p ωψ r b = () R r ψ r L b R r M i s b + r L r t ψ r b + n p ωψ r α = () The torque that is applie to the rotor is evelope accoring to the principle of energy preservation an is given by T = n p M L r (ψ r α i s b ψ r b i s α ) (2) If the motor has to move a loa of torque T L it hols L r J ω = T T L ω = T J T L J ω = n p M (ψ r JL α i s b ψ r b i s α ) T L r J (3) Denoting σ = M2 L s L r, the equations of the inuction motor are finally written as: θ = ω (4) ω t ψ r α t ψ r b t = n p M (ψ r JL α i s b ψ r b i s α ) T L r J t i s α = (5) = R L L r ψ r α n p ωψ r b + R r L r Mi sα (6) = R L L r ψ r b + n p ωψ r α + R r L r Mi sb (7) MR r Lr 2 ψ r α + n p M ωψ r L b r ( M 2 R r + Lr 2 R ) s Lr 2 i s α + v s α (8) ωψ r L α + MR r r Lr 2 ψ r b ( M 2 R r + Lr 2 R ) s Lr 2 i s b + v s b (9) t i s b = n p M Therefore one can efine the state vector x =[θ,ω,ψ r α, ψ r b, i s α, i s b ] T. Uncertainty can be associate with the value of the loa torque T L, or the value of the components of the electric circuits of the stator an the rotor. The following parameters are also efine: α = R r L r, β = σ L M s L r, γ = ( ) M 2 R r + R Lr 2 s, μ = n p M JL r. Therefore, the ynamic moel of the inuction motor can be written as: ẋ = f (x) + g α u s α + g b u s b (2) In state equations form, the ynamic moel of the motor can be written as x 2 μ ( x 6 x 4 x 5 ) T L J α n p x 2 x 4 + α Mx 5 f (x) = n p x 2 α x 4 + α Mx 6 α β + n p β x 2 x 4 γ x 5 n p β x 2 + α β x 4 γ x 6 [ ] T g α =,,,, g b = [,,,,,, Fiel Oriente Control (2) ] T (22) The classical metho for inuction motors control was introuce by Blaschke (97) an is base on a transformation of the stator s currents (i s α ) an (i s b ) an of the magnetic fluxes of the rotor (ψ r α an ψ r b ) to the reference frame q which rotates together with the rotor [4 7]. Thus the controller s esign uses the currents i s an i sq an the fluxes ψ r an 23

4 88 Intell In Syst (25) :85 98 ψ r q. The angle of the vectors that escribe the magnetic fluxes ψ r α an ψ r b is first efine, i.e. ( ) ρ = tan ψrb (23) ψ ra The angle between the inertial reference frame of the stator an the rotating reference frame of the rotor is taken to be equal to ρ. The transition from (i s α, i s b ) to (i s, i sq ) is given by (is ) ( )( ) cos(ρ) sin(ρ) is = α (24) sin(ρ) cos(ρ) i sq The transition from (ψ r α,ψ r b ) to (ψ r,ψ r q ) is given by ( ) ( )( ) ψr cos(ρ) sin(ρ) ψr = α (25) sin(ρ) cos(ρ) ψ r q i s b ψ r b Moreover, it hols that cos(ρ) = ψ ra ψ, sin(ρ) = ψ r b ψ, an ψ = ψr 2 α + ψr 2 b. Using the above transformation ones obtains i s = ψ r αi s α + ψ r b i s b ψ i sq = ψ r αi s b ψ r b i s α ψ ψ r = ψ ψ r q = (26) Therefore, in the rotating frame q of the motor there will be only one non-zero component of the magnetic flux ψ r, while the component of the flux along the axis equals. The new inputs of the system are consiere to be v s,v sq, which are connecte to v sa,v s b accoring to the relation ( vs α v s b ) ( ) ( ) ψra ψ = ψ rb vs ψ rb ψ ra v sq (27) In the new coorinates the inuction motor moel is written as: t θ = ω (28) t ω = μψ r i sq T L J (29) t ψ r = αψ r + αmi s (3) t i s = γ i s + αβψ r + n p ωi sq + αmi sq 2 + v s (3) ψ r t i sq = γi sq βn p ωψ r n p ωi s αmi sqi s ψ + v sq (32) t ρ = n pω + αmi sq ψ r (33) Defining the state vector of the motor ynamics in the q reference frame as x =[θ,ω,ψ r, i s, i sq,ρ] the associate state-space moel becomes ẋ = f (x)+g v s +g q v sq, where x 2 μ x 5 T L J α + αmx 4 f (x) = n p x 2 + αmx 5 [ g =,,,,, g q = γ x 4 + αβ + n p x 2 x 5 + αmx2 5 γ x 5 βn p x 2 n p x 2 x 4 αmx 4x 5 [,,,,, ] T (34) ] T (35) Next, the following nonlinear feeback control law is efine ( vs v sq ) = n pωi sq αmi 2 s q ψ r αbψ r + v n p ωi s + bn p ωψ r + αmi s qi s ψ r + v q (36) The terms in Eq. (36) have been selecte so as to linearize Eqs. (3) an (32) an to prouce first-orer linear ODE. The control signal in the inertial coorinates system a b will be ( vs α v s b ) ( ψs = ψ σ L α s ψ s b ψ s b ψ s α ) n pωi sq αmi 2 s q ψ r αβψ r + v n p ωi s + βn p ωψ r + αmi s qi s ψ r + v q (37) Substituting Eq. (36) into Eqs. (3) an (32) one obtains [6]: θ = ω (38) t ω = μψ r i sq T L J (39) t i sq = γi sq + v q (4) t ψ r = αψ r + αmi s (4) t i s = γi s + v (42) t ρ = n pω + αm i sq ψ r (43) The system of Eqs. (39) to(43) consists of two linear subsystems, where the first one has as output the magnetic flux ψ r an the secon has as output the rotation spee ω, i.e. 23

5 Intell In Syst (25) : t ψ r = αψ r + αmi s (44) t i s = γi s + v (45) t ω = μψ r i sq T L J (46) t i sq = γi sq + v q (47) If ψ r ψ r ref, i.e. the transient phenomena for ψ r have been eliminate an therefore ψ r has converge to a steay state value, then the two subsystems escribe by Eqs. (44) (45) an (46) (47) are ecouple. The subsystem that is escribe by Eqs. (44) an (45) is linear with control input v s, an can be controlle using methos of linear control, such as optimal control, or PID control. For instance the following PI controller has been propose for the control of the magnetic flux ref v (t) = k (ψ r ψ r ) t k 2 (ψ r (τ) ψ ref r (τ)τ (48) If Eq. (48) is applie to the subsystem that is escribe by Eqs. (44) an (45), then one can succee ψ r (t) ψ ref r (t). If ψ r (t) is not sufficiently measurable using Hall sensors then it can be reconstructe using some kin of observer or Kalman Filtering. Now, the subsystem that consists of Eqs. (46) an (47) is examine. The term T = μψ ref r i sq enotes the torque evelope by the motor. The above mentione subsystem is a moel equivalent to that of a DC motor an thus after succeeing ψ r ψ ref r, one can also control the motor s spee ω, using control algorithms alreay applie to the control of DC motors. A first approach to the control of the spee ω is to use neste PI loops, i.e. t v q = K q (T T ref ) K q 2 (T (t) T ref (t))τ t T ref = K q 3 (ω ω ref ) K q 4 (ω(t) ω ref (t))τ (49) From the above it can be seen that fiel oriente (vector control) for inuction motors requires the tuning of the several PID-type controllers an this limits the metho s reliability only roun local operating points. Consequently, the stability an robustness properties of the fiel-oriente control for asynchronous motors are oubtful. More efficient control approaches, of proven stability, have to be searche for. This problem will be solve in the following sections. Linearization of the Inuction Motor s Dynamic Moel As shown in Fiel Oriente Control section, the nonlinear state space equation of the inuction motor, expresse in the q reference frame, is given by ẋ = f (x) + g v s + g q v sq (5) where the state vector has been efine as x =[θ,ω,ψ r, i s, i sq,ρ] while functions f (x), g a (x) an g b (x) have been efine as x 2 μ x 5 T L J α + αmx 4 f (x) = g = g q = γ x 4 + αβ + n p x 2 x 5 + αmx2 5 γ x 5 βn p x 2 n p x 2 x 4 αmx 4x 5 n p x 2 + αmx 5 [,,,,, ] T (5) [ ] T,,,,, (52) Then, the Jacobian matrix of the vector fiel f (x) is: μx 5 μ α αm A = J φ = n p x 5 αβ αmx2 5 γ n x3 2 p x 2 + 2αMx 5 βn p n p x 4 βn p x 2 + αmx 4x 5 n x3 2 p x 2 αmx 5 γ αmx 4 n p αmx 5 αm x3 2 x3 (53) 23

6 9 Intell In Syst (25) :85 98 Moreover, linearization of the motor s ynamics with respect to the control input variables u, u 2 gives the Jacobian matrix B =[J ga J gb ]= (54) Thus, after linearization roun its current operating point, the inuction motor s ynamic moel is written as ẋ = Ax + Bu + (55) Parameter stans for the linearization error in the inuction motor s ynamic moel appearing in Eq. (55). The reference setpoints for the asynchronous motor are enote by x =[x,...,x 6 ]. Tracking of this trajectory is succeee after applying the control input u. At every time instant the control input u is assume to iffer from the control input u appearing in Eq. (55) by an amount equal to u, that is u = u + u ẋ = Ax + Bu + 2 (56) The ynamics of the controlle system escribe in Eq. (55) can be also written as ẋ = Ax + Bu + Bu Bu + (57) an by enoting 3 = Bu + as an aggregate isturbance term one obtains ẋ = Ax + Bu + Bu + 3 (58) By subtracting Eq. (56) from Eq. (58) one has ẋ ẋ = A(x x ) + Bu (59) By enoting the tracking error as e = x x an the aggregate isturbance term as = 3 2, the tracking error ynamics becomes ė = Ae + Bu + (6) The above linearize form of the inuction motor s moel can be efficiently controlle after applying an H-infinity feeback control scheme. The Nonlinear H-infinity Control Mini-max Control an Disturbance Rejection The initial nonlinear moel of the inuction motor is in the form ẋ = f (x, u) x R n, u R m (6) Linearization of the system (asynchronous motor) is performe at each iteration of the control algorithm roun its present operating point (x, u ) = (x(t), u(t T s )). The linearize equivalent of the system is escribe by ẋ = Ax + Bu + L x R n, u R m, R q (62) where matrices A an B are obtaine from the computation of the Jacobians f f f x x 2 x n f 2 f 2 f x x 2 2 x n A = (x,u ) (63) f n x 2 f n x f n u f n x n f f f u u 2 u m f 2 f 2 f u u 2 2 u m B = (x,u ) (64) f n u 2 f n u m an vector enotes isturbance terms ue to linearization errors. The problem of isturbance rejection for the linearize moel that is escribe by ẋ = Ax + Bu + L y = Cx (65) where x R n, u R m, R q an y R p, cannot be hanle efficiently if the classical LQR control scheme is applie. This is because of the existence of the perturbation term. The isturbance term apart from moeling (parametric) uncertainty an external perturbation terms can also represent noise terms of any istribution. In the H control approach, a feeback control scheme is esigne for trajectory tracking by the system s state vector an simultaneous isturbance rejection, consiering that the isturbance affects the system in the worst possible manner. The isturbances effect are incorporate in the following quaratic cost function: 23

7 Intell In Syst (25) : T J(t) = [y T (t)y(t) 2 + ru T (t)u(t) ρ 2 T (t) (t)]t, r,ρ > (66) The significance of the negative sign in the cost function s term that is associate with the perturbation variable (t) is that the isturbance tries to maximize the cost function J(t) while the control signal u(t) tries to mininize it. The physical meaning of the relation given above is that the control signal an the isturbances compete to each other within a mini-max ifferential game. This problem of mini-max optimization can be written as min u max J(u, ) (67) The objective of the optimization proceure is to compute a control signal u(t) which can compensate for the worst possible isturbance, that is externally impose to the system. However, the solution to the mini-max optimization problem is irectly relate to the value of the parameter ρ. This means that there is an upper boun in the isturbances magnitue that can be annihilate by the control signal. H-infinity Feeback Control For the linearize system given by Eq. (65) the cost function of Eq. (66) is efine, where the coefficient r etermines the penalization of the control input an the weight coefficient ρ etermines the rewar of the isturbances effects. It is assume that: It is assume that (i) The energy that is transferre from the isturbances signal (t) is boune, that is T (t) (t)t <, (ii) the matrices [A, B] an [A, L] are stabilizable, (iii) the matrix [A, C] is etectable. Then, the optimal feeback control law is given by u(t) = Kx(t) (68) with K = r BT P (69) where P is a positive semi-efinite symmetric matrix which is obtaine from the solution of the Riccati equation ( A T P + PA+ Q P r BBT ) 2ρ 2 LLT P = (7) where Q is also a positive efinite symmetric matrix. The worst case isturbance is given by (t) = ρ 2 L T Px(t) (7) The iagram of the consiere control loop is epicte in Fig. 2. Fig. 2 Diagram of the control scheme for the train s inuction motor 23

8 92 Intell In Syst (25) :85 98 The Role of Riccati Equation Coefficients in H Control Robustness The parameter ρ in Eq. (66), is an inication of the closeloop system robustness. If the values of ρ>are excessively ecrease with respect to r, then the solution of the Riccati equation is no longer a positive efinite matrix. Consequently there is a lower boun ρ min of ρ for which the H control problem has a solution. The acceptable values of ρ lie in the interval [ρ min, ). Ifρ min is foun an use in the esign of the H controller, then the close-loop system will have increase robustness. Unlike this, if a value ρ > ρ min is use, then an amissible stabilizing H controller will be erive but it will be a suboptimal one. The Hamiltonian matrix H = ( A Q ( ) ) r BB T LL T ρ 2 A T (72) provies a criterion for the existence of a solution of the Riccati equation Eq. (7). A necessary conition for the solution of the algebraic Riccati equation to be a positive semi-efinite symmetric matrix is that H has no imaginary eigenvalues [22]. Lyapunov Stability Analysis Through Lyapunov stability analysis it will be shown that the propose nonlinear control scheme assures H tracking performance for the inuction motor, an that in case of boune isturbance terms asymptotic convergence to the reference setpoints is succeee. The tracking error ynamics for the asynchronous motor is written in the form ė = Ae + Bu + L (73) where in the inuction machine s case L = I R 2 with I being the ientity matrix. Variable enotes moel uncertainties an external isturbances of the motor s moel. The following Lyapunov equation is consiere V = 2 et Pe (74) where e = x x is the tracking error. By ifferentiating with respect to time one obtains V = 2 ėt Pe + 2 epė V = 2 [Ae+ Bu+L ] T P + 2 et P[Ae+ Bu+L ] (75) V = 2 [et A T + u T B T + T L T ]Pe + 2 et P[Ae + Bu + L ] (76) V = 2 et A T Pe + 2 ut B T Pe + 2 T L T Pe + 2 et PAe+ 2 et PBu+ 2 et PL (77) The previous equation is rewritten as V = ( 2 et (A T P + PA)e + 2 ut B T Pe + ) 2 et PBu ( + 2 T L T Pe + ) 2 et PL (78) Assumption For given positive efinite matrix Q an coefficients r an ρ there exists a positive efinite matrix P, which is the solution of the following matrix equation ( A T P + PA= Q + P r BBT ) ρ 2 LLT P (79) Moreover, the following feeback control law is applie to the system u = r BT Pe (8) By substituting Eqs. (79) an (8) one obtains V = ( [ Q 2 et + P r BBT ) ] 2ρ 2 LLT P e ( + e T PB ) r BT Pe + e T PL (8) V = ( 2 et Qe + r PBBT Pe ) 2ρ 2 et PLL T Pe r et PBB T Pe) + e T PL (82) which after intermeiate operations gives V = 2 et Qe 2ρ 2 et PLL T Pe + e T PL (83) or, equivalently V = 2 et Qe 2ρ 2 et PLL T Pe + 2 et PL + 2 T L T Pe (84) Lemma The following inequality hols 2 et L + 2 L T Pe 2ρ 2 et PLL T Pe 2 ρ2 T (85) 23

9 Intell In Syst (25) : Proof The binomial (ρα ρ b)2 is consiere. Expaning the left part of the above inequality one gets ρ 2 a 2 + ρ 2 b2 2ab 2 ρ2 a 2 + 2ρ 2 b2 ab ab 2ρ 2 b2 2 ρ2 a 2 2 ab + 2 ab 2ρ 2 b2 2 ρ2 a 2 (86) The following substitutions are carrie out: a = an b = e T PL an the previous relation becomes 2 T L T Pe + 2 et PL 2ρ 2 et PLL T Pe 2 ρ2 T (87) Equation (87) is substitute in Eq. (84) an the inequality is enforce, thus giving V 2 et Qe + 2 ρ2 T (88) Equation (88) shows that the H tracking performance criterion is satisfie. The integration of V from to T gives T V (t)t 2 2V (T ) + T T e 2 Q t + T 2 ρ2 2 t T e 2 Q t 2V () + ρ2 2 t (89) Moreover, if there exists a positive constant M > such that 2 t M (9) then one gets e 2 Q t 2V () + ρ2 M (9) Thus, the integral e 2 Qt is boune. Moreover, V (T ) is boune an from the efinition of the Lyapunov function V in Eq. (74) it becomes clear that e(t) will be also boune since e(t) e ={e e T Pe 2V () + ρ 2 M }. Accoring to the above an with the use of Barbalat s Lemma one obtains lim t e(t) =. Robust State Estimation with the Use of the H Kalman Filter A Kalman Filter for the linearize moel of the inuction motor that is given in Eq. (55) can be esigne to cope with the case of maximum errors of some linear combination of states for worst case assumptions of process noise, measurement noise an isturbances. This can be useful in state estimation for the inuction motor, as a metho for moel uncertainty compensation. Filters esigne to minimize a weighte norm of state errors are calle H or minimax filters [28,29]. The iscrete-time H filter uses the same state-space moel as the Kalman Filter, which has the form x(k + ) = A(k)x(k) + B(k)u(k) + w(k) z(k) = C(k)x(k) + v(k) (92) E[w(k)] =, E[w(k)w(k) T ] = Q(k)δ ij, E[v(k)] =, E[v(k)v(k) T ] = R(k)δ ij an E(w(k)v(k) T ) =. The upate of the state estimate is again given by ˆx(k) =ˆx (k) + K (k)(z(k) C(k) ˆx (k)) (93) that minimizes the trace of the covariance matrix of the state vector estimation error J = 2 E{ x(k)t x(k)} = 2 tr(p (k)) (94) where x (k) = x(k) ˆx (k) an P (k) = E[ x (k) T x (k)]. TheH filtering approach efines first a transformation (k) = L(k)x(k) (95) where L(k) R n n is a full rank matrix. The use of the transformation given in Eq. (95) allows certain combinations of states to be given more weight than others. Next, efining the estimation error variable (i) = (i) ˆ(i), the cost function of the H filter is initially formulate as k J(k) = (i + ) T S(i) (i + )/b i= b = x () T P () x () k + i= k + i= w T (i + )Q(i + ) w(i + ) v T (i)r(i) v(i) (96) where S i is a positive-efinite symmetric weighting matrix. It can be observe that both matrices S(k) an L(k) appear in the cost function an thus affect the solution ˆx (k + ) of the optimization problem. The objective is to fin state vector estimates ˆx (k) an ˆx(k) that keep the cost function below a given value /θ for worst case conitions, i.e. 23

10 94 Intell In Syst (25) :85 98 J(k) < (97) θ By rewriting Eq. (96) an substituting Eq. (92) a moifie cost functional is obtaine k J a (k) = θ x () T P () x () + Ɣ(i) i= Ɣ(i) = (x(i + ) ˆx (i + )) T W i (x(i + ) ˆx (i + )) an θ (wt (i + )Q(i + ) w(i + ) + (y(i) C(i)x (i)) T R(i) (y(i) C(i)x (i))) (98) W (i) = L(i) T S(i)L(i) (99) This cost function oes not inclue the ynamic moel of the system given in Eq. (92) an this is ae by using a vector of Lagrange multipliers λ(i + ). This gives J(k) = θ x () T P () x () k ) λ(i + )T + (Ɣ i +2 ( A(i) ˆx(i)+ B(i)u(i) θ i= + w(i) x(i + )) + 2λ()T θ x() 2λ()T θ x() () The cost function of the filter given in Eq. () can be use as the basis for the solution. It is aime to fin equations efining ˆx (k + ), or equivalently a measurement weighting matrix (similar to the Kalman gain matrix), that minimizes the cost for worst case assumptions about x(), w(i) an y(i). Thus, the optimization objective is formulate as J (k) = min x i max x(),w(i),y(i) J(k) () It is note that the estimation algorithm has knowlege of the output measurement y(i) but no knowlege about the initial conitions of the system x() an the process noise w(i). Uner this assumption, the estimation shoul be able to compensate for worst case values for the unknown parameters. This is a game theoretic problem that is solve in two steps. In the first step of optimization, partial erivatives of J(k) with respect to x(), w(i) an λ(i) are set to zero so as to maximize the cost function of Eq. (), now being epenant only on the terms ˆx (k + ) an y(k) which are inclue in Ɣ i. In the secon step of optimization, the partial erivatives of J(k) with respect to ˆx (k + ) an y(k) are set to zero, to obtain a conition for the filter s gain matrix that minimizes this cost functional. From the optimization conitions J(k)/ x = T, J(k)/ w(i) = T, J(k)/ λ(i) = T ones obtains an expression of J(k) as function of ˆx (k + ) an y(k). Next, from the optimization conitions J(k)/ ˆx (i + ) = T, an J(k)/ y(i) = T one obtains the filter s equations. The recursion of the H Kalman Filter, for the moel of the inuction motor, can be formulate again in terms of a measurement upate an a time upate part: Measurement upate: D(k) =[I θ W (k)p (k) + C T (k)r(k) C(k)P (k)] K (k) = P (k)d(k)c T (k)r(k) (2) ˆx(k) =ˆx (k) + K (k)[y(k) C ˆx (k)] Time upate: ˆx (k + ) = A(k)x(k) + B(k)u(k) P (k + ) = A(k)P (k)d(k)a T (k) + Q(k) (3) where it is assume that parameter θ is sufficiently small to assure that the term P (k) θ W (k) + C T (k)r(k) C(k) will be positive efinite. When θ = theh Kalman Filter becomes equivalent to the stanar Kalman Filter. It is note that apart from the process noise covariance matrix Q(k) an the measurement noise covariance matrix R(k) the H Kalman filter requires tuning of the weight matrices L an S, as well as of parameter θ. Simulation Tests The performance of the propose nonlinear H control scheme for asynchronous motors is teste in tracking of various setpoints. First setpoints were efine inepenently for the rotation spee an the magnetic flux of the rotor. Next, base on these values, setpoints for the stator currents i s an i sq were also compute. As shown in the simulation experiments these setpoints can vary ynamically an even in that case the propose nonlinear H-infinity controller succees the accurate setpoints tracking. As it can be observe in Figs. 3, 4, 5 the feeback control scheme of the inuction motor enable accurate convergence to the reference setpoints. Yet simple, the consiere H control law succeee precise tracking of the reference signals. In comparison to feeback control methos for asynchronous motors which are base on exact linearization, the nonlinear H control requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. The known robustness features of H control are the ones that permit to compensate for the approximation 23

11 Intell In Syst (25) : ω (p.u.) i r (p.u.) ψ s (p.u).5 i rq (p.u.) (a) (b) Fig. 3 Nonlinear H control of the asynchronous motor. a Convergence of the rotor s angular spee ω an stator s magnetic flux ψ s to setpoint. b Convergence of the stator s currents to the reference setpoints ω (p.u.) i r (p.u.) ψ s (p.u).5 i rq (p.u.) (a) (b) Fig. 4 Nonlinear H control of the asynchronous motor. a Convergence of the rotor s angular spee ω an stator s magnetic flux ψ s to setpoint 2. b Convergence of the stator s currents to the reference setpoints errors which were inuce to the linearize moel of the inuction motor. The tracking performance of the control metho is shown in Tables an 2. It can be observe that the tracking error for all state variables of the inuction motor was extremely small. Besies, in the simulation iagrams one can note the excellent transient performance of the control algorithm, which means that convergence to the reference setpoints was succeee in a smooth manner, while also avoiing overshoot an oscillations. Moreover, the performance of the nonlinear H-infinity control scheme was teste in the case of functioning of the asynchronous motor uner isturbances. It was assume that aitive input isturbances affecte the inuction motor. These were escribe by sinusoial voltages of amplitue equal to % of the mean value of the control inputs. The 23

12 96 Intell In Syst (25) :85 98 ω (p.u.) i r (p.u.) ψ s (p.u).5 i rq (p.u.) (a) (b) Fig. 5 Nonlinear H control of the asynchronous motor. a Convergence of the rotor s angular spee ω an stator s magnetic flux ψ s to setpoint 3. b Convergence of the stator s currents to the reference setpoints Table Tracking RMSE without isturbances RMSE ω RMSE ψr RMSE is Setpoint Setpoint Table 2 Tracking RMSE uner isturbances RMSE ω RMSE ψr RMSE is Setpoint Setpoint obtaine results, shown in Table 2, confirm that espite the effects of perturbation inputs the tracking accuracy for the motor s state variables was satisfactory. Finally, the suitability of the H-infinity Kalman Filter for estimating non-measurable state variables of the asynchronous motor is shown if Fig. 6. The measure state variables of the motor where x = θ, x 4 = i r an x 5 = i rq. The estimate state variables, which were finally use in the feeback control loop where ˆx 2 = ˆω an ˆ = ˆψ r. It can be notice that, espite the missing sensory information, accurate tracking of the reference setpoints was succeee. Remark Comparing the propose nonlinear H-infinity control approach against backstepping nonlinear control for inuction motors one shoul take into account that backstepping control is a special case of flatness-base control an actually it is a global linearizing metho [24,3,3]. Thus for the nonlinear backstepping control of inuction motors hol the same remarks which have been state in the introuction of the article, in the comparison between global linearizationbase control methos an nonlinear H-infinity control [32]. Yet, conceptually more simple, nonlinear H-infinity control can perform equally well to global linearization-base control methos. On the other han, it shoul be note that backstepping control is applicable to a limite class of systems, that is systems written in the integral backstepping (triangular) form. Consequently, the propose nonlinear H- infinity control metho is applicable to a wier range of electric machines an traction systems. Remark 2 Comparing the propose nonlinear H-infinity control approach against sliing moe controllers an sliing-moe observers for inuction machines, it can be note that the latter control an estimation approaches exhibit specific rawbacks. First, ue to the use of a switching control term, sliing-moe controllers an sliing-moe observers exhibit chattering which means vibratory ynamics an an unesirable transient performance for the control loop [33]. On the other han, H-infinity control succees smooth variations of the control input an goo transient characteristics for the control loop. Secon, in sliing-moe control it is necessary to know beforehan the uncertainty bounaries for the system s ynamics. Unlike this, in the esign of H-infinity control no such assumption is mae while the suitable selection of the attenuation coefficient ρ appearing in the Riccati Eq. (7) can finally provie the control loop with maximum robustness to moel uncertainty an external perturbations. 23

13 Intell In Syst (25) : ω est (p.u.).6.4 ψ s est (p.u.) t (a) t (b) Fig. 6 Nonlinear H-infinity control of the asynchronous motor through estimation of non-measurable state variables with the use of the H- infinity Kalman Filter. a Estimation (green line)ofstatevariablex 2 = ω (blue line) an convergence to the reference setpoint (re line). b Estimation (green line) of state variable = ψ r (blue line) an convergence to the reference setpoint (re line) Conclusions A new nonlinear feeback control metho has been evelope for inuction motors, base on approximate linearization an the use of H control an stability theory. It has been shown that the propose inuction motor control scheme enables the state vector elements of the electrical machine to track accurately all reference setpoints. The first stage of the propose control metho is the linearization of the motor s ynamic moel using first orer Taylor series expansion an the computation of the associate Jacobian matrices. The errors ue to the approximative linearization have been consiere as isturbances that affect, together with external perturbations, the motor s moel. At a secon stage the implementation of H feeback control has been propose. Using the linearize moel of the inuction motor an H-infinity feeback control law is compute at each iteration of the control algorithm, after previously solving an algebraic Riccati equation. The known robustness features of H-infinity control enable to compensate for the errors of the approximative linearization, as well as to eliminate the effects of external perturbations. The efficiency of the propose control scheme for inuction motors is shown analytically an is confirme through simulation experiments. Comparing to other nonlinear control methos which are base on the exact linearization of the electrical machine s moel it can be state that the propose H control uses the approximately linearize moel of the inuction motor without implementing elaborate state transformations (iffeomorphisms) that finally bring the system to a linear form. Of course the computation of Jacobian matrices an the nee to solve at each iteration of the algorithm a Riccati equation is also a computationally cumbersome proceure, especially for state-space moels of large imensionality. Moreover, this approximate linearization introuces aitional perturbation terms which the H controller has to eliminate. The continuous nee for compensation of such cumulative linearization errors brings the H controller closer to its robustness limits. References. Guzinski, J., Abu-Ru, H., Digues, M., Krzeminski, Z., Lewicki, A.: Spee an loa torque observer application in high-spee train electric rive. IEEE Trans. In. Electr. 57(2), (2) 2. Guzinski, J., Digues, M., Krzeminski, Z., Lewicki, A., Abu-Ru, H.: Application of spee an loa torque observer in high-spee train rive for iagnostic purposes. IEEE Trans. In. Electr. 56(), (29) 3. Shibaya, H., Kono, K.: Designing methos of capacitance an control system for a iesel engine an EDLC hybri powere railway traction system. IEEE Trans. In. Electr. 58(9), (2) 4. Deraa, D., Merbaki, A., Geraa, C., Cavagnino, A., Boglietti, A.: High-spee electrical machines: technologies, trens an evelopments. IEEE Trans. In. Electr. 6(6), (24) 5. Marino, R., Scalzi, S., Tomei, P., Verrelli, C.M.: Fault-toleran cruise control of electric vehicles with inuction motors. Control Eng. Pract. 2, (23). (Elsevier) 6. Ciccarelli, F., Ianuzzi, D., Tricoli, P.: Control of metro-train equippe with on-boar supercapacitors for energy saving an 23

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