Sliding Mode Control Part 5 pot

Sliding mode power control of variable-speed wind energy conversion systems, IEEE Transactions on Energy Conversion 232: 551–558.. High-order sliding mode control of a DFIG-based wind tu

control signals produced by the proposed 2-SMC correspond to continuous voltage direct andquadrature components to be applied to the rotor by means of the RSC, which implies theuse of intermediate SVM modulation Furthermore, providing an additional procedure forbumpless transition between the algorithms devoted to synchronization and power control isindispensable for the case of the 2-SMC, but it is not required for the 1-SMC scheme.

Regarding parameter tuning, only the c constants included in the four switching functions

considered need to be tuned for the case of the 1-SMC It therefore turns out that satisfactoryparameter adjustment is easily achieved by mere trial and error However, in addition to those

c constants, the λ and w gains present in the STAs must also be tuned for the 2-SMC variant.

Even though, as stated in (Bartolini et al., 1999), it is actually the most common practice,trial and error tuning is not particularly effective in this latter case, as it may become highlytime-consuming Therefore, it is believed that there exists a strong need for development ofalternative methods for STA-based 2-SMC tuning

Concerning the switching frequency of the RSC IGBTs, it is fixed at 5 kHz in the case of the2-SMC On the contrary, it turns out to be variable, within the range from 0 to 20 kHz, forthe 1-SMC algorithm This feature complicates the design of both the back-to-back converterfeeding the DFIG rotor and the grid-side AC filter, since broadband harmonics may be injectedinto the grid As a result of the 25-μs sample time selected for the 1-SMC scheme, whichleads to the aforementioned maximum switching frequency of 20 kHz, chatter observable

in stator-side active and reactive powers is somewhat lower than±3% of the DFIG 660-kWrated power Even a lower level of chatter arises from application of the SVM-based 2-SMCalgorithm put forward Furthermore, that chatter, or at least great part of it, is caused by theSVM, not by the 2-SMC algorithm itself

Apart from the superior optimum power curve tracking achieved with both alternative SMCdesigns, the dynamic performance resulting from realization of the proposed 1-SMC scheme

is noticeably better than that to which application of its 2-SMC counterpart leads In effect,focusing on the state in which the DFIG stator is disconnected from the grid, HIL emulationresults demonstrate that synchronization is reached faster by employing the 1-SMC algorithm

On the other hand, the power exchange between the DFIG and the grid taking place at theinitial instants after connection is significantly lower when adopting the 1-SMC algorithm putforward, hence evidencing that its dynamic performance is also better for the stage duringwhich power control is dealt with The excellent dynamic performance reachable by means ofits application supports the 1-SMC approach as a potential candidate for DFIG control undergrid faults, where rapidity of response becomes crucial

The main conclusions drawn from the comparison conducted in this section are summarized

in Table 4

1-SMCALGORITHM 2-SMCALGORITHM

ALGORITHM COMPLEXITY Relatively simple More complex

SWITCHING FREQUENCY Variable from 0 to 20 kHz Fixed at 5 kHz

Table 4 Comparison between the two SMC algorithms put forward

6 References

Abo-Khalil, A G., Lee, D.-C & Lee, S.-H (2006) Grid connection of doubly fed induction

generators in wind energy conversion system, Proceedings of the CES/IEEE 5 th International Power Electronics and Motion Control Conference (IPEMC 2006), Shanghai,

China, vol 3, pp 1–5

Arnaltes, S & Rodríguez, J L (2002) Grid synchronisation of doubly fed induction generators

using direct torque control, Proceedings of the IEEE 28 th Annual Conference of the Industrial Electronics Society (IECON 2002), Seville, Spain, pp 3338–3343.

Åström, K J & Hägglund, T (1995) PID Controllers: Theory, Design and Tuning, Instrument

Society America, USA

Bartolini, G., Ferrara, A., Levant, A & Usai, E (1999) On second order sliding mode

controllers, in K Young & ¨U ¨Ozg ¨uner (eds), Variable Structure Systems, Sliding Mode and Nonlinear Control, Springer Verlag, London, UK, pp 329–350.

Beltran, B., Ahmed-Ali, T & Benbouzid, M E H (2008) Sliding mode power control

of variable-speed wind energy conversion systems, IEEE Transactions on Energy Conversion 23(2): 551–558.

Beltran, B., Ahmed-Ali, T & Benbouzid, M E H (2009) High-order sliding-mode

control of variable-speed wind turbines, IEEE Transactions on Industrial Electronics

56(9): 3314–3321

Beltran, B., Benbouzid, M E H & Ahmed-Ali, T (2009) High-order sliding mode control

of a DFIG-based wind turbine for power maximization and grid fault tolerance,

Proceedings of the IEEE International Electric Machines and Drives Conference (IEMDC 2009), Miami, USA, pp 183–189.

Ben Elghali, S E., Benbouzid, M E H., Ahmed-Ali, T., Charpentier, J F & Mekri, F (2008)

High-order sliding mode control of DFIG-based marine current turbine, Proceedings

of the IEEE 34 th Annual Conference of the Industrial Electronics Society (IECON 2008),

Orlando, USA, pp 1228–1233

Blaabjerg, F., Teodorescu, R., Liserre, M & Timbus, A V (2006) Overview of control and

grid synchronization for distributed power generation systems, IEEE Transactions on Industrial Electronics 53(5): 1398–1409.

Ekanayake, J B., Holdsworth, L., Wu, X & Jenkins, N (2003) Dynamic modeling of

doubly fed induction generator wind turbines, IEEE Transactions on Power Systems

18(2): 803–809

Kuo, B C (1992) Digital Control Systems, Oxford University Press, New York, USA.

Levant, A (1993) Sliding order and sliding accuracy in sliding mode control, International

Journal of Control 58(6): 1247–1263.

Ogata, K (2001) Modern Control Engineering, Prentice Hall, Englewood Cliffs (New Jersey),

USA

Peña, R., Cárdenas, R., Proboste, J., Asher, G & Clare, J (2008) Sensorless control of

doubly-fed induction generators using a rotor-current-based MRAS observer, IEEE Transactions on Industrial Electronics 55(1): 330–339.

Peña, R., Clare, J C & Asher, G M (1996) Doubly fed induction generator using back-to-back

PWM converters and its application to variable-speed wind-energy generation, IEE Proceedings - Electric Power Applications 143(3): 231–241.

Peresada, S., Tilli, A & Tonielli, A (2004) Power control of a doubly fed induction machine

via output feedback, Control Engineering Practice 12(1): 41–57.

Rashed, M., Dunnigan, M W., MacConell, P F A., Stronach, A F & Williams, B W.

(2005) Sensorless second-order sliding-mode speed control of a voltage-fed

induction-motor drive using nonlinear state feedback, IEE Proceedings-Electric Power Applications 152(5): 1127–1136.

Susperregui, A., Tapia, G., Zubia, I & Ostolaza, J X (2010) Sliding-mode control of

doubly-fed generator for optimum power curve tracking, IET Electronics Letters

46(2): 126–127

Tapia, A., Tapia, G., Ostolaza, J X & Sáenz, J R (2003) Modeling and control of a

wind turbine driven doubly fed induction generator, IEEE Transactions on Energy Conversion 12(2): 194–204.

Tapia, G., Santamaría, G., Telleria, M & Susperregui, A (2009) Methodology for smooth

connection of doubly fed induction generators to the grid, IEEE Transactions on Energy Conversion 24(4): 959–971.

Tapia, G., Tapia, A & Ostolaza, J X (2006) Two alternative modeling approaches for the

evaluation of wind farm active and reactive power performances, IEEE Transactions

on Energy Conversion 21(4): 901–920.

Utkin, V., Guldner, J & Shi, J (1999) Siliding Mode Control in Electromechanical Systems, Taylor

& Francis, London, UK

Utkin, V I (1993) Sliding mode control design principles and applications to electric drives,

IEEE Transactions on Industrial Electronics 40(1): 23–36.

Vas, P (1998) Sensorless Vector and Direct Torque Control of AC Machines, Oxford University

Press, New York, USA

Xu, L & Cartwright, P (2006) Direct active and reactive power control of DFIG for wind

energy generation, IEEE Transactions on Energy Conversion 21(3): 750–758.

Yan, W., Hu, J., Utkin, V & Xu, L (2008) Sliding mode pulsewidth modulation, IEEE

Transactions on Power Electronics 23(2): 619–626.

Yan, Z., Jin, C & Utkin, V I (2000) Sensorless sliding-mode control of induction motors, IEEE

Transactions on Industrial Electronics 47(6): 1286–1297.

Zhi, D & Xu, L (2007) Direct power control of DFIG with constant switching frequency

and improved transient performance, IEEE Transactions on Energy Conversion

22(1): 110–118

Sliding Mode Control of Electric Drives

John Cortés-Romero1, Alberto Luviano-Juárez2and

Hebertt Sira-Ramírez3

1Universidad Nacional de Colombia Facultad de Ingeniería, Departamento de Ingeniería

Eléctrica y Electrónica Carrera 30 No 45-03 Bogotá

1,2,3Cinvestav IPN, Av IPN No 2508, Departamento de Ingeniería Eléctrica, Sección de

of using induction motors over permanent magnet DC motors for position control tasks; thus,efforts aimed at improving or simplifying feedback controller design are well justified

There exists a variety of control strategies that depend on difficult to measure motor parameters

while their closed loop behavior is found to be sensitive to their variations Even adaptiveschemes tend to be sensitive to speed-estimation errors, yielding to a poor performance in theflux and torque estimation, especially during low-speed operation, Harnefors & Hinkkanen(2008)

Generally speaking, the designed feedback control strategies have to exhibit a certainrobustness level in order to guarantee an acceptable performance It is possible to (on-line oroff-line) obtain estimates of the motor parameters, Hasan & Husain (2009); Toliyat et al (2003),but some of them can be subject to variation when the system is undergoing actual operation.Frequent misbehavior is due to external and internal disturbances, such as generated heat,that significantly affect some of the system parameter values An alternative to overcomethis situation is to use robust feedback control techniques which take into account thesevariations as unknown disturbance inputs that need to be rejected In this context, slidingmode techniques are a good alternative due to their disturbance rejection capability (see forinstance, Utkin et al (1999))

In this chapter, we consider a two stage control scheme, the first one is devoted to the control ofthe rotor shaft position This analog control is performed by means of the stator current inputs,

in a configuration of an observer based control The mathematical model of the rotor dynamics

is a simplified model including additive, completely unknown, lumping nonlinearities andexternal disturbances whose effect is to be determined in an on-line fashion by means of linearobservers The gathered knowledge will be used in the appropriate canceling of the assumedperturbations themselves while reducing the underlying control problem to a simple linearfeedback control task The control scheme thus requires a rather reduced set of parameters to

be implemented

Sliding Mode Control Design for Induction

Motors: An Input-Output Approach

7

The observation scheme for the modeled perturbation is based on an extension of theGeneralized Proportional Integral (GPI) controller, Fliess, Marquez, Delaleau & Sira-Ramírez(2002) to their dual counterpart: the GPI observer which corresponds to a class of extendedLuenberger-like observers, Luviano-Juárez et al (2010) Such observers were introduced

in, Sira-Ramirez, Feliu-Batlle, Beltran-Carbajal & Blanco-Ortega (2008) in the context ofSigma-Delta modulation observer tasks for the detection of obstacles in flexible robotics.Under reasonable assumptions, the observation technique consists in viewing the measuredoutput of the plant as generated by an equivalent perturbed pure integration dynamics with

an additive perturbation input lumping, in a single function, all the nonlinearities of theoutput dynamics The linear GPI observer, is set to approximately estimate the states of thepure integration system as well as the evolution of the, state dependent, perturbation input.This observer allows one to approximately estimate, on the basis of the measured output, thestates of the nonlinear system, as well as to closely estimate the unknown perturbation input.The proposed observation scheme allows one to solve, rather accurately, the disturbanceestimation problem

Here, these observers are used in connection with a robust controller design application withinthe context of high gain observation This approach is prone to overshot effects and may bedeemed sensitive to saturation input constraints, specially when used in a high gain orienteddesign scheme via the choice of large eigenvalues Such a limitation is, in general, an importantweakness in many practical situations However, since our control scheme is based on a linearobserver design that can undergo temporary saturations and smooth “clutchings" into thefeedback loop, its effectiveness can be enhanced without affecting the controller structure andthe overall performance We show that the observer-based control, overcomes these adversesituations while enhancing the performance of the classical GPI based control scheme.The linear part of the controller design is based on the Generalized Proportional Integraloutput feedback controller scheme established in terms of Module Theory

In the second design stage, the designed current signals of the first stage are deemed asreference trajectories, and a discontinuous feedback control law for the input voltages issought which tracks the reference trajectories Since the electrical subsystem is faster than themechanical, we propose a sliding mode control approach based on a class of filtered slidingsurfaces which consist in regarding the traditional surface with the addition of a low passfilter, without affecting the relative degree condition of the sliding surface The “chatteringeffect" related to the sliding mode application is eased by means of a first order low-pass filter

as proposed in, Utkin et al (1999)

GPI control has been established as an efficient linear control technique (See Fliess et al.,Fliess, Marquez, Delaleau & Sira-Ramírez (2002)); it has been shown, in, Sira-Ramírez &Silva-Ortigoza (2006), to be intimately related to classical compensator networks design.The main limitation of this approach lies in the assumption that the available output signal

coincides with the system’s flat output (See Fliess et al.Fliess et al (1995), and also Sira-Ramírez

and Agrawal, Sira-Ramírez & Agrawal (2004)) and, hence, the underlying system is, both,controllable and, also, observable from this special output Nevertheless, this limitation islifted for the case of the induction motor system

The controller design is carried out with the philosophy of the classical field orientedcontroller scheme and implemented through a flux simulator, or reconstructor (see Chiasson,Chiasson (2005)) The methodology is tested and illustrated in an actual laboratoryimplementation of the induction motor plant in a position trajectory tracking task

The rest of the chapter is presented as follows: Section 2 describes each of the methodologies touse along the chapter such as the sliding mode control method, the Generalized ProportionalIntegral control and the disturbance observer The modeling of the motor and the problemformulation are given in Section 3, and the proposed methodologies are joined to solve the

problem in Section 4 The results of the approach are obtained in an experimental framework,

as depicted in Section 5 Finally some concluding remarks are given

2 Some preliminary aspects

2.1 Sliding mode control using a proportional integral surface: Introductory example

Consider the following first order system:

where y is the output of the system, ξ(t)can be interpreted as a disturbance input (which may

be state dependent) and u ∈ {− W, W }is a switched class input We propose here to take as a

sliding surface coordinate function the following expression in Laplace domain s:

σ = − s+z

e=y − y ∗ with z >0

The switched control is defined as

since the term− σξ(t) +σ ˙y ∗ − zeσ does not depend on the input, by setting W in such a way

that we can ensure that ˙V <0, the sliding condition forσ is achieved.

The classical interpretation of the output feedback controller suggests, immediately, thefollowing discontinuous feedback control scheme:

where n(s) =s+z regulates the dynamic behavior of the tracking error and d(s) =s acts as

a “filter" of the sliding surface

The equivalent control is obtained from the invariance conditions:

σ=σ˙ =0i.e,

in other words, the proposed sliding surface has, in the equivalent control sense, the samebehavior of the traditional proportional sliding surface of the formσ1 = ze However, the

closed loop behavior of the system with the smooth sliding surface, presents some advantages

as shown in, Slotine & Li (1991) Since this class of controls induce a “chattering effect", toreduce this phenomenon, we insert in the control law output a first order low-pass filter,which, in some cases, needs and auxiliary control loop (as shown in the integral sliding modecontrol design, Utkin et al (1999)) In our case, the architecture of the control system based ontwo control loops and disturbance observers will act as the auxiliary control input

2.2 Generalized Proportional Integral Control

GPI control, or Control based on Integral Reconstructors, Fliess & Sira-Ramírez (2004), is arecent development in the literature on automatic control Its main line of development restswithin the finite dimensional linear systems case, with some extensions to linear delayed

differential systems and to nonlinear systems (see Fliess et al., Fliess, Marquez, Delaleau

& Sira-Ramírez (2002), Fliess et al., Fliess, Marquez & Mounier (2002) and Hernández and

Sira-Ramírez, Hernández & Sira-Ramírez (2003))

The main idea of this control approach is the use of structural reconstruction of the statevector This means that states of the system are obtained modulo the effect of unknown initialconditions as well as constant, ramp, parabolic, or, in general, polynomial, additive externalperturbation inputs The reconstructed states are computed solely on the basis of inputs andoutputs These state reconstructions may be used in a linear state feedback controller design,provided the feedback controller is complemented with a sufficient number of iterated output,

or input, integral error compensation which structurally match the effects of the neglectedperturbation inputs and initial states

To clarify the idea behind GPI control, consider the following elementary example,

value of ˙y(0)

Let e yy − y ∗(t)be the reference trajectory tracking error and let u ∗be a feed-forward input

nominally given by ¨y ∗(t) =u ∗(t) The input error is defined as e uu − u ∗(t) =u − ¨y (t).Integrating equation (7) we have,

The relation between the structural estimate of ˙y of the velocity and the actual value of the

velocity state is given by,

ˆ˙y= ˙y − ˙y(0) − ξt (10)The presence of an unstable ramp error between the integral reconstructor of the velocity andthe actual velocity value, prompts us to use a complementary double integral compensatingcontrol action on the basis of the position tracking error We have the following result:

Proposition 1. Given the perturbed dynamical system, described in (7), the following dynamical feedback control law

u = ¨y − k3(ˆ˙y − ˙y ∗ ) − k2e y(t ) − k1

 t

0 e y(τ)dτ

− k0

t0

 τ

0 e y(σ)dσdτ=k3(˙y(0) +ξt) (13)Taking two time derivatives in (13) the introduced disturbance due to the integralreconstructor is annihilated as follows:

e(4)y +k3e(3)y +k2¨e y+k1˙e y+k0e y=0 (14)

The justification of this last step is readily obtained by defining the following state variablesalong with their initial conditions,

to zero as a function of time

Remark 2. Notice that the GPI controller (11) can also be written as a classic compensation network (expressed in the frequency domain) From (11) and (9),

2.3 Generalized proportional integral observers

Consider the following n-th order scalar nonlinear differential equation,

y (n)=φ(t, y, ˙y, ¨y, · · · , y (n−1)) +ku (20)whereφ is a smooth nonlinear scalar function, k ∈ R and u is a control input We state the

following definitions and assumptions:

Definition 3. Define the following time function:

ϕ : t → φ(t, y(t), ˙y(t), ¨y(t),· · · , y (n−1)(t)) (21)

i.e., denote by ϕ(t), the value of φ for a certain solution y(t) of (20) for a fixed set of nite initial conditions In other words; ϕ(t) = φ(t, y(t), ˙y(t), ¨y(t),· · · , y (n−1)(t)), where y(t) is a smooth bounded solution of Eq (20) from a certain set of nite initial conditions.

2.3.1 A GPI observer approach to state estimation of unknown dynamics

We formulate the state estimation problem for the system (20) via GPI observers as follows:

Under the above assumptions, given the noise-free measurement of y(t), u(t), it is desired to estimate the natural state variables (or: phase variables) of the system (20), given by y(t), ˙y(t),

¨y(t), , y (n−1)(t), via the use of the natural equivalence of system (20) with the simpli ed uncertain system given by (23).

The solution to the simultaneous state and perturbation estimation problem can be achievedvia the use of an extended version of the traditional linear Luenberger observer, that weaddress here as GPI observer, as follows

Proposition 5. Luviano-Juárez et al (2010) Under the assumptions given above For a system of the form (23), the following observer

˙ˆy1=λ p +n−1(y − ˆy1) + ˆy2

˙ˆy2=λ p +n−2(y − ˆy1) + ˆy3

an asymptotically exponentially fashion, the perturbation input ϕ(t)and its time derivatives ˙ ϕ(t), modulo a small error, uniformly bounding the reconstruction error ε=y − ˆy(t) =y − ˆy1, and its first

n − 1 - th order time derivatives provided the design parameters, λ0,· · ·,λ p +n−1 are chosen so that the roots of the associated polynomial in the complex variable s:

P(s) =s p +n+λ p +n−1 s p +n−1+λ p +n−2 s p +n−2+ .+λ1s+λ0 (25)

are all located deep in the left half of the complex plane.

Proof Define, as suggested in the Proposition, the estimation error as follows:

ε(t) y(t ) − y1(t) (26)

taking p+n time derivatives in last equation, and using the reconstruction error dynamics

forε, derivable from the observer equations, leads to the following perturbed reconstruction

error dynamics:

ε (p+n)+λ p +n−1 ε (p+n−1)+λ p +n−2 ε (p+n−2) + · · · + λ1ε˙+λ0ε=ϕ (p)(t) (27)

which is a perturbed n+p - th order linear time invariant system, whose perturbation

input is given byϕ (p)(t) Given that the characteristic polynomial P(s), corresponding to theunperturbed output reconstruction error system, has its roots in the left half of the complexplane, then the Bounded Input Bounded Output (BIBO) stability condition is assured, Kailath

(1979) since, uniformly in t, ϕ (p)(t) < γ

p Thus, the output reconstruction error,ε, and its first n+p −1 time derivatives are ultimately constrained to a disk in the reconstructionerror phase space of arbitrary small radius which is further decreased as the roots of thedominating characteristic polynomial are chosen farther and farther into the left half of thecomplex plane

3 Problem formulation

Consider the following dynamic model describing the two-phase equivalent model of a

three-phase motor controlled by the phase voltages u Sa and u Sb with state variables givenby:θ, describing the rotor angular position, ω being the rotor angular velocity, ψ Raandψ Rb,

representing the unmeasured rotor fluxes, while i Sa and i Sbare taken to be the stator currents

dθ

dt =ω dω

R R and R S are, respectively, the rotor and stator resistances, L R and L Srepresent, respectively,

the rotor and stator inductances, M is mutual inductance constant, J is the moment of inertia and n pis the number of pole pairs The signal τ Lis the unknown load torque perturbationinput We adopt the complex notation like in Sira-Ramirez, Beltran-Carbajal & Blanco-Ortega(2008) Define the following complex variables:

whereψ Rdenotes the complex conjugate of the complex rotor fluxψ R.

We have, thus, established explicit, separate, dynamics for the squared rotor flux magnitude

and for the rotor flux phase angle This representation, clearly exhibits a decoupling property of

the model which allows one to, independently, control the square of the flux magnitude andthe angular position by means of the stator currents acting as auxiliary control input variables.This representation also establishes that the complex flux phase angle is largely determined

by the manner in which the angular position is controlled by the stator currents

The problem formulation is as follows: Given the induction motor dynamics, given a desiredconstant reference level for the rotor flux magnitude| ψ ∗ R | >0, and given a smooth referencetrajectory θ ∗(t) for the angular position of the motor shaft, the control problem consists in

finding a feedback control law for the phase voltages u Sa and u Sbin such a way thatθ is forced

to track the given reference trajectory,θ ∗, while the rotor flux magnitude stabilizes around the

desired value,| ψ ∗

R | Such objectives are to be achieved in spite of the presence of unknownbut bounded perturbation inputs represented by 1) the load torque,τ L(t), in the rotor shaftdynamics and 2) the effects of motor nonlinearities acting on the current dynamics throughpossibly unknown parameters

4 Control strategy

The GPI observer-controller design considerations will be based on the following simplified,linear, models lumping the external load disturbances and the system nonlinearities in theform of components of an unknown perturbation input vector, as follows:

additive dissipation terms, depending on the stator currents i Sa , i Sband the angular velocity

The currents i Sa and i Sb can be directly measured; on the other hand, rotor fluxes must

be estimated For the flux estimation, we used a real time simulation of the rotor fluxequation dynamics Parametersη, n p , M need to be known; on the other hand, the lumped

parameter μ must be estimated Nevertheless, in our control scheme, such a task is not

entirely necessary due to the remarkable robustness of the scheme and a reasonable guesscan be used in the controller expression for such parameters The disturbance functions

ξ1(t), ξ(t) can be envisioned to contain the rest of the system dynamics, including someun-modeled dynamics (which can be of a rather complex nonlinear character) In these terms,

we also lump disturbances of additive nature such as frictions and the effects generated byparameter variations during the system operation and even the effects of inaccurate parameterestimations These perturbation inputs, however, do not contain any control terms

For the correct tracking of angular position, it is necessary to provide additional control loopsfor other variables As it is customary, the flux modulus has to be regulated to a certain value

in order to assure the efficient operation of the induction machine avoiding possible saturationeffects.

The proposed control scheme consists in a two stage feedback controller design The first stagecontrols the angular position of the motor shaft to track the reference signalθ ∗(t)by means

of the stator currents taken as auxiliary control inputs As a collateral objective it is desired tohave the flux magnitude converging towards a given constant value| ψ ∗ R |1 For this stage, thecontrol strategy is implemented by means of a GPI based observer controller, Cortés-Romero

et al (2009) As a result of the first stage a set of desirable current trajectories is synthesized.The obtained currents are thus taken as output references for the second multi-variable stage.The second stage designs a discontinuous feedback controller to force the actual currents totrack the obtained current references in the first stage In the second stage the stator voltagesare the control inputs The following section deals with the flux reconstructor

de

dt = −( η+jn p ω)e

whose unique eigenvalue has a strictly negative real part (and a time varying complex part)

Thus the complex error, e, satisfies e → 0 in an exponentially asymptotic manner Thus,henceforth, when we useψ in the expressions it is implicitly assumed that it is obtained from

the proposed reconstructor undergoing the exponential convergence process ψ → ψ.

4.2 Outer loop controller design stage

For this first design stage we consider the following dynamics:

with the complex stator current i Sacting as auxiliary control input

We propose the following complex controller:

1 Inaccurate parameters may cause minimal variations in the flux regulation, however, the angular position remains unaffected due to the robustness of the controller.

3 Problem formulation

Consider the following dynamic model describing the two-phase equivalent model of a

three-phase... of

the model which allows one to, independently, control the square of the flux magnitude andthe angular position by means of the stator currents acting as auxiliary control input variables.This... throughpossibly unknown parameters

4 Control strategy

The GPI observer-controller design considerations will be based on the following simplified,linear, models lumping the external load