Adaptive Field-oriented Control of Synchronous

European Journal of Control (2008)3:177–195 # 2008 EUCA DOI:10.3166/EJC.14.177–195

Adaptive Field-oriented Control of Synchronous Motors with Damping Windings R. Marino, P. Tomei, and C.M. Verrelli Dipartimento di Ingegneria Elettronica, Universitá di Roma \Tor Vergata", Via del Politecnico 1, 00133 Roma, Italy

Two different nonlinear dynamic control algorithms are presented for synchronous motors with damping windings: (i) an adaptive speed-sensorless controller for rotor position tracking in the presence of unknown constant load torque, on the basis of rotor angle, stator and field windings currents measurements; (ii) an adaptive control law for rotor speed tracking in the presence of uncertain constant load torque and motor inertia, which is based on measurements of mechanical variables (rotor angle and speed) and stator windings currents but does not require field current. As in classical field oriented control, the three voltage inputs are designed so that the direct axis component of the stator current vector is driven to zero; the controllers generate, as an intermediate step, the reference signals for the field current and for the quadrature axis component of the stator current vector, which respectively determine the direct axis component of the damping winding flux vector and the electromagnetic torque. Simulation results are provided for a 20-KW synchronous machine, which show the effectiveness of the two proposed control algorithms. Keywords: Nonlinear control design; output feedback control; output tracking; synchronous motors

E-mail: [email protected] E-mail: [email protected] Correspondence to: C.M. Verrelli, E-mail: [email protected]

1. Introduction Salient pole wound rotor synchronous motors with damping windings (see [15,16]) consist of a stator containing armature windings (electrically connected to the a.c. supply system) and a rotor equipped with a field winding carrying a d.c. current and one or more damping windings so that oscillations in the power grid are damped. Due to their higher efficiency and overload capability, to their smaller physical size and to their lower maintenance and ventilation requirements along with the absence of brush commutation at stator side (so that no power limitation is imposed), synchronous machines gradually replaced d.c. motors in applications such as: mine-shaft winders, rolling mill and gearless cement-mill drives, ship propulsion drives, large compressors, excavators and belt conveyor systems. The problem of controlling a salient pole wound rotor synchronous motor is however rather difficult. It is a nonlinear, highly coupled and multivariable problem with three control inputs (stator and excitation windings voltages) and three outputs to be controlled (rotor position or speed, damping winding flux modulus and stator current vector angle), which do not coincide with the measured outputs. Although rotor angle and speed and stator currents can be measured, flux sensors are usually not available and no field current information can be obtained in applications in which slip rings are excluded; moreover, damping winding currents are also not available.

Received 16 June 2007; Accepted 20 February 2008 Recommended by E. Mosca, D.W. Clarke

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The basics of field oriented control for synchronous motors with damping windings can be found in [11] (see also [3,4] and references therein), while theoretical and experimental results in [18] show that, due to the rotor salient pole effect, the air-gap flux oriented control does not allow for decoupled control of synchronous motors with dampers. In [17], a magnetizingflux oriented control method is presented: the stator current vector is divided into two components which independently determine the magnetization and the electromagnetic torque. In [2], a method of estimating the currents in the damping windings is proposed in order to allow their inclusion within a field oriented controller while a vector control theory is developed in [12] for a current-fed salient pole synchronous machine with dampers. A nonlinear control scheme consisting of two nested loops for speed and currents control is experimentally validated in [1]: the references for stator and field currents are computed for desired internal and torque angles (the field oriented control can be obtained as a special case) while unmeasurable damper winding currents are estimated by a reduced order observer. A novel and simple method of achieving a field-oriented self-controlled synchronous motor is presented in [14]: an additional field winding is located in the quadrature axis of the rotor. Two different methods for estimating the field current when brushless excitation is employed are proposed in [5]. The aim of this paper is to design two nonlinear adaptive output feedback controls with stability analysis for synchronous motors with damping windings: rotor position or speed tracking of arbitrary smooth bounded reference signals is to be guaranteed along with tracking for direct axis component of damping winding flux vector; direct axis component of the stator current vector is to be regulated to zero so that field orientation is achieved. Two different adaptive algorithms are proposed: (i) a speed-sensorless rotor position tracking controller, which is based on rotor angle and both stator and excitation windings currents measurements; (ii) a rotor speed tracking control law relying on measurements of both rotor angle and speed and stator windings currents. While the former does not require rotor speed sensors and achieves exponential position tracking for any unknown constant load torque (global exponential position tracking when flux initial values are known), the latter is adaptive with respect to both constant uncertain load torque (within known bounds) and motor inertia (within restricted bounds) and guarantees asymptotic speed tracking for any motor initial condition without requiring the availability of field current measurements. As in classical field oriented control, reference

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signals for quadrature axis component of stator current vector and field current are generated as an intermediate control step: they are responsible for the tracking of rotor angle or speed and for the tracking of direct axis component of damping winding flux vector, respectively. Numerical simulations with reference to a two pole pairs 20-KW synchronous motor with damping windings illustrate the effectiveness of the two proposed control algorithms.

2. Dynamical Model and Field Orientation Assuming linear magnetic circuits, the dynamics of a balanced synchronous motor with damping windings in the ðd; qÞ reference frame rotating at the electrical angular speed p! and identified by the electrical angle p in the fixed ða; bÞ reference frame attached to the stator, are given by the seventh-order nonlinear model (see for instance [6,11,17]) _ ¼ ! 3p ½ sd isq 2J ¼ Rs isd þ p!

!_ ¼ _ sd

_ sq ¼ Rs isq p! _ wd ¼ Rwd iwd

sq isd

TL : Te TL ¼ J J J

sq

þ usd

sd

þ usq

ð1Þ

_ wq ¼ Rwq iwq _ f ¼ Rf if þ uf in which: p is the number of pole pairs, is the rotor angle, ! is the rotor speed, ð sd ; sq Þ are the stator fluxes, ð wd ; wq Þ are the damping winding fluxes, f is the excitation winding flux, ðusd ; usq Þ are the stator voltages, uf is the excitation winding voltage, J 2 ½Jm ; JM is the motor inertia while ðRs ; Rwd ; Rwq ; Rf Þ are the resistances in stator, damping and excitation windings, respectively. Te is the electromagnetic torque while the load torque TL 2 ½TLm ; TLM is typically an uncertain step disturbance. The magnetic equations are assumed to be linear and given by sd

¼ Lsd isd þ Lmd iwd þ Lmd if

wd

¼ Lsq isq þ Lmq iwq ¼ Lmd isd þ Lwd iwd þ Lmd if

wq

¼ Lmq isq þ Lwq iwq

sq

f

ð2Þ

¼ Lmd isd þ Lmd iwd þ Lf if

in which: ðisd ; isq Þ are the stator currents, ðiwd ; iwq Þ are the damping winding currents, if is the excitation current, ðLsd ; Lsq Þ, ðLwd ; Lwq Þ, Lf are the inductances

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in stator, damping and excitation windings, respectively, while ðLmd ; Lmq Þ are the mutual inductances between stator and rotor windings. The vectors ð sd ; sq ÞT , ð wd ; wq ÞT , ðisd ; isq ÞT , ðiwd ; iwq ÞT , ðusd ; usq ÞT are obtained multiplying the corresponding vectors ð sa ; sb ÞT , ð wa ; wb ÞT , ðisa ; isb ÞT , ðiwa ; iwb ÞT , ðusa ; usb ÞT in the fixed ða; bÞ reference frame by the matrix: cos p sin p RðÞ ¼ : sin p cos p By eliminating the unmeasured variables ð sd ; sq ; f Þ in Eq. (1) by using Eq. (2), the dynamics of the synchronous motor with damping windings in the ðd; qÞ reference frame are given by [note that Lsd Lsq ¼ Lmd Lmq according to [11]] _ ¼ ! 3p !_ ¼ ½ð!1 !2 Þisq isd þ !1 if isq 2J Lmq Lmd TL þ wd isq wq isd Lwd Lwq J R R L wd wd md _ wd ¼ ðisd þ if Þ wd þ Lwd Lwd _ wq ¼ Rwq wq þ Rwq Lmq isq Lwq Lwq

ð3Þ

d d isd ¼ L1 dt if f " disq Lwd ðRs Lwq 2 þRwq Lmq 2 Þ ¼ q isq dt Lwq 2 ðLsd Lwd Lmd 2 Þp!isd Lmd ðLwd Lmd Þp!if þ # Lmd p!

Rwq Lwd Lmq Lwq 2

wq

wd þ Lwd usq

ðRs Lwd 2 þRwd Lmd 2 Þ Rwd Lmd 2 isd if 2 Lwd Lwd 2 ðLsq Lwq Lmq 2 Þ Rwd Lmd þ p!isq þ wd Lwq Lwd 2 Lmq þ p! wq þ usd ð4Þ Lwq

d ¼

in which ð; !; isd ; isq ; if ; wd ; wq Þ constitute the state variables and ðusd ; usq ; uf Þ are the control inputs. To simplify notation, in Eq. (3) we have used the reparametrization: Lmd 2 >0 Lwd Lmq 2 !2 ¼ Lmq >0 Lwq Lwq >0 q ¼ ðLsq Lwq Lmq 2 ÞLwd 2 3 2 2

!1 ¼ Lmd

L¼4

Lsd Lwd Lmd Lwd

Lmd Lwd Lmd Lwd

Lmd Lwd L2md Lwd

Lf Lwd Lmd 2 Lwd

5 > 0:

According to [11], the torque per ampere can be maximized by imposing the stator current to be all in the q-axis, i.e., isd ¼ 0: this condition is usually referred to as field orientation. Even though one drawback of choosing isd ¼ 0 is the inescapable fact that the terminal power factor will be lagging (which may be unacceptable in large hp drives), this choice is advantageous in terms of decoupling the transient response of the system. In fact, if field orientation is performed, the electromagnetic torque is 3p Lmd !1 if þ Te ¼ wd isq Lwd 2 while the dynamics of the damping winding flux vector d-component are given by _ wd ¼ Rwd wd þ Rwd Lmd if Lwd Lwd so that, while in induction machines the electromagnetic torque production requires the stator currents to contain both the flux- and the torque-producing components, in a synchronous machine the excitation is provided by the field current and the main torque is produced by its interaction with the quadrature axis component of the stator current vector, leading to a complete decoupling of the d-damper circuit from the stator windings. If we rewrite, according to Eq. (2), the torque equation as 3p Lmd ðif þ iwd Þisq Lmq iwq isd 2 þðLmd Lmq Þisd isq

Te ¼

it is divided in the three torque components: ðRf Lwd þRwd Lmd Þ Rwd Lmd if isd Lwd 2 Lwd 2 Rwd Lmd þ ð5Þ wd þ uf Lwd 2

f ¼

2

2

2

TeðfdÞ ¼ Lmd ðif þ iwd Þisq (field plus d-axis damper winding torque); Teq ¼ Lmq iwq isd (q-axis damper winding torque); Ter ¼ ðLmd Lmq Þisd isq (reluctance (saliency) torque).

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3p Lmd if isq 2 which is similar to the one of the compensated separately excited d.c. machine.

Following the field oriented control strategy [11], our goal is to design a dynamic output feedback compensator [only ð; isd ; if ; isq Þ measurements are available] 2 3 2 32 3 usa usd cosp sinp 0 4 usb 5 ¼ 4 sinp cosp 0 54 usq 5 uf uf 0 0 1

3. Rotor Position Tracking

so that the asymptotic properties

Under steady-state conditions and field orientation, wq are constant and isd ; iwd ; iwq are zero, so that the torque expression reduces to wd ,

Te ¼

In this section, assuming that rotor angle and both stator and field currents measurements are available for feedback, a fifth order dynamic nonlinear adaptive control algorithm is designed for the seventh order nonlinear model (Eqs. 3–5) of a synchronous motor with damping windings: asymptotic rotor position tracking of arbitrary smooth bounded reference signals, damping winding flux vector d-axis component tracking of smooth bounded reference signals belonging to a certain class of time functions (including positive constant reference values) and stator current vector d-axis component regulation to zero are to be guaranteed in the presence of unknown constant load torque. The controller includes an observer and an estimator for both the unmeasured rotor speed and damping winding fluxes and for the unknown load torque and guarantees, when the damping winding flux estimator is properly initialized, boundedness and exponential tracking and estimation for any initial condition and for any unknown constant load torque; a local result holds when flux initial values are not exactly known. The solution presented in this section is suitable in positioning applications such as mine-shaft winders and excavators, in which it is desiderable to eliminate speed sensors. In fact, in addition to cost considerations, the two common methods for obtaining velocity information (direct tachometer readings or differentiation of the rotor position measurements) may fail to provide accurate and noise-free velocity estimates atlow speed and may deteriorate the positioning performance. 3.1. Problem Statement

Let us denote by ðtÞ and ðtÞ the smooth bounded reference signals with bounded time derivatives ðiÞ ðtÞ, ðjÞ ðtÞ, 1 i 3, 1 j 2, for the output variables to be controlled, which are the rotor position (t) and the damping winding flux vector d-component ðtÞ and ð1Þ ðtÞ wd ðtÞ, respectively and assume that satisfy, for all t 0, the following inequality:

ðtÞ þ

ðLwd Lmd Þ Rwd

ð1Þ

ðtÞ c > 0

in terms of a positive scalar c .

ð6Þ

lim ½ðtÞ ðtÞ ¼ 0

t!þ1

lim ½

t!þ1

wd ðtÞ

ðtÞ ¼ 0

lim ½isd ðtÞ ¼ 0

t!þ1

are guaranteed for any unknown constant load torque TL . 3.2. Control Algorithm (I) The proposed control algorithm consists of a current loop: " L2mq 2 5 2 usd ¼ d kd isd kp isd 2 ^wq Lwq 4 # ðLsq Lwq Lmq 2 Þ2 2 þ isq L2wq uf ¼ f kf ðif if Þ 1 h usq ¼ q Lwd q þ kq ðisq isq Þ 5 kp2 q2 ðisq isq Þ ðLsd Lwd Lmd 2 Þ2 i2sd 4 i 2 þL2md ðLwd Lmd Þ2 i2f þ L2md ^wd ð7Þ with the known functions " # 0 d d ¼ L dif f f dt ðRs Lwd 2 þRwd Lmd 2 Þ Rwd Lmd 2 i if sd Lwd 2 Lwd 2 ðLsq Lwq Lmq 2 Þ Rwd Lmd ^ ^ sq þ þ p!i wd Lwq Lwd 2 Lmq ^ ^wq þ p! Lwq

d ¼

ðRf Lwd 2 þRwd Lmd 2 Þ Rwd Lmd 2 if isd 2 Lwd Lwd 2 Rwd Lmd ^ þ wd Lwd 2

f ¼

181


" q ¼ q

Lwd ðRs Lwq 2 þRwq Lmq 2 Þ isq Lwq 2

^ sd ðLsd Lwd Lmd 2 Þp!i ^fþ Lmd ðLwd Lmd Þp!i # disq ^ ^wd : Lmd p! dt

Rwq Lwd Lmq ^ wq Lwq 2

It depends on the intermediate reference signals ðif ; isq Þ for ðif ; isq Þ (which are responsible for tracking of rotor angle and direct axis component of damping winding flux vector, respectively) Lwd Rwd þ ð1Þ if ¼ Lmd Rwd Lwd 2J ðLwd Lmd Þ ð1Þ 1 isq ¼ þ 3p Rwd h k! !^ ðk! k þ 1Þð^ Þ þ k! ð1Þ ^ ð1Þ Þ þ þð2Þ k ð!

T^L i J

ð8Þ

^ T^L Þ for the unmeasured and on the estimates ð!; variable ! and the unknown load torque TL , respectively, which are provided by the third order observer including the auxiliary variable ^ _ ^ ^ ¼ !^ þ k1 ð Þ h 3p ^_ ¼ ! ð!1 !2 Þisq isd þ !1 if isq 2J i Lmq ^ Lmd ^ þ wd isq wq isd Lwd Lwq T^L ^ þ k2 ð Þ J ^ T_^L ¼ Jk3 ð Þ;

and robust feedback terms; (ii) two outer control loop (Eq. 8) for rotor angle and damping winding flux vector d-component; (iii) an observer (Eq. 9) and an estimator (Eq. 10) for the unmeasured rotor speed and damping winding fluxes and the unknown load torque. The control algorithm (Eqs. 7–10) depends on: the available (, isd , if , isq ) measurements; the smooth bounded reference signals ðtÞ, ðtÞ and their bounded time derivatives ðiÞ ðtÞ, ðjÞ ðtÞ, 1 i 3, 1 j 2; the known motor parameters p, Rs , Rwd , Rwq , Rf , Lsd , Lsq , Lwd , Lwq , Lf , Lmd , Lmq , J; the positive control parameters k , k! , kd , kf , kq , k, k1 , k2 , k3 . 3.3. Stability Analysis ~ !ðtÞ, ~wd ðtÞ, ~wq ðtÞ, isd ðtÞ, ~if ðtÞ, ~isq ðtÞ, ~ Let yðtÞ ¼ ½, T e ðtÞ, e! ðtÞ, eT ðtÞ be the solution of the closed loop system (Eqs. 3–5 and 7–10) with ~ ¼ , ~ ~wd ¼ wd , ~wq ¼ wq rq , ~ ¼ ! ð1Þ þ k , ! ^ e! ¼ ! !, ~if ¼ if i , ~isq ¼ isq i , e ¼ , ^ eT ¼ sq f T^L TL where rq satisfies the first order differential J equation Rwq Rwq Lmq l rq þ i ð11Þ r_q ¼ Lwq Lwq sq in which 2J ðLwd Lmd Þ ð1Þ 1 l þ isq ¼ 3p Rwd h ðk! þ k Þe! þ ðk! k þ 1Þe TL i þ eT þ ð2Þ þ : J Define

ð9Þ

the estimates of the unmeasured damping winding fluxes are obtained by the second order open loop observer

2J ðLwd Lmd Þ ð1Þ 1 l ¼ i i ¼ þ inl sq sq sq 3p Rwd : 2 ~ ðk! þ k Þ!~ þ ðk 1Þ~ ¼ g1 ~ þ g2 !:

The main result of this section is stated in the following theorem.

ð10Þ

Theorem 1: If the damping winding flux estimator (10) is properly initialized to the damping winding flux values wd ð0Þ; wq ð0Þ, then the closed loop system solution yðtÞ of Eqs. (3–5), (7–10) is bounded and tends exponentially to zero:

The control law (Eqs. 7–10), which is designed by using back-stepping and robust techniques (involving the choice of suitable Lyapunov functions as it is shown in the next section) consists of: (i) an inner control loop (Eq. 7) for stator and field currents containing feed-forward actions and suitable stabilizing

(i) for any initial condition yð0Þ 2 > > > > ^ > > < if TL < TLm and 0 ProjT ½ ; T^L ¼ if T^L > TLM and 0 > > > > T1 if T^L < TLm and < 0 > > > : T2 if T^L > TLM and > 0 " T1 ¼ 1 " T2 ¼ 1

TLm 2 T^L

2

#

TLm 2 ðTLm "Þ2 2 T^L TLM 2

#

ðTLM þ "Þ2 TLM 2

0 TLm ðTLm "Þ;

0 TLM ðTLM þ "Þ

185


^ ¼ ProjJ ½ ; J

"

j1 ¼ 1 "

j2 ¼ 1

8 > > > > > > >
JM and 0

> > > > > > j1

> :

j2

if J^ < Jm and < 0 if J^ > JM and > 0

Jm 2 J^

Jm 2 ðJm Þ2 J^ JM 2 2

ðJM þ Þ JM

2

ky12 ðtÞk 1 ky12 ð0Þke 2 t ;

#

h

:

4.3. Stability Analysis Define 0 1 ; 2Rwd Lmd 2 Lwd

lim ky11 ðtÞk ¼ 0

t!þ1

The control law (Eqs. 18–22), which is designed by using back-stepping and robust adaptive techniques (involving the choice of suitable Lyapunov functions as it is illustrated in the next section), consists of: (i) an inner control loop (Eq. 18) for the stator and field currents containing feed-forward actions and stabilizing terms; (ii) two outer control loops (Eq. 19) for rotor speed and damping winding flux vector d-component; (iii) a closed loop observer (Eq. 20) for the unmeasured damping winding fluxes; (iv) two closed loop identifiers (Eq. 22) for the uncertain load torque and motor inertia; (v) a damping winding flux vector q-component reference signal generator (Eq. 21). The control algorithm (18–22) depends on: the available (!, isd , isq ) measurements; the bounded reference signals ! ðtÞ, ðtÞ and their bounded time derivatives !ðiÞ ðtÞ, ðjÞ ðtÞ, 1 i; j 2; the known motor parameters p, Rs , Rwd , Rwq , Rf , Lsd , Lsq , Lwd , Lwq , Lf , Lmd , Lmq ; the known bounds TLm , TLM , Jm , JM ; the positive control parameters k! , , k!e , a , d ; q , kd , kq , , ", , ; the saturation function sat ðxÞ (a C1 odd function whose derivative is always positive and has the finite limit as x goes to þ1).

c1 ¼

Theorem 3: The closed loop system solution yðtÞ of Eqs. (3–5) and (18–22) is bounded and satisfies [ 1 , 2 are suitable positive reals]

#

2

2

The main result of this section is stated in the following theorem.

c2 ¼

0 þ 1 2Rwd Lmd 2 Lwd

where 0 ¼ 4 Lwd 2 Rf þ 2Rwd Lmd 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ 4 Lwd 2 Rf ðLwd 2 Rf þ Rwd Lmd 2 Þ: Let us denote by yðtÞ ¼ ½y1 ðtÞT , y2 ðtÞ, y3 ðtÞT the solution of the closed loop system (3–5) and (18–22) with ^ !T , y12 ¼ ½ wd y1 ¼ ½yT11 ; yT12 T , y11 ¼ ½! ! ; ! ; wq rq ; isd ; if if ; isq isq ; wd ^wd ; wq ^wq T , ^ y2 ¼ TL T^L , y3 ¼ J J^ and define ’ðtÞ ¼ ½!ðtÞ !ðtÞ; y12 ðtÞT ; y2 ðtÞT .

i

8t 0

lim y2 ðtÞ þ y3 ðtÞ!ð1Þ ðtÞ ¼ 0

t!þ1

(i) for any initial condition y1 ð0Þ 2

i 2 hTLM þ " TLm 1 ð1Þ þ k! ðÞk1 : Jm k! 2

Proof: Let us define the tracking and estimation ^ !ðtÞ, ~wd ðtÞ ¼ ~ ¼ !ðtÞ ! ðtÞ, e! ðtÞ ¼ !ðtÞ errors: ! ~ ðtÞ ðtÞ, ðtÞ ¼ ðtÞ rq ðtÞ, ~isd ðtÞ ¼ isd ðtÞ, wq wd wq ~if ðtÞ ¼ if ðtÞ i ðtÞ, ~isq ðtÞ ¼ isq ðtÞ i ðtÞ, ewd ðtÞ ¼ sq f ^ ewq ðtÞ ¼ wq ðtÞ ^wq ðtÞ, T~L ðtÞ ¼ wd ðtÞ wd ðtÞ, ~ ¼ J JðtÞ. ^ The adaptive control TL T^L ðtÞ, JðtÞ algorithm (18–22) substituted in model (3–5), leads to the closed loop system T~L ~ þ e! Þ ~_ ¼ k! sat ð! ! J 3p h Lmd ~ þ isq wd ð!1 !2 Þisq ~isd þ!1 isq ~if þ Lwd 2J Lmq ~ ~ Lmq ~ Lmd ~ rq isd þ isd isq Lwq wq Lwq Lwd i J~ þ !1 if ~isq ½k! sat ð!^ ! Þ þ !ð1Þ J ð23Þ

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T~L 3p h ð!1 !2 Þisq ~isd J 2J Lmq ~ ~ Lmd þ !1 isq ~ isq ~wd if þ isd Lwd Lwq wq i Lmq ~ Lmd ~ rq isd þ isq þ!1 if ~ isq Lwq Lwd i h J~ þ k! sat ð!^ ! Þ þ !ð1Þ J n 2 a a ^ ! Þ þ !ð1Þ þ e! k! sat ð! 4 4 9p2 a h þ ð!1 !2 Þ2 isq 2 þ !1 2 ðisq 2 þ if 2 Þ 16 io Lmq 2 Lmd 2 þ ði 2 þ 2 Þ þ ði 2 þ rq 2 Þ 2 sq 2 sd Lwd Lwq he i ! ^ T~_L ¼ ProjT ; TL i h ~ ! Þþ!ð1Þ e! ; J^ J_~¼ ProjJ 1 k! sat ð!þe e_! ¼ k!e e! þ

ð24Þ R R L _~ ¼ wd ~ þ wd md ð~ isd þ ~ if Þ wd Lwd wd Lwd _~ ¼ Rwq ~ þ Rwq Lmq ~ ð25Þ isq wq Lwq wq Lwq " # h i _~ isd ð4Þ 1 ~ þ ¼ L _~ if ð5Þ " # kd ~ isd ¼ Rwd Lmd 2 ~ isq isd þq Lmd ðLwd Lmd Þp! ~ L 2 2

~¼4

wd

2 L p!ewq i~f þ RLwd L2md ewd þ Lmq RwdL Lmd 2 wq wd

wd

ðRf Lwd 2 þRwd Lmd 2 Þ ~ if Lwd 2

h

þ RLwd L2md ewd

3 5

wd

_~ isq þq Lmd ðLwd Lmd Þp! ~if isq ¼ kq ~ i Rwq Lwd Lmq þ e L p!e wq md wd Lwq 2 Rwd Rwd Lmd ~ 1 h Rwd Lmd ~ ewd þ if isd e_wd ¼ Lwd Lwd d Lwd 2 i q Lmd p! i~sq Rwq 1 h Lmq ewq p! ~ isd Lwq q Lwq Rwq Lwd Lmq ~ i þ q isq : Lwq 2

e_wq ¼

ð26Þ

Consider the quadratic positive definite function 1 Vi ¼ ðd ewd 2 þq ewq 2 Þ 2 1 1 2 ð27Þ þ ½i~sd ; i~f L½i~sd ; i~f T þ i~sq 2 2

whose time derivative along the trajectories of the closed loop subsystem (26) is given by [L ¼ LT ] Rwq Rwd ewd 2 q ewq 2 V_ i ¼ d Lwd Lwq Rwd Lmd 1 ~ þ ðd þ Þ if ewd Lwd Lwd ðRf Lwd 2 þRwd Lmd 2 Þ ~ 2 if Lwd 2 2 2 kd ~isd kq ~isq

ð28Þ

which is negative definite for any d satisfying the condition ðvÞ: Eqs. (27) and (28) imply that ðewd ; ewq ; ~isd , ~if ; ~isq Þ tend exponentially to zero from any initial condition. Subsystem (25) may be rewritten as 2 3 wd 0 RLwd 5

_ ¼ 4 Rwq 0 Lwq 2 3 Rwd Lmd Rwd Lmd 0 Lwd Lwd 5 w þ4 ð29Þ Rwq Lmq 0 0 Lwq ð15Þ with ¼ ½ ~wd ; ~wq T and w ¼ ½~isd ; ~if ; ~isq T . Since w tends exponentially to zero for any initial condition, subsystem (25) complies with the hypotheses of Lemma III.1 in [8]: ð ~wd ; ~wq Þ tend globally exponentially to zero. Consider the quadratic positive definite function 1 Vr ¼ e! 2 : 2

ð30Þ

By computing the time derivative of function Vr along the e! -trajectories and by completing the squares, we obtain 1 h ~ 2 ~2 2 2 TL þ J þ2 ~isd þ ~ if V_ r k!e e! 2 þ a J2 i 2 2 2 þ 2 ~isq þ ~wd þ ~wq : ð31Þ ~ ~ ~ ~isd , ~if , ~isq , wd , wq Þ are bounded [recall Since ðT~L , J, that the adaptation laws for T^L and J^ are designed using projection algorithms], from (30) and (31) it follows that e! ðtÞ is bounded for all t 0. Since ð~isq ; ~wq Þ are bounded, and ðisq ; rq Þ are bounded, ðisq ðtÞ; wq ðtÞÞ are bounded and, consequently, since ~_ and e_! ðtÞ are bounded for all t 0; e! is bounded, !ðtÞ therefore e! ðtÞ is uniformly continuous for all t 0. ^_ ~_ e_! Þ are bounded, !ðtÞ is bounded for Since ð!; all t 0. Consider the quadratic positive definite function 1 1 1 2 2 Va ¼ e! 2 þ T~L þ J~ : 2 2J 2J

ð32Þ

187


By computing the time derivative of function Va along the trajectories of the closed loop subsystem (24) and by completing the squares, we finally obtain 1 h ~ 2 ~2 2 2 isd þ if þ2 i~sq V_ a k!e e! 2 þ a J2 i 2 2 þ ~wdþ ~wq : ð33Þ isd ; ~ if ; ~isq ; ~wd ; Since e! is uniformly continuous and ð~ ~wq Þ are exponentially decaying signals, by integrating (33) we have Z t k!e e! 2 ðÞd Va ð0Þ lim t!þ1 0 h 1 Z th 2 2 2 þ lim if ðÞ þ 2 ~isqðÞ 2~ isdðÞ þ ~ t!þ1 a J2 0 i i 2 2 þ ~wd ðÞ þ ~wq ðÞ d < þ1 which implies, by virtue of Barbalat’s lemma (see for instance [9]), that lim e! ðtÞ ¼ 0:

ð34Þ

t!þ1

~_ isq ; wq ; if Þ are bounded, ði~sd ; i~f ; ~ isq ; ~wd ; ~wq Þ Since ð!; tend exponentially to zero, e! tends asymptotically to ~ J (by virtue of ðviÞ), if we choose the zero and jJj 2 positive control parameter according to ðviiÞ then, ~ is bounded recalling Eq. (23), we can establish that !ðtÞ ~ (therefore !) and ð~isd ; ~if ; for all t 0. Since ! ~ isd ðtÞ; _~ if ðtÞ; _~ isq ðtÞÞ are isq ; ~wd ; ~wq Þ are bounded, ð_~ di ðtÞ _ bounded and, consequently, since isq is bounded, sqdt is di _~ ^_ T_~L ; J, isd ; ~ if ; ~ isq , !; bounded for all t 0. Since ðe_! ; dtsq , ~ _ ;~ _ ; r_ , i_ Þ are bounded, e€ ðtÞ is bounded for all t ~ wd

wq

q

f

!

0 and, consequently, e_! is uniformly continuous. By virtue of Barbalat’s lemma, we can establish that lim e_! ðtÞ ¼ 0: t!þ1

Recalling the first equation in (24), since e! tends asymptotically to zero, ðisq ; rq ; if Þ are bounded and ð~isd , ~ isq , ~wd , ~wq Þ tend exponentially to zero, we have if , ~ h T~ ðtÞ L

~ JðtÞ ^ ! ðtÞÞ ðk! sat ð!ðtÞ þ t!þ1 J J i ð35Þ þ !ð1Þ ðtÞÞ ¼ 0: lim

Recalling Eq. (23), since e! tends asymptotically to isd , ~ if , ~ isq , ~wd , ~wq Þ zero, ðisq ; wq ; if Þ are bounded and ð~ tend exponentially to zero, we can finally establish, according to Eq. (22), that !~ tends asymptotically to zero for any initial condition. Remark: When the excitation current if is measured, if may be replaced by if in the functions d and q while the control inputs uf and the estimation law for ^wd may

be modified according to (kf is a positive control parameter) uf ¼ f kf ðif if Þ ^_ wd ¼ Rwd ^wd þ Rwd Lmd ðisd þ if Þ Lwd Lwd i 1 n Rwd Lmd h þ isd þ ðif if Þ 2 d Lwd o q Lmd P! ðisq isq Þ so that no restrictions on the positive control parameter d are to be imposed. When motor inertia is known, a simplified version of the controller, which guarantees the additional properties of exponential load torque estimation and local exponential rotor speed tracking, can be derived according to the following theorem. Theorem 4: The simplified version of the controller, obtained from Eqs. (18)–(22) by setting J^ J and by modifying the rotor speed observer according to ^ !Þ k! sat ð! ^ ! Þ þ !ð1Þ ^_ ¼ k!e ð! ! h 3p þ ð!1 !2 Þisq isd þ !1 if ðisq isq Þ 2J i Lmd ðisq isq Þ ð36Þ þ Lwd guarantees [hðÞ 0 and ðÞ > 0 are suitable functions, 1 , 2 are suitable positive reals] j!ðtÞ ! ðtÞj is bounded 8t 0 and lim j!ðtÞ ! ðtÞj ¼ 0 t!þ1

ky12 ðtÞk 1 ky12 ð0Þke 2 t ; 8t 0 ^ !ðtÞ; y2 ðtÞk k½!ðtÞ hðk’ð0ÞkÞeðkð0ÞkÞt ; 8t 0 (i) for any initial condition y1 ð0Þ 2