Progress In Electromagnetics Research B, Vol. 34, , 2011

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1 Progress In Electromagnetics Research B, Vol. 34, , 211 INFLUENCE OF CONSTANT VALUES AND MOTOR PARAMETERS DEVIATIONS ON THE PERFORMANCE OF THE ADAPTIVE SLIDING-MODE OBSERVER IN A SENSORLESS INDUCTION MOTOR DRIVE P. J. Costa Branco * an J. Ferraz Technical University of Lisbon, Instituto Superior Técnico (IST, DEEC-Energia, Av. Rovisco Pais, Lisboa 149-1, Portugal Abstract The aaptive sliing-moe observer has been wiely use to estimate the rotor flux an rotor spee in inverter-fe sensorless inuction motor rives. However, the technique requires setting a priori the sliing-moe observer constants an also knowlege of the inuction motor parameters. This particular aspect can cause significant errors in the estimation of the rotor spee use in sensorless control schemes. Changes in the inuction machine parameters ue to temperature or ifferent saturation levels will affect the ynamic operation of the observer espite its aaptive nature. In this context, a sensitivity stuy of the aaptive sliing-moe observer is presente an iscusse in this paper. Various experiments are performe on a sensorless inirect vector-controlle inuction motor rive uner a variety of conitions to verify the observer robustness. 1. INTRODUCTION Several applications of the simple aaptive sliing-moe observer [1 or its improvements for sensorless inuction motor rives have been evelope in the literature in the last years. References [2 18 resume some of the research propose in the last years, together with other techniques as, for example, using high gain observers [19, 2. However, the influence of the constants an motor parameters eviation on the aaptive observer performance has been little consiere. It is in this context that this paper presents the constants an motor parameters sensitivity stuy an several experiments of the aaptive sliing-moe observer in a sensorless inuction motor rive. Receive 16 August 211, Accepte 1 September 211, Scheule 14 September 211 * Corresponing author: Paulo José a Costa Branco (pbranco@ist.utl.pt.

2 226 Costa Branco an Ferraz 2. INDUCTION MOTOR MODEL The inuction motor moel with the stator currents an rotor flux as state variables can be written in the stationary-coorinate system by [ [ [ isψr isψr us = A + B, (1 where i s correspons to DQ components of stator currents, u s is the vector of DQ components of stator voltages, an ψ r is the vector of DQ components of rotor flux, all efine as i s = [ ids i Qs, u s = Matrices A an B are given by where an A = A 11 = ηi, [ uds, ψ u r = Qs [ A11 A 12, B = A 21 A 22 [ B1 A 12 =β ( τ 1 r I pωj, A 21 =τ 1 r MI, A 22 =τ 1 r I pωj, B 1 = 1 σl s I σ = 1 M 2, β = M, L s L r σl s L r [ [ 1 1 I =, J = 1 1 η = M 2 R r + L 2 rr s σl s L 2 r, [ ψdr. (2 ψ Qr, (3, τ r = L r R r, where M is the mutual inuctance coefficient, L s an L r are respectively the stator an rotor inuctance coefficients, R s an R r are respectively the stator an rotor phase resistances, ω being the motor spee, an p being the number of pole pairs. Using (2 to (5 in (1 yiels i Ds i Qs ψ Dr ψ Qr = ( η i Ds + ( ( βτr 1 1 ψdr + (βpω ψ Qr + σl s = ( η i Qs (βpω ψ Dr + ( βτ 1 r ( 1 ψqr + σl s (4 (5 u Ds, (6a u Qs, (6b = ( τr 1 M i Ds τ 1 r ψ Dr (pω ψ Qr, (6c = ( τr 1 M i Qs + (pω ψ Dr τ 1 r ψ Qr, (6

3 Progress In Electromagnetics Research B, Vol. 34, which is the inuction motor moel in DQ coorinates in stationarycoorinate system. The electromagnetic torque is given by T e = 3 pm (ψ Dr ψ Qs ψ Qr ψ Ds. (7 2 L r 3. THE ADAPTIVE SLIDING-MODE OBSERVER: REVIEW 3.1. Rotor Flux Estimation The aaptive observer can be expresse by (8 where ˆx is the vector compose by DQ components of estimate stator current an estimate rotor flux (9, K is a gain matrix efine by (1 an (11. Currents ˆι Ds an ˆι Qs are estimate an compare with those ones measure to prouce an error variable for the sliing observer estimate the rotor flux components an thus the rotor spee. This is why the sliing error variable is use to guie the observer. ˆx = Aˆx+Bu s+k sgn (Îs i s (8 ˆx = K = K 1 = [ Îsˆψr = ˆι Ds ˆι Qs ˆψ Dr ˆψ Qr [ K1 LK 1 [ k1 L = k 2 (9 [ l11 l 12 l 21 l 22 (1 (11 Developing (8, it is written as in (12. = A 11 Î s +A 12 ˆψr +B 1 u s +K 1 sgn (Îs i s (12a Îs ˆψ r = A 21 Î s + A 22 ˆψr LK 1 sgn (Îs i s (12b The matrix K 1 values are set by trial-an-error to optimize the observer performance. To obtain the coefficients of matrix L, the following expressions are efine [2: x = (q 1 ε + γ τ r ε, y = γω (13 ε,

4 228 Costa Branco an Ferraz with q an γ being constants assigne experimentally with ε efine as: ε = σl sl r M. (14 The matrix L is thus calculate as in (15. l 11 = x l 12 = y l 21 = y l 22 = x (15 Expaning vectors Îs = [ˆι [ T Ds, ˆι Qs an ˆψr = ˆψDr, ˆψ T Qr, as well as the matrices A 11, A 12, A 21, A 22, K 1, LK 1 gives the set of ifferential equations for stator currents an rotor flux inicate in (16. The system has as input variables the voltages u Ds an u Qs, currents i Ds an i Qs an the rotor spee measure ω or estimate ˆω. The variables to be estimate are the components of the stator current in stationary reference frame, ˆι Ds an ˆι Qs an the rotor flux components in the same reference, ˆψ Dr an ˆψ Qr ˆι Ds ˆι Qs = ηˆι Ds + β τ r ˆψDr +βpω ˆψ Qr + 1 σl s u Ds k 1 sgn(ˆι Ds i Ds (16a = ηˆι Qs βpω ˆψ Dr + β τ r ˆψQr + 1 σl s u Qs k 2 sgn(ˆι Qs i Qs (16b ˆψ Dr ˆψ Qr = M ˆι Ds 1 ˆψDr pω τ r τ ˆψ Qr +l 11 k 1 sgn (ˆι Ds i Ds r +l 12 k 2 sgn(ˆι Qs i Qs (16c = M ˆι Qs +pω τ ˆψ Dr 1 ˆψQr +l 21 k 1 sgn (ˆι Ds i Ds r τ r +l 22 k 2 sgn(ˆι Qs i Qs ( Rotor Spee Estimation Consier the error Equation (17 an the Lyapunov function (18 where e ψ is the matrix [e Dψ e Qψ T (Appenix A an function W must be such as to ensure convergence of the rotor spee estimation accoring to the theory of Lyapunov stability. e ψ = ΛK 1 sgn (Îs i s (17 V = e T ψ e ψ + W (18

5 Progress In Electromagnetics Research B, Vol. 34, The erivative of Lyapunov function (18 is given by V = ( e T ψ eψ + e T ψ (e ψ + W. (19 Defining a variable z as in (2, substituting Equations (17 an (2 in (19, an after performing the erivatives the result is given by (21. z = K 1 sgn (Îs i s (2 V = zt Λ T A 1 12 z+zt Λ T A 1 ω r 12 ε Jψ r + W (21 The erivative of Lyapunov function V with respect to time can be expresse as: V = V 1 + V 2 (22 Therefore, equating expressions (21 an (22 gives { V 1 = z T Λ T A 1 12 z, V 2 = z T Λ T A 1 12 ω ε Jψ r + W. (23 Accoring to the theory of Lyapunov stability, to be ensure stability, equation (22 must be negative efinite. For that, conitions (24 an (25 must be impose V 1 < (24 V 2 = (25 In orer to achieve the first conition (24, we must establish the equality of (26 where constant γ must have a positive value. Using (26 provie (24 results in (27. This equation will always be negative efinite since it is always impose positive values for γ Λ T = γa 12, γ > (26 V 1 = z T γz (27 Substituting (26 in the secon equation of (23, gives V 2 = γz T ω ε Jψ r + W. (28 Applying the conition (25 in (28 we obtain the result expresse by (29. W = γz T ω ε Jψ r (29

6 23 Costa Branco an Ferraz Thus, from (29, function W can be chosen equal to (3, in which parameter µ shoul also have a positive value Deriving (3 with respect to time, yiels W = ω 2µε, µ > (3 W = 2 ω 2µε ω Developing (29 using (2, we obtain W =γ [k 11sgn (ˆι Ds i Ds k 22 sgn (ˆι Qs i Qs ω ε Now rewriting (32, gives (31 [ [ 1 ˆψDr. 1 ˆψ Qr (32 W = γ ω [k 111 sgn (ˆι Ds i Qs ε ˆψ Qr k 222 sgn (ˆι Qs i Qs ˆψ Dr. (33 Finally, equalizing (31 an (33, we obtain the equation of the rotor spee estimator efine by ˆω [ = µγ k 11 sgn (ˆι Ds i Ds ˆψ Qr k 22 sgn(ˆι Qs i Qs ˆψ Dr (34 Accoring to Equations (26 an (3, positive parameters µ an γ in (34 have to be chosen in orer to optimize the performance of the rotor spee estimator. At this time, there is no way to attribute their value using an analytical equation. Therefore, their values must be attribute by trial-an-error, being tune for a stable an fast sliing observer. Their influence is istinct because, while parameter µ only irectly influences the progress of the estimate spee, parameter γ irectly affects not only the estimate spee, but also the stator current an rotor flux estimates, as parameter γ is inclue in matrix L. Notice that, in general, the increase of these parameters results in a faster transient response of the observer. However, this increase reuces the robustness of the observer when in stationary regime, showing some oscillations in the estimate spee as will be iscusse in the next section. 4. IMPLEMENTATION OF THE ADAPTIVE SLIDING-MODE OBSERVER: EXPERIMENTAL RESULTS The experimental setup use for our sensitivity stuy is shown in Fig. 1(a. It consists in an electrical rive system with a 37 W cage

7 Progress In Electromagnetics Research B, Vol. 34, Loa resistance DC generator Inuction motor DSP TMS32F2835 Braking resistance IGBT inverter (a is Flux Estimator ^ Ψr i s Current Estimator ^i s ^ ω ^ Ψr i s Spee Estimator ^i s ^ ω Estimate Spee (b Figure 1. (a DSP-base inuction motor rive system use to verify the parameter sensitivity of the moifie aaptive sliing-moe observer. The experimental setup consists in an inuction motor, an IGBT inverter, a DC generator use as loaing machine, an a DSP (TMS32F2835 boar. (b Block iagram of the sliing-moe observer implemente in the DSP. inuction motor (parameters liste in Table 1, an IGBT inverter controlle by a Texas Instruments DSP TMS32F2835 using a SVPWM (Space Vector PWM algorithm, an a DC generator use as motor loa. The braking resistance is use by the inuction motor rive when braking. The IGBT inverter inverts the electric power irection an this is issipate in the braking resistor to ecrease the motor spee. The inuction motor is equippe with a spee sensor (incremental encoer an also has two current sensors (Hall Effect sensors. Fig. 1(b shows in a block iagram the observer structure, which was implemente in the DSP.

8 232 Costa Branco an Ferraz Table 1. motor. Parameters of the 37 W one-pole Pair 5-Hz inuction Rate power 37 W Rate voltage 4 V Rate torque 1.3 Nm Rate spee 282 rpm Rate current 1.7 A Magnetizing inuctance M 1.46 H Stator inuctance L s 1.48 H Rotor inuctance L r 1.48 H Stator resistance R s 16.1 Ω Rotor resistance R r 24.6 Ω Total moment of inertia.35 kgm The rive system topology consists in an inirect vector spee control of the inuction motor. Generally, this control works best for rives that use a spee sensor to measure the rotor spee. In sensorless moe, the spee information is not available by irect measurement an in our case it must be estimate by the moifie sliing-moe observer. The scheme of the implemente sensorless inuction motor control use in this research is shown in Fig. 2. The controller an observer were all programme in the DSP boar together with the SVPWM current control for the IGBT inverter. The influence of constants values an the motor parameters eviation in the aaptive observer performance was extensively teste using this experimental setup Stator Currents an Rotor Flux Estimation In this section, results concerning the estimation of the components of the stator currents (ˆι Ds, ˆι Qs an rotor flux ( ˆψ Dr, ˆψ Qr are presente both in steay-state an in transient regime. In this experiment, it was not use any spee sensor with the inuction motor rive. The observer uses the reference voltages u Ds an u Qs, employe in the inirect fiel oriente control, an uses the currents i Ds an i Qs obtaine from the transformation abc/dq of the measure stator currents i a an i b.

9 Progress In Electromagnetics Research B, Vol. 34, ω ref ωˆ PI spee controller i qs_ ref i s_ ref PI current controllers u qs _ ref u s _ ref q abc u a_ref u b_ref u c_ref S V P W M Vc i a i b i c Spee sensor compensav Slip o compensation e escorregamento M r RΨr ωˆ Aaptive i Qs sliing-moe spee observer i Ds i s i qs q αβ i Ds i Qs DQ abc ω Figure 2. Block iagram of sensorless inirect vector spee control of the inuction motor. Is D component (measure/estimate [A i Ds î Ds Time [s (a Is Q component (measure/estimate [A i Qs î Qs Time [s (b Figure 3. (a The D components of the measure i Ds an estimate î Ds stator currents. (b The Q components of the measure i Qs an estimate ˆι Qs stator currents Stator Currents Figure 3(a shows the estimate current ˆι Ds. Fig. 3(b shows the estimate current ˆι Qs. As can be verifie in Figs. 3(a an 3(b the estimate currents using the observer follow the measure one with low significant error. All results were obtaine in low spee values, with a reference spee of 9 rpm following to 3 rpm. Figure 4(a shows the measure current i Ds an the estimate current ˆι Ds when the rotor spee is equal to 45 rpm. The current components i Qs an ˆι Qs are not shown since they have similar trens.

10 234 Costa Branco an Ferraz Is D component (measure/estimate [A i Ds î Ds Time [s (a Is D component (measure/estimate [A 1.5 î Ds Time [s (b i Ds Figure 4. Measure current i Ds an estimate current ˆι Ds for a (a rotor spee equal to 45 rpm an (b for a spee equal to 15 rpm. At last Fig. 4(b shows for a higher rotor spee equal to 15 rpm the measure i Ds an the estimate current ˆι Ds. Both results show the goo performance of the implemente observer in a wie spee range Rotor Flux The experimental results regaring the estimation of rotor flux DQ components were obtaine for a low spee value of 3 rpm (Fig. 5(a, an average spee of 45 rpm (Fig. 5(b, an a high spee of 15 rpm (Fig. 5(c. We verify in all three results that the DQ estimate rotor flux components have equal amplitues an a ephasing angle of 9 as expecte. Furthermore, each component has a frequency relate to the rotor spee Performance of the Observer When in the Sensorless Inirect Fiel Oriente Controller This section presents the performance of the inirect fiel oriente control without using the spee sensor, but using the aaptive observer an its estimate rotor spee. Results were obtaine for various spee reference profiles. Because the evolution of the real spee was very near to the estimate one, figures only show the estimate rotor spee. Figure 6(a shows the spee reference ω ref which is characterize by a ramp spee from to +75 rpm in 5 secons, an then for a constant spee equal to +75 rpm. This figure also presents the evolution of the estimate spee ˆω. From the results obtaine, it appears that for the first 2 secons the spee controller works to bring the estimate spee to the reference one. Here is also foun that the elay was ue to the convergence process inherent to the spee observer. In the following instants, the estimate spee follows the

11 Progress In Electromagnetics Research B, Vol. 34, Rotor flux [Wb ψˆ Dr ψˆ Qr Rotor flux [Wb ψˆ Dr ψˆ Qr Time [s (a Time [s (b Rotor flux [Wb ψˆ Dr ψˆ Qr Time [s (c Figure 5. Rotor flux estimate components ˆψ Dr an ˆψ Qr for a spee equal to (a 3 rpm, (b 45 rpm, an (c 15 rpm, using the moifie observer. spee reference with very goo approximation, showing an average error of about +3% to reach the steaystate. Follow, the average error stays significantly lower in +1%. A secon test was carrie out but being opposite to that anterior. It uses now a reference spee ramp with a negative slope an a steaystate spee equal to 75 rpm, as shown in Fig. 6(b. The evolution of the estimate spee is similar to the anterior results in Fig. 6(a. Figure 6(c shows the results obtaine when a pyrami type reference is applie to the spee controller. After the initial secons that are associate with the control system an the observer convergence, the error between the reference an estimate spee ecrease to a value aroun 2%. This confirms the goo performance of the sensorless spee control implemente An interesting test was performe which require a motor acceleration from to 75 rpm, maintaining this spee uring 2 secons, an finally braking to rpm. Fig. 6( shows the spee reference signal an the evolution obtaine by the estimate spee. During acceleration an eceleration, the error shows an average value of +3%. However, uring the two secons of steay state, the

12 236 Costa Branco an Ferraz Motor spee [rpm Motor spee [rpm ω ref ωˆ 75 rpm Time [s ω ref ωˆ (a Time [s (c 75 rpm Motor spee [rpm Motor spee [rpm ωref -75 rpm Time [s ωˆ ωˆ ω ref (b Time [s ( -75 rpm Figure 6. Inirect fiel oriente control without spee sensor but using the spee estimate by the moifie observer. (a Spee reference ω ref in ramp with positive slope an the estimate spee ˆω. (b Spee reference ω ref in ramp with negative slope an the estimate spee ˆω. (c Spee reference ω ref in pyrami form an estimate spee ˆω. ( Reference spee ω ref accelerates from rpm to 75 rpm, maintaine for 2 secons an then slows up again at rpm, an estimate spee ˆω. error rops significantly to a mean value less than +1%, showing the excellent performance of inirect fiel-oriente spee sensorless. 5. PARAMETER SENSITIVITY OF THE ADAPTIVE OBSERVER In orer to ientify the aaptive sliing-moe observer parameters that have the most impact on its performance, a parameter sensitivity analysis was performe. Experimental results in this section analyze the sensitivity to the observer constants an also motor parameters variation.

13 Progress In Electromagnetics Research B, Vol. 34, Observer Sensitivity Uner its Constants The set of parameters of the observer, whose values were assigne, is liste in Table 2. Since most parameters are assigne by trial-anerror, it is important to check the sensitivity of each parameter in orer to etermine which ones are the most significant an also efining what shoul be the first parameter to have assigne a value. Figure 7(a shows the results for parameter µ. Note that this parameter irectly influences the evolution of estimate spee ˆω, as can be seen from Eq. (34. The results in Fig. 7(a show that this parameter affects the convergence time of the estimate spee to the measure one. Taking as reference the results obtaine with µ =, it appears that when parameter µ has its value increase by 5% ( µ = +5% the evolution of estimate spee to the measure one becomes more fast. However, we observe that in steay-state, the estimate spee starte to show significant fluctuations aroun the value of measure spee. In the opposite conition, where parameter µ ha its value reuce by 5% ( µ = 5%, the convergence time became greater, as shown in Fig. 7(a. However, the estimate spee starte to show a steaystate without significant fluctuations Figure 7(b shows the variation of parameter γ, which is also irectly use in Eq. (34 to obtain the estimate spee. The results are similar to those presente with parameter µ, also showing the great sensitivity of the estimator for parameter γ. The last parameters teste were k 1, k 2, inclue in the iagonal Table 2. Constants use in the aaptive observer. k k l l l l µ γ.19 ε H q.65

14 238 Costa Branco an Ferraz matrix equations K 1 an K 1m present in observer Equation (12. Fig. 7(c shows the results when the parameters change. For a variation of 5% ( k 1 = k 2 = 5%, the observer ynamics becomes more slow in its convergence to the measure spee, with a small oscillatory behavior in its steay-state after reaching the measure spee. For an opposite variation of +5% ( k 1 = k 1m = k 2 = k 2m = +5%, the observer ynamics becomes faster, acquiring however a significant oscillatory behavior after reaching the measure spee, as shown in Fig. 7(c. For parameters l 11, l 22, l 12, l 21 of matrix L in (12, which are inclue in the equations for estimating the rotor flux, an parameter q, the tests performe with both positive an negative variations of these parameters le to the conclusion that a variation of the same oes not significantly affect the ynamics of the observer. For this reason the results were not inclue. 6 6 Motor spee [rpm µ = µ = 5% µ =+5% Motor spee [rpm γ= γ= 5% γ=+5% Time [s (a Time [s (b 6 Motor spee [rpm k1, m= k2, m= 5% k1, m= k2, m= + 5% 1 k1, m = k2, m = Time [s (c Figure 7. (a Parameter µ. The measure spee ω (black an estimate ˆω (color with ifferent levels of gray for the following changes in the value of µ µ =, µ = +5%, an µ = 5%. (b Parameter γ. The measure spee ω (black an estimate ˆω (color with ifferent levels of gray for the following changes in the value of γ γ =, γ = +5%, e γ = 5%. (c Parameters k 11, an k 22, from matrix equations K 1. The measure spee ω (black an estimate ˆω (color with ifferent levels of gray for the following changes: ( k 11 = k 22 = +5%.

15 Progress In Electromagnetics Research B, Vol. 34, Motor spee [rpm R r = R r =+25% R r =+5% Time [s Motor spee [rpm 1 (a 8 R s =+ 2% R s = R s =+ 1% Motor spee [rpm R r = 25% L r = L r = 1% Time [s (b Time [s (c Figure 8. (a Rotor resistance R r variation. The measure spee ω (black an estimate ˆω (color with ifferent gray levels for the following R r increase: R r =, R r = +25%, e R r = +5%. (b Inuction rotor coefficient L r variation. The measure spee ω (black an estimate ˆω (color with ifferent gray levels for the following L r ecrease: L r =, L r = 1%, e L r = 25%. (c Rotor resistance R s variation. The measure spee ω (black an estimate ˆω (color with ifferent gray levels for the following R s increase: R s =, R s = +1%, e R s = +2% Observer Sensitivity Uner Motor Parameters Variation This section evaluates the performance of the observer uner changes in the inuction motor parameters. Experimental results are shown for variations in the values of rotor resistance R r, rotor inuctance L r, an the stator resistance R s. For rotor resistance, its variation was only contemplate for an increase of its value cause by a possible temperature augmentation. Fig. 8(a shows the experimental curves of the estimate spee. The results show that as the rotor resistance increases, the convergence of the observer to measure spee becomes slower. Moreover, in steay-state, the estimator behaves increasingly oscillatory for a rotor resistance increase. To stuy the behavior of the observer for an L r variation, it was consiere only a reuction of its value since it is associate with the

16 24 Costa Branco an Ferraz motor operating with magnetic saturation. There were establishe two ecreases in this parameter: 1% an 25% of its initial value. The curves of the estimate spee in Fig. 8(b show not only a consierable ecrease in the convergence time of the observer for ecreasing of the rotor inuction coefficient, but also an excessively oscillatory behavior aroun the measure spee. Next, we analyze the observer sensitivity when there was an increase in the value of the stator resistance R s. We consiere two increases: +1% an +2%. Fig. 8(c shows the results. In general, one can say that the observer is insensitive to a stator resistance variation, either in their time of convergence as in steay-state. 6. CONCLUSION Parameter variations in an inuction motor rive uner sensorless operation base on the aaptive sliing-moe observer inuce flux eviations from the comman value an oscillations uring ynamic operation. More significant is the observer sensitivity to the values attribute by the user to the constants use by the observer. Base on the experimental verification of the observer parameter sensitivity, automatic tuning methos can be applie to change the observer constants an aapt it to ifferent spee an also acceleration levels. APPENDIX A. Defining ω = ˆω ω, an rewriting it as ˆω = ω+ ω, it is easily verifie that  = A + A, which A arises ue to the ifference between estimate an measure spee. This new matrix is then constructe as A =  A. Observing the elements of A, matrices A 11 an A 21 o not epen on ω an, since its elements are all constants, we conclue that these are equal to Â11 an Â21, respectively, since A 11 = A 21 =. Matrix Â12 is ecompose as: [ [ τ 1  12 = β r pˆω τ pˆω τr 1 = β r 1 p(ω + ω p(ω + ω τr 1 [ [ τ 1 = β r pω p ω β, (A1 pω p ω τ 1 r concluing that, as expecte, Â12 = A 12 + A 12, where [ p ω A 12 = β = βp ωj. (A2 p ω

17 Progress In Electromagnetics Research B, Vol. 34, Matrix Â22 can be also ecompose as [ [ τ 1 Â 22 = r pˆω τ pˆω τ 1 = r 1 r p (ω + ω [ [ τ 1 = r pω p ω + pω p ω τ 1 r p(ω + ω τ 1 r, (A3 where Â22 = A 22 + A 22 an thus A 22 yiels [ p ω A 22 = = p ωj (A4 p ω At last, the matrix A becomes [ [ A11 A A = 12 βp ωj = A 21 A 22 p ωj βp ω βp ω = p ω p ω (A5 Subtracting (6 from (12 results in Îs i s =A 11 (Îs i s +Â12 ˆψ r A 12 ψ r +K 1 sgn (Îs i s, (A6 ˆψ r ψ r =A 21 (Îs i s +Â22 ˆψ r A 22 ψ r +K 2 sgn (Îs i s. (A7 Defining the stator current an rotor flux errors as e i = Îs i s (A8 e ψ = ˆψ r ψ r (A9 an using them in (A6 an (A7, results in the following erivative error equations: e i =A 11 e i +A 12 e ψ + A 11 Î s + A 12 ˆψr +K 1 sgn (Îs i s, (A1 e ψ =A 21 e i + Â22 ˆψ r A 22 ψ r + K 2 sgn (Îs i s. (A11 To implement the sliing-moe ynamics, we have the conitions in (31. e i = e i = Using this conition in (A1 an (A11, results in = A 12 e ψ + A 11 Î s + A 12 ˆψr + K 1 sgn (Îs i s e ψ = A 22 e ψ + A 21 Î s + A 22 ˆψr + K 2 sgn (Îs i s (A12 (A13 (A14

18 242 Costa Branco an Ferraz Since A 11 = A 21 =, an using (A2 an (A4 in (A13 an (A14 yiels = A 12 e ψ + β ωj ˆψ r + K 1 sgn (Îs i s (A15 e ψ = A 22 e ψ ωj ˆψ r + K 2 sgn (Îs i s (A16 Now, (A15 an (A16 can be rewritten using DQ coorinates as [ [ [ [ [ τ 1 = β r pω edψ p ω ˆψDr pω τr 1 +β e Qψ p ω ˆψ Qr +K 1 sgn (Îs i s (A17 [ [ [ [ [ edψ τ 1 = r pω edψ p ω ˆψDr e Qψ pω τ 1 r e Qψ p ω ˆψ Qr +K 2 sgn (Îs i s (A18 Expaning (A17 an (A18 gives e Dψ e Qψ =βτr 1 e Dψ +βpωe Qψ βp ω ˆψ Qr + k 11 sgn (ˆι Ds i Ds, (A19 = βpωe Dψ +βτr 1 e Qψ +βp ω ˆψ Dr +k 11 sgn (ˆι Qs i Qs (A2 = τ 1 r e Dψ pωe Dψ + p ω ˆψ Qr + k 22 sgn (ˆι Ds i Ds, (A21 =pωe Dψ τ 1 r e Qψ p ω ˆψ Dr +k 22 sgn (ˆι Qs i Qs. (A22 Diviing expressions (A19 an (A2 by β, an summing them with (A21 an (A22 yiels ( e Dψ k11 = β + k 22 sgn (ˆι Ds i Ds, (A23 ( e Qψ k11 = β + k 22 sgn (ˆι Qs i Qs. (A24 Therefore, one verifies that (A23 an (A24 can be rewritten as [ ( edψ K1 = e Qψ β + K 2 sgn (Îs i s. (A25 Since K 2 = LK 1, (A25 can be written as [ ( edψ I = e Qψ β L K 1 sgn (Îs i s. (A26

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