ISSN 10683712, Russian Electrical Engineering, 2011, Vol. 82, No. 6, pp. 319–323. © Allerton Press, Inc., 2011. Original Russian Text © V.V. Pankratov, V.V. Vdovin, G.G. Sitnikov, S.S. Domanov, 2011, published in Elektrotekhnika, 2011, No. 6, pp. 42–46.

Globally Stable Adaptive Observer for Systems of GeneralPurpose Industrial Asynchronous Electric Drives V. V. Pankratov, V. V. Vdovin, G. G. Sitnikov, and S. S. Domanov Received May 23, 2011

Abstract—A structure and methods for the parametric synthesis of an adaptive observer that is formally stable in the entire plane of mechanical characteristics of the electric drive and enables one to identify the electric rotation frequency of the rotor, the support vector of flux linkage, and the active stator resistance are described. Results of modeling a generalpurpose industrial electric drive with the developed identification algorithm are presented. Keywords: Asynchronous electric drive, sensorless vector control, adaptive observer, parametrical adaptation. DOI: 10.3103/S1068371211060101

The implementation of a controlled electric drive built mainly based on asynchronous motors (AMs) with shortcircuit rotors is a common way of solving energy supply problems in industry, housing and community services, and transportation. A largescale, general purpose, industrial asynchronous electric drive (AED) must have high reliability and a relatively low price, and the modernization of the machines and mechanisms must not require serious changes in their construction. Socalled sensorless EDs, which only require informa tion on the internal electrical values of a frequency con verter connected to the motor by a power cable meets all of these requirements to a great exten. The best characteristics of sensorless AEDs are achieved in systems with vector control of an AM, the development of which requires one to indirectly deter mine information on the online values of nonmeasur able coordinates of the motor state, i.e., the support vector of flux linkage and electric rotation frequency of rotor. This problem is solved using the algorithm of the online identification of coordinates (observation). At present, passiveidentification algorithms based on adaptive models of the electromagnetic process in an AM are the most theoretically proven; however, the algorithms of this class described in the literature are not applicable at low frequencies of the stator supply voltage [2], which, in particular, restricts the range of AED control in breaking mode. Moreover, all algo rithms for identifying AM state coordinates are more or less sensitive to the deviations of machine parame ters from their calculated values [3]. Thus, as a rule, the preliminary identification of AM parameters is carried out during adjustments or directly before each start of the EAD. Furthermore, in practice, the tem perature drift of the active stator and rotor resistances appears to be the most critical parametrical distur bance. If the deviation of the active rotor resistance

(resulting in time constant deviation) commonly causes static errors in speed control, then the drift of the stator winding resistance may lead to the unstable operation of sensorless AEDs at low speeds. The present paper describes a structure and tech nique for the parametrical synthesis of a fullorder adaptive observer that is formally stable in the entire range of AED mechanical characteristics and enables one to identify the electric rotation frequency of the rotor, the support vector of the flux linkage, and the active AM stator resistance. Some results of AED modeling with the developed algorithm of identifica tion are presented. STRUCTURAL SYNTHESIS OF ALGORITHM The mathematical model of the observed object, i.e., the electromagnetic processes in an AM in fixed Cartesian coordinates, may be written as follows [2]: ⎧ x· = Ax + BU s ; (1) ⎨ ⎩ y = Cx , where x = [isαisβψrαψrβ]T is the vector of the object state coordinates, i.e., the stator and rotor currents and rotor flux linkage; Us = [UsαUsβ]T is the vector of the control input, i.e., the stator voltages; C = [E 0]T is the output matrix; y = Cx = Is = [isαisβ]T is the column vec tor of the directly measured variables; A is the inherent matrix of the object

319

2

R kr k Rr kr –  E – s   E – rω e D L L L T L σe σe σe r σe ; A = Lm 1  E –  E + ω e D Tr Tr

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is the control matrix; E = 1 0 is the 01

1 B =  E 0 L σe

unit matrix; D = 0 – 1 is the matrix of vector rota 1 0 tion by an angle of π/2; Rr, Rs, kr, Lσe, Tr, Lm are the parameters of the motor; and ωe is the electric rotation frequency of the rotor. The mathematical model of the fullorder observer (FOO) for object (1) has the following form [2]: ˆ xˆ + BU + L ( yˆ – y ); ⎧ xˆ = A s (2) ⎨ ⎩ yˆ = Cxˆ , where ^ is an estimator of the corresponding values, L = [Li Lψ]T is the matrix of the stabilizing additive, dimL = 4 × 2, and 2 ˆ kr k ˆ Rr kr R –  E – sE   E – rω eD ˆ = L σe L σe L σe T r L σe . A Lm 1 ˆ eD  E –  E + ω Tr Tr

Subtracting object equation (1) from observer equation (2) yields the following equation of observer dynamics in deviations: ε· = ( A + LC )ε + A τ τ, where ε = xˆ – x is the vector of observer errors, τ = ˆe – [δρ]T is the vector of parameter deviations, δ = ω ωe is the deviation of the electric rotation frequency of ˆ – R is the deviation of the the motor rotor, ρ = R s s active stator resistance, Aτ is the matrix of influence caused by parameter deviation vector on the error vec tor, and Aτ =

k 1 – rDΨ r – I s L σe . L σe 0

DΨ r

According to the method of Krasovskii [4], the synthe sis of the FOO adaptation algorithm requires one to introduce a quadratic function as a candidate to the Lyapunov function V = ε δ

T

Hε 0 0 Hδ

ε , δ

where Hε, Hδ are symmetrical, positively defined matrixes of the weights and matrices of weights as follows: 2

Hε =

2

–1

2

hi E h E 2

h E Hψ E

,

Hτ

λδ

0 –1

0 λρ

,

where hi, h, hψ, λδ, λρ are some positive constants. To achieve the negative definiteness of the full derivative of the Lyapunov function and implement the adaptation algorithm in terms of a sensorless elec tric drive, we suggest to use following relations for weights and the stabilizing additive, which have not been used before: 2 R L 2 2 L σe⎞ ⎛ h i = h ψ ⎛   1 + s r⎞ ; ⎝ kr ⎠ ⎝ R r L σe⎠

2 2 k h ψ = h r; L σe

ˆ L ˆ eDR L = –ω s r . R r L σe At the same time, the laws of FOO adaptation to the electric rotation frequency of the rotor and active stator resistance are written as follows: ˆ dt + k ε T DΨ ˆ , ˆ = k ε T DΨ ω e

P

∫

r

i

r

I i

R S = k ρ ε i ˆI s dt + R s0 ,

∫

T

where kP and kI are the transition factors of the pro portional and integral parts of the controller, i.e., the generator of the electric rotation frequency estimator; Rs0 is the initial approximation of the estimator of the active stator resistance; kρ is the transition factor of the estimate of the controller of the active resistance of the stator in an AM. PARAMETRICAL SYNTHESIS OF IDENTIFIER OF ELECTRIC ROTATION FREQUENCY OF AM In our case, the direct Lyapunov method allows one to obtain the structure of the adaptation law, but not the numerical values of the controller, i.e., the adaptor coefficients. The following approach to their determination is proposed. Let us consider the vector product involved in the adaptation law of the electric rotation frequency of the rotor T ˆ = ( ˆI T – I T )DΨ ˆ = ˆI ⋅ ε i DΨ r r s s s ˆ sin ( α – δ ) = ˆI ⋅ – I ⋅ ψ s

r

i

s

ˆ r sin ( α ) ψ ˆ r sin ( α ) ψ

ˆ r sin ( α ) cos ( δ i ) – Is ⋅ ψ ˆ r sin ( δ i ) cos ( α ), + ˆI s ⋅ ψ where ˆI s is the absolute value of the vector of the ˆ r is the absolute estimation of the stator current; ψ value of the vector of rotor flux linkage, I s is the absolute value of the stator current vectors, α is the angle between stator current estimator vector and rotor flux linkage estimator vector (Fig. 1), and δi is the angle between the stator current vector and vector of their estimators.

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GLOBALLY STABLE ADAPTIVE OBSERVER ψy

321

^ψy θΨ δψ

Ψr2nom

e

^ ω s ^ ω Ψ

∫

+

^ θ Ψ

Fig. 2. Block diagram of linearized circuit for identifying electric frequency of rotor rotation.

^I s δi

^ Isq Fig. 1. Relative position of vectors of state observers in motor.

Assuming that ˆI s ≈ I s and δi ≈ δψ are negligible angles, we obtain

The equation of the steadystate mode for the FOO in the coordinate system (1), (2) oriented to the vector estimator of the flux linkage of the rotor taking into consideration the stabilizing additive has the following form: ˆ ˆi – ω L ˆi – R ˆ T (ˆi – i ) ( ω – ω ˆ s ) = u s1 ; ⎧R s s1 s r s2 ψ σe s2 s2 ψ ⎪ˆ ˆ T (ˆi – i ) ˆ r) + R ⎪ R sˆi s2 + ω ψ ( L σeˆi s1 + k r ψ s r s1 s1 (3) ⎨ ˆ s ) = u s2 ; ⎪ × ( ωψ – ω ⎪ ˆ r ; L mˆi s2 – ω ˆ s Tr ψ ˆ r = 0, ⎩ L mˆi s1 = ψ ˆ r = Ψrref is the assigned rotor flux linkage and where ψ ωs is the slip frequency. The equation of the steadystate mode for the AM in the coordinate system (1), (2) oriented to the rotor flux linkage vector estimator considering stabilizing additive has the following form:

T ˆ ≈ ˆI ⋅ ψ ˆ r sin ( α ) – ˆI s ⋅ ψ ˆ r sin ( α ) ⋅ 1 ε i DΨ r s ˆ r δ i cos ( α ) = I s cos ( α ) ψ ˆ r δi + ˆI s ⋅ ψ

1 ψ ˆ r δ i =  ˆ r 2 δψ . = ˆI sd ψ Lm For an electric drive operating in the first control zone, one may assign a flux linkage with the same ˆ r ≈ ψrnom. In this case, value as the nominal value ψ the block diagram of the linearized circuit of FOO adaptation to the rotation frequency has the following form shown in Fig. 2. Adjusting the closed circuit to characteristics of the secondorder increase with the eigen frequency ΩO and shape coefficient AO, we obtain the following expression for the controller coefficients: 2 Lm k I = Ω O  ; 2 ψ rnom

Lm

^ ω

Is

α I^sd

δ + Ψ (–)

^ εTi DΨr

Lm k P = A O Ω O  . 2 ψ rnom

⎧ R s i s1 – ω ψ L σe i s2 – ω ψ k r ψ r2 = u s1 ; ⎪ ⎪ R s i s2 + ω ψ ( L σe i s1 + k r ψ r1 ) = u s2 ; ⎨ ⎪ L m i s2 + T r ω s ψ r2 = ψ r1 ; ⎪ ⎩ L m i s1 – T r ω s ψ r1 = ψ r2 .

(4)

Because the algorithm of the identification of the electric rotation frequency of the rotor reduces the product between the current error and the estimator vector of flux linkages to zero, T ˆ = ε ψ ˆ –ε ψ ˆ = ε ψ ˆ = 0. ε DΨ i

r

i2

r1

r2

i1

i2

r1

Consequently, in the steadystate mode, εi2 = 0 and i = ˆi s2 . (5) s2

SYNTHESIS OF IDENTIFIER OF ACTIVE RESISTANCE OF STATOR IN AM Because thermal processes are the main reason for the drift of the active stator resistance and their rate is much lower than that of the electromagnetic and elec tromechanical processes in an AM, the identification subsystem for the stator resistance may be adjusted to the much lower performance compared to circuits of the vector control system and rotationfrequency adapter. This allows one to neglect the response rate of the latter factors during the synthesis of the identifica tion algorithm for the active resistance of the stator. RUSSIAN ELECTRICAL ENGINEERING

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Then, the law of identifying the active resistance of the stator an be written as ˆ = R + k ε ˆi dt. R s s1 s0

ρ

∫

1

By coherently solving the systems of equations (3) and (4) with allowance for (5) relative to the slip fre quency, we obtain 2

αω s + bω s + c = 0,

(6)

where a, b, c are the coefficients that depend on the parameters of the AM, the location of the operating

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PANKRATOV et al. Rs0

Rs

–ρ + (–)

εi1 ·is1

K

∫

Kρ

+

^

Rs

0.4 0.2 0 –0.2

Fig. 3. Block diagram of linearized circuit for identifying active resistance of AM stator.

point of the AED, and the deviation of the active stator resistance ˆ – ρ) – L ρ) a ( . ) = Tr ( ωψ ( kr Lm ( R s σe ˆ ˆ ˆ ( 2R – ρ ) ); – T ρ(ω R – ω 2

r

ψ

s

s

s

ˆ T + L )ω b ( . ) = ωψ kr Lm Tr ( ωψ ( R s r s ψ ˆ ˆ ); – 2 ( R – ρ )T ω s

s

r

ˆ Tω ˆ s Ls + R ˆ s) ) c ( . ) = –kr Lm Tr ωψ ( ωψ ω s r ˆ s ( ωψ – ω ˆ – ρ ) ( ω – 2ω ˆ ) – (R ˆ ) ). – ρL ω – T ρ ( ρ ( ω – ω s

ψ

r

ψ

s

s

ωe*, o.e.

ψ

s

The value of the slip frequency is the solution of (6) as follows:

×104

δ*, o.e. 1 0 –1 –2 Is*, o.e. 1.5 1.0 0,5 0 –0.5 Is*, o.e. 1 0 –1 Us*, o.e. 0,5

isd

isq

0 –0.5 0

0.2

0.4

0.6

0.8

1.0

t,s

2

– 4a ( . )c ( .; ) b ( . ) + b ( . ) ω s ( . ) = – 2a ( . )

Fig. 4. Transients in sensorless system of vector control ωe*.

at the same time, the integrand value included in the adaptation law of the active resistance of the stator will have the following form:

lowing expression for coefficient of stator active resis tance controller:

ε i1 ( . )iˆs1

Ωρ ˆ ,ω ,ω ˆ s, ˆi s1 ) =  . kρ ( R s ψ ˆ ˆ s, ˆi s1 ) K ( R s , ω ψ, ω

ˆ s ) – ρ ( 1 + T 2r ω 2s ( . ) ) 2 ωψ kr Tr Lm ( ωs ( . ) – ω ˆ . = i s1  ˆ – ρ ) ( 1 + T 2 ω 2 ( . ) ) + ω ω ( . )k T L (R s r s ψ s r r m Thus, the expression ε i1ˆi s1 for the steadystate modes is a function of both the operating position of the point and the deviation of the active resistance of the stator. The results of calculations show that, at any operating point of the AED, the function ε i1ˆi s1 ( ρ ) can be approximated by a straight line. We suggest that the linearization of this relation be carried out according to the formula ˆ ,ω ,ω ˆ ,ω ,ω ˆ s, ˆi s1, ρ ) = K ( R ˆ s, ˆi s1 )ρ. ε i1ˆi s1 ( R s s ψ ψ The corresponding block diagram is presented in Fig. 3. Adjusting the closed loop of identifying Rs as an aperiodic link with bandwidth Ωρ, we obtain the fol

Thus, using the obtained expressions, one may undertake the continuous tuning of the active resis tance identifier of the stator, which yields the required quality of calculations of the stator resistance in the entire range of mechanical characteristics of the AED. RESULTS OF MODELING The digital modeling of the developed algorithm integrated in the system of a sensorless AED based on a 4A225M4U3 electric drive with vector control was car ried out using MATLAB Simulink software. Figure 4 presents curves of the transients of the electric rotation frequency of the rotor, current reference signals in the coordinates (d, q) oriented to the vector of rotor flux linkage estimators, threephase currents of the AM and phase voltages inputs, and the error in identifying the electric frequency of the rotor rotation. The elec tric drive operates in the following modes: motor mag netization, acceleration to a frequency of ωenom/2

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GLOBALLY STABLE ADAPTIVE OBSERVER Rs*, o.e. 1.06

adding nondiagonal elements to the matrix of the weight coefficients of the Lyapunov function and the purposeful choice of the matrix of the stabilizing addi tive providing the positive definiteness of the Lypunov function and the negative definiteness of its derivative. (2) The identifier does not contain open integrating links and uses only information on the phase points and voltages at the output of the FC and parameters of the AM equivalent circuit obtained by preliminary identification. (3) The described techniques for calculating the coefficients of adaptation laws provide the necessary quality of the transients and can be easily implemented in digital systems of ED control.

Rs*

1.04 ^

Rs*

1.02

323

1.0 0.98 0.96 0.94 0

5

10

15

20

25

30

t, s

Fig. 5. Processes of identifying active resistance of AM stator.

without loading, loading with nominal torque, and load dropping. All of the variables in Fig. 4 are pre sented in relative units. The transits of identifying the active stator resistance with the forced stepbystep deviation of the resistance by 5% in both directions and the operation of the electric drive at a frequency with the nominal load are presented in Fig. 5. CONCLUSIONS (1) The developed algorithm for identifying the state coordinates and the AM stator resistance is for mally stable at all operating points of the plane of ED mechanical characteristics. This result is achieved by

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REFERENCES 1. Hottz, J., Sensorless Control of Induction Motor Drives, Proc. IEEE, 2002, vol. 90, no. 8, pp. 1359– 1394. 2. Kubota, H., Sato, I., Tamura, Y., and Matsuse, K., Regenerating–Mode Low–Speed Operation of Sen sorless Induction Motor Drive with Adaptive Observer, IEEE Trans. Ind. Appl., 2002, vol. 38, no. 4, pp. 1081– 1086. 3. Hinkkanen, M. and Luomi, J., Parameter Sensitivity of Full–Order Flux Observers for Induction Motors, IEEE Trans. Ind. Appl., 2003, vol. 39, no. 4, pp. 1127– 1135. 4. Kim, D.P., Teoriya avtomaticheskogo upravleniya (The Theory of Automatic Control), vol. 2: Mnogomernye, nelineinye, optimal’nye i adaptivnye sistemy: Ucheb noe posobie (Multidimentional, Nonlinear, Optimal and Adaptive Systems: Student’s Book), Moscow: FIZMATLIT, 2004.

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