Reduced-order Observer With Parameter Adaption For Fast Rotor Flux Estimation In Induction Machines

Reduced-order observer with parameter adaption for fast rotor flux estimation in induction machines R. Nilsen, PhD Prof. M.P. Kazmierkowski, PhD

Indexing terms: Control equipment and applications, Algorithms, Mathematical techniques, Matrix algebra, Induction motors ~

~~

~

Abstract: In the paper a new reduced-order observer with parameter adaption is presented. The observer is based on the ‘current’ model in field-oriented co-ordinates. The theoretical principles of the developed algorithm are discussed. Some results of comparative investigations are presented, which illustrate the steady-state and dynamic properties of the speed control system with observer and conventional current models.

Main symbols

D, do

= set of parameter vectors describing the model set

fk

= frequency of a freely oriented co-ordinate system

= coupling

f,

= stator

f,

=h-n

matrix

frequency

= frequency of rotor flux vector =current k = sample number m = load torque me = machine torque n = p.u. rotational speed ne = speed limit for calculation of gain factor in the observer and parameter adaption loop R = resistor T = sampling interval T, = mechanical time constant T, = rotor time constant x = state-space vector x = reactance z = measurement vector a = winding axis of the fictitious a-winding, basis vector fl = winding axis of the fictitious /3-winding, basis vector St, = angle between real and predicted rotor flux vector St,br = difference between real and predicted rotor flux amplitude E = error vector 0 = parameter vector of unknown coefficients (dimension d x 1)

f i

Paper 6362D (C9, Pl), first received 3rd February and in revised form 11th July 1988 Dr. Nilsen is with the Norwegian Institute of Technology, Trondheim, Norway Prof. Kaimierkowski is with the Politechnika Warszawska, Instytut Sterowania i Elektroniki Przemyslowej, ul. Koszykowa 75 g.E, p. 322, 00-662 Warszawa, Poland

IEE PROCEEDINGS, Vol. 136, Pt. D , No. I , J A N U A R Y 1989

5

= rotor position with respect to as = angle of flux linkages vectors

o

= leakage coefficient

8

= rotor leakage coeficient = flux linkages = electrical angular velocity

(T,

t,b o

Subscripts e = electromagnetic g = limit value h = main k = general co-ordinate system M = model m = mechanical n =nominal r =rotor ref = reference value ra = a-component of rotor (current, vector r/3 = 8-component of rotor (current, vector s =stator sa = a-component of stator (current, vector s/3 = /3-component of stator (current, vector pr = rotor magnetising (T =leakage

voltage or flux) voltage or flux) voltage or flux) voltage or flux)

Superscripts d = dimension of parameter vector 0 k = with respect to a common freely oriented coordinate system s = stator oriented quantities, fixed in stator T = transpose t,br = rotor flux oriented quantities - 1 = inverse = estimated value - = predicted value = value calculated from measured and predicted quantities A

-

1

Introduction

Recently, advanced control strategies for inverter-fed induction motor drives have been based on field-oriented control philosophy [l, 4, 81. For realisation of such control systems, it is important to have exact information about the rotor flux space vector (i.e. amplitude and angle). Particularly, if the position of the rotor flux vector cannot be defined with high accuracy, the main idea of field orientation, the correct decomposition of the stator current space vector into flux and torque producing components, will not be satisfied. This leads to poor dynamic properties and an incorrect stationary point of operation. 35

Unfortunately, the rotor flux vector is not measurable directly and, therefore, has to be calculated from a model or a state observer. Primarily, the machine models (voltage model and current model [2, 8, 141) were used, but they have the disadvantage that operational parameter changes have an influence on the estimated rotor flux vector. Therefore, to compensate for the parameter changes, an additional parameter correction loop was applied [3, 6, 8, 131. More recently, observers have been used in the estimation of rotor flux [ 5 , 7,9, 11, 141. In this paper, a new algorithm for a reduced-order observer with parameter adaption is presented. This observer is based on the 'current' model in field-oriented co-ordinates. 2

In the model given by eqn. 7 the input quantities are vectors of stator voltage U," and current i,". The model described by eqn. 6 is called the 'current model' and that by eqn. 7 the 'voltage model'. These two types of models are those mostly used in practical implemented drive systems. Similarly, from eqns. 1-5, some other models for rotor flux estimation can be obtained [2, 8, 151. Analysis of eqns. 6 and 7 in stator, rotor or flux-oriented synchronous co-ordinate systems gives different variants of rotor flux estimators. For microcomputer-based implementation, a rotor flux-oriented co-ordinate (field co-ordinates) system is preferable [SI. In this case, see Fig. 1, =

$r

(8)

=f#r

-fk

Mathematical description of an induction machine

The mathematical description of an induction machine is based on space vectors which can be represented in coordinate system k rotating with speed [2, 81. In per unit the induction machine equations can be written as follows:

x

1 d*f uf = r, . if + -on dt

+ do f,+f Fig. 1 vectors

1 d*f + do(f k - n)+$ +wn dt 11.5 = x , . if + xh . if #f = x, . if + xh . if uf = r, . if

Representation of the rotor frux $, and stator current is space

a' - 8"= reference frame fixed to stator a+' - 8*' = reference frame fixed to the rotor flux vector $, which rotates with

speed& [P.u.] (field co-ordinates)

and

dn- - 1 (me-4

T,

dt

(4)

3

*,

The vector is described by its polar co-ordinates with respect to the stator:

*:

where

do=[;

= [$r, 5 2' (10) Therefore, the 'current' model given by eqn. 6 can be written in field co-ordinates a*' - /I* as,follows , :

-3

Conventional models for rotor flux estimation

where

5; = 5:

Assuming induction motor with squirrel cage rotor, U$ = 0; from eqns. lb and 2b, a differential equation for the rotor flux vector can be expressed as follows:

where the rotor time constant T, = x,/(r,o,). In the rotor flux model described in eqn. 6 the input quantities are speed n and stator current vector i,". From eqns. la, 2a and 2b, we obtain

=

where the total leakage factor 0 = 1 - x ~ / ( x s x , ) .

+8

cos 5; sin 5; -sin 5; cos 5;

and

1

speed of rotor flux-oriented ondt -"' = co-ordinate system

1 d5; ----d'' ondt

36

(9)

-A

= rotor

frequency

-1_d8- - n = speed of the machine shaft ondt

IEE PROCEEDINGS, Vol. 136, Pt. D, No. I , J A N U A R Y 1989

The 'current' model depends on the parameters T, and xh . However, only the time constant T, influences the calculation of The magnetising reactance x,, affects the amplitude of the rotor flux $,. Both parameters should be adapted online, xh due to saturation effects. T, due to both saturation effects and temperature changes. For the 'current' model, in field co-ordinates, the classical Euler method can be used to give a discrete form. This method guarantees acceptable accuracy for the actual sampling interval T = 1 ms:

<:.

good dynamic behaviour of the observer. In general, the gain matrix K is chosen so that the observer dynamic is faster than the dynamic of the system, including controllers [ l o ] . It is possible to design different types of observers, because an observer essentially is based on the combination of conventional models and corrective prediction error feedback. Some examples are given in References 5 and 14. 4.1 Rotor flux observer based on the 'current' model in field co-ordinates

A block scheme of the observer developed by the authors is shown in Fig. 4 . This observer is based on the discrete

py l

w[kI

The block scheme of a discrete 'current' model in field co-ordinates is shown in Fig. 2.

I

L--

Fig. 2

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - --- -

I J

I

e

A discrete 'current' model infield co-ordinates Fig. 4

4

obsever I

L

System with rotorjux observer

Reduced-order observer for rotor flux vector

When !sing a flux model only, the predicted rotor flux vector *,[k] may deviate from the system output *,[k]. Some error sources can be : (i) an incorrect initial state in the model (ii) incorrect model parameters (iii) unmeasureable inputs exciting the system. To compensate for these effects an observer can be used. In general, an observer can be described as shown in Fig. 3.

system

current model in field co-ordinates (Fig. 2), presented in Section 3. It should be noted that the transformed output vector r?f'[k], from the system, is a function of the predicted state-space vector $ y [ k ] and that g ( ., .) is a function of C [ k ] . The measurement vector is, then

cm1,

(14)

rCk1 = iXk1, nCkllT The predicted state vector is chosen as

(15) i [ k ] = $f'[kl = [$rCkl, CCkllT Such an observer is called the 'reduced-order observer', because it does not attempt to estimate all the state variables in a model of the system (not stator current is). The prediction error for the observer shown in Fig. 4 is given as

e[k] = i i r [ k ] - ii%[k] (16) The predicted output i i p [ k ] can, because the rotor flux model is oriented along the predicted rotor flux, be written as $[k]

= ?$p[k]

+ WnT [ i r [ k ]

-

i r [ k - 111

+f*,[k]jZ,d,ip[k] Fig. 3

System with observer

The prediction error e[k] is used to add a correction term K e [ k ] in the difference equation. This correction term is different from zero, as long as the prediction error is nonzero. Usually, to obtain small prediction errors in the steady state, large gain factors K are required. However, the gain factors have to be restricted to give IEE PROCEEDINGS, Vol. 136, Pt. D, No. I , J A N U A R Y 1989

+ ii$;[k]

(17)

where the magnetising voltage is

the transformed quantities used in the observer are iip[k]

=

T::<[k]

ip[k]

=

Tt:i:[k]

(19) 37

The prediction error may also be interpreted as the difference between magnetising voltages, where the measured magnetising voltage is ii$[k] = i i r [ k ] - i , i r [ k ] - x ,

where f, =f & . Assuming ?, = r, , 2, = x, and 8, = a, we obtain

WnT

x [i?[k] - ir[k- l ] ] - f * r [ k ] 2 , d o i r [ k ]

(20) where it is assumed that ii$@ = 0 in the steady state. From Fig. 5, it can be seen that c1 should be used for correction of 5; while e2 can be _used for correcting the amplitude of predicted rotor flux $ r . The gain coefficient matrix K is given as

=[E: ::1 \US

Fig. 5

-stator

Incorrect orientation of the predicted rotorflux

If the observer model parameters 8,]' differ from the real one: @ =

[:

- 9

xh,

r s , ar

r

6 = [1/2,i h ,

%,,

?,

(21)

the orientation of the predicted system is incorrect. From Fig. 5, we can see the connections between 'real' and 'predicted' field-oriented quantities : C"Ck1 = Tss(dtr)i!"[k] iir[k] where

sg,

=

T,~'(dC,)#"[k]

= 5; -

= 5; -

Before choosing the gain coefficient matrix K , it should be noted that, for incorrect parameter values, the observer will always give some transient errors and sometimes even errors in the steady state. If the observer is ,very sensitive to incorrect model parameters, these parameters should be adapted online. In inverter-fed induction motor systems, however, the calculation time needed for implementation of an observer and parameter adaption algorithm may be too large compared with the sampling time needed for the controllers. Therefore, to reduce the calculation time needed, a simplified gain matrix can be applied. In this case, the gain coefficients are chosen as

(224 (22b)

(23)

is the orientation angle error.

+

1 8r k21 = - f;.I k3

k22 = 0 The correction term in the observer (assuming i s ,2, and 8, are correct) would be :

4.2 Choice of gain coefficient matrix K It can be made by help of investigation of the prediction error ~ [ k which, ] for the steady state, may be expressed as

KE=[

]

0 k 3 $ , sindt, .

The model rotor frequencyi is then given by

I

I I I I

"rei

I

I *I

I

I I I

'Ldiscrete - _ _ _ -controller - - _ - - with - _ _observer _ _ _ -/model -_____ Fig. 6

38

I -1

The simulated system IEE PROCEEDINGS, Vol. 136, Pt. D, No. I , J A N U A R Y 1989

This means that the observer will only besorrecting for the angle error St,, by help of adjusting f,. The amplitude may, however, be corrected by help of Ah-adaption. For calculation of the factor kll the speed n is used 1

1 2 ?

08-

a

.

a, U 7

c

0.4

O

o

o 00

02

v 06

04

/

00 0 0

00

02

04

0 6

08

06

08

time, s

lime, s

0

0

O

’I U

e

0 8

..3 x L

0

c

2 04

”-

0

-0, C, 0 0

0.0

t

00 0.0

02

06

04

08

time,s b

Fig. 7

Initial valuefor S<,[O] = x / 4

a Amplitude $, b Error angle Sc, ....... model

02

Or, t1me.s b

Fig. 8 Initial oalue for $,CO] = 4/3$,[0] a Amplitude $, b Error angle 65, ....... model observer ~

__ observer

instead of-&, because of large transients in&r. To avoid incorrect sign of k21,the gain factor is set to zero when the absolute value of the speed is below n, (in P.u.). The gain factor kll is thus chosen as

Because of the nonlinearity of the system, the coefficient k, is found by help of simulation. 5

Comparative investigations of the observer and current’ model

The block scheme of the simulated field-oriented control system is shown in Fig. 6. The inner current control loops are each represented with a first-order system. The current-controlled inverter and induction machine are simulated with a time step h = 10 (in ps), while the discrete controllers and observer (or current model) have the sampling time T = 1.0 (in ms). The controller design is based on the criterion of symmetrical optimum [Z, 81. The investigation is performed for different values of the gain factor k, in the observer, but, in this paper, only IEE PROCEEDINGS, Vol. 136, Pt. D, No. I , J A N U A R Y I989

shown in Figs. 7 and 8. It can be seen that the convergence time constant of the observer is considerably smaller than for the current model alone. In Fig. 9 the dynamic behaviour of the control system with the current model and observer is shown, when all the parameters are correct. The behaviour in both cases is identical. Results from a worst-case study is shown in Figs. 10 and 11 for the _current model and for the observer, respectively, where T, = 0.5 T, and )2h = 1.5 xh. It can be seen that, if choosing an appropriate gain factor, the behaviour of the observer is preferable to the current model. In simulation with observer presented in Fig. 11, the correction term in the observer is kept constant when entering the speed region I n I n, . When comparing the steady-state errors for incorrect parameters, the observer was to be preferred instead of the current model. Because of the applied choice of gain factors in the observer (simplified algorithm), error in predicted amplitude $, is not compensated for. This disadvantage can be compensated for, if 2, is adapted online. The sensitivity for error in 2h may, however, be reduced for other choices of gain factors. Generally, however, in spite of better stationary and dynamical behaviour of the observer than the current model, errors occur for incorrect parameters. Therefore appropriate

-=

39

and

parameter adaption is necessary. Both be adapted online. 6

f h

should

three basic problems should be considered : criterion function, search direction and gain sequence. 6.1 Choice of criterion function In parameter identification algorithms for induction motors, mostly quadratic criterion functions are used.

Reduced-order observer with parameter adaption

When designing a recursive online identification algorithm based on the reduced-order observer, the following

o

4

p

,

~

-

~

~

-

00 I

-0 4

-n

- - - - - - --

- - - - - - -1

1-

-1 O F

V'rref

08

n'k

08

04

00

-0 8

,

6

-

00

F

L-.

r

.

-

r

.

-6kr - e

:

-1

ot IO

00

20

time, s

Fig. 9

Current model:

T = T, andi, = x h

Similar for the observer

20r

6

-

a:

-0 4 -0 8 :

-65,

0000

IO

2 0

(334

time,s

Fig. 10

40

Current model:

2 = 0.5 T, and i, = 1.5 x, IEE PROCEEDINGS, Vol. 136, Pt. D, No. I , J A N U A R Y I989

n

~

This can be seen directly from the eqns. 25 and 29 and Fig. 5. 6.3 Choice of gain factors matrix For adapting the parameter vector 0, a discretised integrator is applied: ~

6[k]

=

[ 6 [ k - 11 + L[k]&[k]]D,

(34)

' 0O0 0 L

-1001 L

.

-08-04

Fig. 12

.

I

.

00 0 4

I

.

08

.

-04-02

.

[

.

.

00 0 2 0 4

PU

PU

0

b

The gain matrix L[k] is chosen as a diagonal matrix, based on physical interpretation :

[L'glkl

(35) L[k] = Lz;k]] The prediction errors are proportional with the frequency fJlr (eqn. 25). Therefore, to give frequency-independent correction terms, the gain factors should be inversely proportional to this frequency. To avoid rapid tranients in the gain factors, motor speed n i's used instead of fJlr when calculating L , and L, : L , [k] = - is^ func (T$)n inv (n)k (364 L,[k] = n inv (n)k, (36b) The gain functions are shown in Fig. 12. 6.4 Choice of initial values For the observer with parameter adaption, the initial values of X and 6 have to be chosen, i.e.

Gainfunctions of parameter adaption loop

a n ~ n (n) v

(374

b rsp func (i$)

During magnetising of the machine to $, angle 5; is kept equal to zero :

$,Col

Lz/

'

40

'

50

'

$0,' 7'0

'

8'0

'

90

1 / T, , s

Fig. 13

Parameter domain D ,

-4 system

=

the

(384

= $,ref

e=0 (38b) The choice of initial values for the parameter vector 6 should be based on prior knowledge of the system. Good initial values for the user can be calculated based on machine parameters given by the manufacturer. The manufacturer, however, may use initial values based on physical interpretation, i.e. modelling. The machine data may also be based on the classical machine tests. From these classical tests, the set D, is defined. Th_e parameter ?,, is assumed to be witkin (1.5, 3.0), while T, should be in the range 0.75 T,, < T, < 1.5 T o ,see Fig. 13. The time constant T,, is the rotor time constant at rated load and temperature of 75°C. 6.5 Total algorithm for rotor flux estimation The total recursive algorithm for estimation of the rotor flux , with the help_ of a reduced-order observer with online adaption of (l/T,) and 2,,,can be expressed as

(electromagnetic p a r t )

+,

ii?[k]

= z(4[k],

@[k]

= g(X[k], C[k],

A

6 [kl

E

[kl

I

__ -1

I

I

-

- -- - - - -

-

-- -

i[k

X[k])

(394

6 [ k - 13)

(39W

&[k] = UP[k] - ii?[k]

(394

LCkl

=

L(nCk1)

(394

@[k]

=

[ 6 [ k - 11 -k L(n[k])E[k]]~,

+ 11 =.mCkl,

(394

m1,nCkl, ml) (39f)

Fig. 14

Reduced-order observer with parameter adaption

IEE PROCEEDINGS, Vol. 136, Pt. D, No. I , J A N U A R Y I989

The function z(.,.) performs the transformation between stator oriented co-ordinate system and the co-ordinate system oriented along the predicted rotor flux vector

+,,

41

while g(.,.,.)is the function given by eqn. 17. The function f(.,.,.,.) is the flux model in field co-ordinates shown in Fig. 2. A block scheme of the algorithm is shown in Fig. 14. 7

spite of magnetic saturation effects, which makes the method superior to those presented in References 8 and 13. This is of major importance for large power machines,

Simulation results

The effect of introducing parameter adaption is clearly shown in Fig. 15. Until t = 1.1 (in s), where the adaption "ref

---

-

:

-0.8 O

i

-

A

-1 0

O

.

-.--.-,

'

.

"

'

*

65r

18.0

8

65r '

4.01

Qrref

2 ,

,

.

.

.

,

,

,

.

0

.

.

.

. .

-l'fr

.

~

,

"

1 .o

'8 :

'

4 0 00

0.0

0.0

2 .o 3 .0 time, s Coupling between adaption mechanism: 10

m p t i o n on

Fig. 10

40

f

= 0.75T, and

P, =

xh

0 1 :0

:

00

10

OdaDtlon on

Fig. 15

20

t

30

40

t1me.s

Obseroer with adaption: T[O] = OST, andP,[O] = ISx,

mechanism is tyrned on, only an observer is used. As shown, both 1/T, and ?h converge to correct values and SY, and St,br go to zero, when the adaption mechanism is switched on. To check the coupling between the adaption mechanism for 1/T, and % h , respectively, the simulations shown in _Fig. 16 were made. No significant influence of the (l/T,)-adaption on the ?,-estimate is shown. Similarly (not shown in the Figure) the adaption of t h does not exert a significant influence on the estimate 1 / T , . As shown in Fig. 17, an incorrect value of PS will influence the adaption of parameters and the errors SY, and S$, . These errors are not tending to zero, but fluctuate around this value (similar influence has an incorrect ?2,-value).However, the adaption algorithm is stable. 8

"

Conclusion

The new recursive algorithm, for estimati_onof the rotor flux !br and adaption of the parameters 1/T, and ?h developed in this work, shows superior behaviour to other known methods. The algorithm is based on a reduced-order observer. This observer is more rapidly correcting for incorrect initial values of +r, and is less sensitive to incorrect model parameters than the very often applied current models [8, 131. Because of the adaption of ? h , this algorithm makes it possible to keep constant rotor flux in the machine, in

-

42

:

00

:

10 00 00

IO

20 30 time , s Parameter sensitioity.' i, = 1.2rs

adoption on

Fig. 17

40

t

which hardly work in saturation at rated magnetising current. The adaption of 1/T, provides correct orientation of the predicted rotor flux $ r , i.e. S<, = 0. 1EE PROCEEDINGS, Vol. 136, Pt. D, No. I , J A N U A R Y 1989

The parameter adaption mechanism has excellent stability for the actual initial values of the parameter estimates. The convergence rate of these parameter estimates can easily be adjusted by appropriate choice of the gain matrix L [ k ] . However, in practice, it is necessary to consider the noise caused by the measurement systems. In particular, the stator current and voltage measurements contain harmonics effected by the switch mode of the transistor inverter. Nevertheless, the structure of the observer and the applied sampling time (T = 1 ms) have some filtering functions. The parameter adaption loop must be blocked in the region near zero speed I n I < n, = 0.05, . . ., 0.1 (for stator frequencies If,l < 3, . .., 5 Hz), because of the noise influence that occurs in the voltage measurement signal. The described observer with parameter adaption will be implemented with the help of a signal processor solution, as a part of a rotor flux-oriented induction machine drive [lo]. 9

References

1 BLASCHKE, F.: ‘The principle of field orientation as applied to the new TRANSVECTOR closed loop control system for rotating field machines’, Siemens Rev., 1972,39, (5), pp. 217-229 2 BUHLER, H.: ‘Einfuhrung in die Theorie geregelter Drehstromantriebe. Vol. 1. Grundlagen, Vol. 2. Anwendungen’ (Birkhauser, Basel-Stuttgart, 1977) 3 GARCES, L.J.: ‘Parameter adaption for the speed controlled static AC drive with squirrel cage induction motor’, IEEE Trans., 1980, IA-16, pp. 173-178

IEE PROCEEDINGS, Vol. 136, Pt. D, No, I , J A N U A R Y 1989

4 HASSE, K.: ‘Drehzahlregelverfahren fur schnelle Umrichterantriebe

mit stromrichtergespeisten Asynchron-KurzschluO Laufermaschinen’, Regelungstechnik, 1972.20, (2), pp. 60-67 5 HORI, Y., COTTER, V., and KAYA, Y.: ‘A novel induction machine flux observer and its application to a high performance AC-drive system’. IFAC Symposium, Munich, FRG, 1987, pp. 355-360 6 KAAMIERKOWSKI, M.P., and SULKOWSKI, W.: ‘Transistor 7

8 9 10 11

inverter-fed induction motor drive with vector control system’, IEEEfIAS Annual M E E T . Con$ Rec., Denver 1986, pp. 162-168 KUBOTA, H., MATSUSE, K., and FUKAO, T.: ‘New control method of inverter-fed induction motor drive by using state observer with rotor resistance identification’, IEEEIIAS Annual Meet. Con$ Rec., 1984, pp. 601-606 LEONHARD, W.: ‘Control of electrical drives’ (Springer Verlag, Berlin, 1985) LOSER, F., and SATTLER, P.K.: ‘Identification and compensation of the rotor temperature of AC drives by an observer’, IEEEIIAS Annual Meet. Con$ Rec., 1984, pp. 532-537 NILSEN, R.: ‘Modeling, identification and control of an induction machine’. PhD thesis, NTH, Trondheim, Norway, 1987 OKUYAMA, T., et al.: ‘High performance AC motor speed control system using GTO-converters’, IPEC-Tokyo, Con$ Rec., 1983, pp.

720-731 12 PAVLIK, E.: ‘Anschauliche Darstellung des Beobachters nach Luenbexger’, Regelungstechnik, 1978,26, (2), pp. A5-A11 13 SCHUMACHER, W.: ‘Mikrorechner-geregelter Asynchron-

stellantrieb’. Dissertation, Brunswick Technical University, FRG, 1985 14 VERGHESE, G.C., and SANDERS, S.R.: ‘Observers for faster flux estimation in induction machines’, 16th Ann. IEEE Power Electr. Spec. Con$, Toulouse, 1985, pp. 751-760 15 ZAGELEIN, W.: ‘Ein Beobachter mit geringer Param-

eterempfindkeit fur die FluDkomponenten der Asynchronmaschine’, Automatisierungstechnik, 1986,34, (3), pp. 102-1 10

43