VARIABLE - FRACTIONAL ORDER DEAD-BEAT CONTROL OF AN ELECTOMAGNETIC SERVO - PART I

Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19-21, 2006

VARIABLE - FRACTIONAL ORDER DEAD-BEAT CONTROL OF AN ELECTOMAGNETIC SERVO - PART I Tomasz Rybicki, Piotr Ostalczyk Institute of Automatic Control, Technical University of Lodz, Łódź, ul. Stefanowskiego18/22, Poland

Abstract: In this paper a simple dead-beat control of an electromagnetic servo is proposed. An electromagnetic servo is identified as a third-order linear continuous-time plant. A method of a dead-beat control of its discrete equivalence is proposed. In a control algorithm a non-integer variable-order backward difference is used. Copyright © 2006 IFAC Keywords: Difference electromagnetic servo.

equations,

discrete-time

1. INTRODUCTION The area of practical applications for Fractional Calculus (Miller and Ross, 1993; Oldham and Spanier, 1974; Podlubny, 1999) grows wider and wider. It expands from identification and modelling of different physical plants and phenomena (Bologna and Grigolini, 2003; Sjöberg and Kari, 2003; ReyesMelo, et al., 2004) to fractional order control and controllers (Ferreira and Machado, 2003; Oustaloup, 1991, 1994, 1995; Suarez, et al., 2003) The paper presents another application and practical implementation of Fractional Order Controller in industrial knitting machine. The idea of the paper combines well-known theory of dead-beat controllers and fractional calculus to make a variable-fractionalorder dead-beat servo controller. From one hand the proposed controller advances from dead-beat control in fast and settling time predictable responses of the system (Åström and Wittenmark 1984). From the other the fractional order control strategy ensures the step response maximal overshot and damping factor due to the parameter changes of the system (Oustaloup, 1991, 1994, 1995). As it will be shown the two aspects presented above are essential in control of an electromagnetic servo used in circular knitting machine. The whole paper consists of two parts. Part one reveals the requirements for the system that come from the industrial application. In concentrates on

systems,

dead-beat

control,

building a precise model of the electromagnetic servo used in knitting machine and presents the practical aspects of the electromagnetic servo fractional-order dead–beat control circuit built for the experiments. Part two covers the theoretical bases of the variablefractional-order dead–beat control strategy as well as the comparison between simulation and practical results of the closed loop system with such a controller. 2. AN ELECTROMAGNETIC SERVO In general, the technological cycle of forming the stitch using knitting machines requires the appropriate sequence of raising and lowering the needles. These operations are realised by a finite number of steel sections powered up by electromagnetic servos. The group of steel sections creates a lock of the knitting machine which is shown in Fig.1 (Kornobis, 1997). It consists of several different steel sections for different purposes: • • • •

Lower raising steel sections – 1, 2, Upper raising steel sections – 4, 5, Lowering steel sections – 6, 7, 8, Helping steel sections – 3, 9 and 10.

Fig. 1. The classical lock of weft knitting machine. As it can be seen in Fig. 1 the edges of steel sections form the passage which leads the needle’s knee. Lower raising steel sections (1, 2) raise the needles to the stitch gathering position and upper (4, 5) – until the maximal position. Lowering steel sections (6, 7, 8) move the needles in two reverse stages: from the maximal position to stitch gathering position and finally to initial position. The process of switching on and off the appropriate steel section is made by its fast, linear movement towards and from the knitting machine cylinder, respectively. This movement is caused by the specialised electromechanical devices called electromagnetic servos. Fig. 2 shows the electromagnetic servo used in circular knitting machine.

Fig. 3. The arrangement of servos in circular knitting machine – top view. 2.1 An electromagnetic mathematical model

servo

continuous-time

The description of the electromagnetic servo presented in section 2 reveals two main groups of physical phenomena taking place during its motion: electrical phenomena and mechanical phenomena. The description of electrical phenomena could be made using Kirchhoff’s law in the form:

u (t ) = L

di (t ) dx(t ) + Ri (t ) + e(t ) + B dt dt

(1)

dx(t ) dt

(2)

where:

e(t ) = Ce Fig. 2. The electromagnetic servo The electromagnetic servo shown in Fig. 2 consists of the stator, firmly screwed down to the knitting machine body and of the movable part which is able to make the reciprocating motion. At the end of the movable part the steel section which enables the raising (lowering) of the needles is situated. The electromagnetic servo consists of the strong magnet connected to the stator and an induction coil connected to the movable part. Magnetic fields of the magnet and solenoid coil react together creating the driving force. This force moves the steel section towards and from the knitting machine cylinder depending on the current flow direction. The movable part of the servo is equipped with the small additional core which is the part of the linear variable differential transformer (LVDT). The LVDT enables accurate position measurement of the movable part. Typical circular knitting machine is equipped with several electromagnetic servos (between 3 and 6). Fig. 3 shows the arrangement of 4 servos in the circular knitting machine.

and: u (t ) - external supply voltage causing the movement of the coil, i(t ) - coil’s current, x(t ) - displacement of the moving part, L - inductance, B - induction, e(t ) - electromotive force. The movement of the movable part can be described using the mechanical phenomena Newton’s law:

m

d 2 x(t ) dt

2

= −k

dx + Bi(t ) dt

(3)

where: m - mass of the movable part, k - damping factor resulting from air flow resistance. When creating the model of electromagnetic servo based upon equations (1), (2) and (3) several simplified assumptions were made (Dębowski, et al., 2004): - constancy of resistance and inductance (position and temperature independence), - constancy of damping factor k and induction. Making several elementary conversions using equations (1), (2) and (3) leads to the transfer

function in s-domain which shows the linear displacement of the movable part as a function of external supply voltage:

(

)

(4)

where:

Lm kL + Rm a3 = , a2 = , B B kR a1 = + B + Ce B

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

(5)

0

0.005

0.01

0.015

0.02

0.025

tim e [s]

Fig. 5. Step response of the electromagnetic servo model.

To identify parameters a 3 , a 2 , a1 in equation (4) the least square method was used (Masłowski, 1993). It was helpful in finding the best step response of the model (4) compared with the step response of the electromagnetic servo acquired experimentally. Fig. 4. shows the results of the step response acquired experimentally. To perform this experiment the electromagnetic servo was powered with 10V DC power supply and position signal (CH1) and current signal (CH2) were measured. A digital channel called START marks the duration of the experiment. It can be seen that at the end of the experiment the position signal suddenly stops and the current signal raises which marks the servo’s movable part has reached the end of its range. Fig. 5. shows the step response of the model (4) which was obtained for the following values of the parameters:

a 3 = 2.146316799214005 x 10 -6 a 2 = 7.285585546532878 x 10 -5

displacem ent [V]

X (s ) 1 = 2 U (s ) s a3 s + a2 s + a1

0.4

To compare the two step responses (Fig.4 and Fig.5) the error signal was evaluated. It was defined as the difference of the step response of real servo presented in Fig.4 and step response of its model presented in Fig.5. 0.015

0.01

step response error [V]

G p (s ) =

0.5 0.45

0.005

0

-0.005

-0.01

-0.015

(6)

a1 = 4.651158465393928 x 10 -2

0

0.005

0.01

0.015

0.02

0.025

time [s]

Fig.6. The difference of the step responses presented in Fig.4. and Fig.5. Fig. 6 shows that the fitting process of the model is very accurate and that the real difference oscillates very close to zero. 2.2 An electromagnetic mathematical model

servo

discrete-time

To enable the design of a variable non-integer order deadbeat controller for the electromagnetic servo the discretized state-space model is required. Firstly, the conversion to the analog state-space model was performed using Matlab environment. As a result the model in form (7) was obtained: Fig.4. Step responses of the electromagnetic servo acquired experimentally.

x' (t ) = Ax(t ) + Bv(t ) y (t ) = Cx(t ) + Dv(t ) where the four matrixes are:

(7)

− 33,945 − 169,3 0 A =  128 0 0  0 16 0 16 B =  0   0 

L 0 0  L 0  0 L 0 ,  I P, k =  L 0  a0  a1 M   0 1 a2

0   0  0 0  a0 

D P, k

1 a2 0 1  0 0 = 0 0 M M  0 0

a1 a2 1 0 M 0

a0 a1 a1 1 M 0

b1 b2

N P, k

0 b2 0 0  0 0 = 0 0 M M  0 0

b0 L 0  vk   yk  b1 L 0   y  b1 L 0 ,  k −1  , v = vk −1  = y k k   M   M  0 L 0     M M M  y0   v0   0 0 0

0 0 0 a0 a1

(8)

C = [0 0 14,219] D = [0] Next, the conversion to discrete time model was evaluated using sample time Ts = 0.005s . This could be done by one of the commonly used discretization methods such as ZOH (zero order hold), FOH (first order hold), Bilinear, Pole-Zero match and Forward etc. Several experiments were made to obtain the best analogue model approximation. The most accurate result gives the ZOH discretization method. As a result the model in the form (9) was obtained:

0 0 M 0

where y k , v k denote output and control vectors, respectively and y 0 = [ y−1 of plant initial conditions.

y− 2

y− 3 ]T is a vector

Fig.7 shows the step response of the analogue and digital model of electromagnetic servo. Two step responses using ZOH and FOH discretization methods were plotted.

x(k +1) = Ax(k ) + Bv(k ) y (k ) = Cx(k ) + Dv(k )

0.5

displacem ent [V]

0,61266 − 0,71018 0 A =  0,53693 0,75505 0 0,02315 0,072356 1 (10)

0

0.005

0.01

0.015

0.02

0.025

tim e [s]

3.VARIABLE-FRACTIONAL ORDER DEADBEAT CONTROL REALIZATION

(11)

where: a2 = −2.3677 , a1 = 2.21162 , a0 = 0.8439 , b2 = 0.0926 , b1 = 0.34232 , b0 = 0.0821 . Taking into account all the equations (11) for k − 1, k − 2,K,1,0 and collecting together in a matrixvector form we get:

where:

0.2

Fig.7. Step responses of analog and digital model (ZOH, FOH) of the electromagnetic servo.

According to (9), the plant is described by a difference equation:

DP,k y k + I P,k y 0 = N P,k v k

0.3

0

D = [0]

= b2 v k −1 + b1v k − 2 + b0 v k −3

0.4

0.1

C = [0 0 145,219]

y k + a 2 y k −1+ a1 y k − 2 + a 0 y k −3 =

ZOH A NA LOG FOH

0.6

where the four matrixes are:

 0,067116  B =  0,02315  0,0006374

0.7

(9)

(12)

The control strategy used was based upon the evaluation of a variable-order (VO) (Coimbra, 2003, Lorenzo and Hartley 1998, 2002), fractional-order (FO) control strategy realisation. The control was realised in a closed-loop system presented in Fig. 8.

dk

vk

ek VOFO Controller

yk

 d k  1 d    1 d k =  k −1  =    M  M     d 0  1 is a reference (a discrete step function) signal.

Plant

Sensor Fig.8. Block diagram of a closed-loop system with an electromechanical servo and a discrete variableorder controller One assumes that the VOFO controller is described by a simple difference equation (Ostalczyk and Rybicki, 2006)

k p 0 ∆(knk )v k = ek

(13)

(n )

where k p is a constant term, and 0 ∆ k k v k denotes a variable-order backward difference (Ostalczyk 2000, 2001) defined below ( nk ) 0 ∆k fk

=

k

∑ b( i

nk )

f k −i

where k h denotes a sensor coefficient and

Combining equations (12), (16) and (17) we get the input/output description of the closed-loop system: y k = (1k + G O, k )−1G O , k d k +

(18)

− (1k + G O, k )−1 D−P1,k I P , k y 0

where G O , k = D −P1,k N P , k D −R1,k N R , k , and matrix 1k is

(k + 1) × (k + 1)

unit matrix.

Fig.8. can be redrawn as Fig.9 to show all the different components of the closed loop control system including the effect of sampling and hold (Choudhury, 2005).

(14)

i =0

(nk ) can be calculated as

where coefficients bi

bi(nk − l ) =  0 for i = −1,−2, L   (15) = 1 for i=0  (nk − l )  n k − l + 1  i = 1,2, L  for 1 − bi i    Equation (13) may be expressed similarly to the plant description: D R, k v k = N R, k e k

(16)

where

D R, k

N R, k

b0( n k )   0 = M   0  0  1 k  p = M  0 

b1( n k ) b0( n k −1 ) M 0 0 L

L bk( n−k1) L bk( n−k2−1 ) L M ( n1 ) L b0 L 0

bk( n k )   bk( n−k1−1 )  M ,  b1( n1 )  b0( n0 ) 

  ek   vk  0 e     v M  , v k =  k −1  , e k =  k −1   M   M  1       e0   y0  k p 

An adder in Fig.8 is described as ek = d k − khy k

(17)

Fig.9. Digital control (Choudhury, 2005).

loop

block

diagram

In Fig.9. one can name several blocks which are always present while the control algorithms are realised using microcontrollers or DSPs: - GC (z ) - denotes the control algorithm - ADC represented by an ideal sampler with time period TS - k h – proportional element, which denotes the sensor - Cd - denotes computational delay - ZOH - denotes the holding device. As it can be seen in Fig.9. blocks: G p (s ) , k h , Cd , ZOH and the sampling device together represent the discrete transfer function G p (z ) of the plant G p (s ) .

4. PRACTICAL ASPECTS OF THE ELECTROMAGNETIC SERVO CONTROL To enable the practical tests of variable non-integer order dead-beat control the special test bed was created. The simplified schematic diagram of this equipment which is based on Fig.9 is shown in Fig. 10.

Fig.10. Simplified schematic diagram electromagnetic servo control circuit.

of

the

It can be seen in Fig. 10 that the test bed consists of the electromagnetic servo, fully described in sections 1 and 2, power stage and two sensor sub circuits. Power stage was built using four DMOS transistors in “H” bridge circuit which enables forward and backward movement of the steel section. The switching of the transistors is controlled using the pulse width modulation (PWM) by the supervisory microcontroller – SAB517. The PWM module acts as a hold device as presented in Fig.9. The variable non-integer order dead-beat control algorithm was implemented using C language. The sensor sub circuits enable the continuous measurement of a linear displacement of movable part and the current of its coil. The displacement signal was used as the feedback signal in the closed-loop control circuit. The theoretical considerations of variable fractional order deadbeat control as well as the practical results using described closed loop automatic control system with electromagnetic servo are presented in the second part of this article. REFERENCES Åström K.J., B. Wittenmark (1984). Computer Controlled Systems: Theory and Design, Prentice-Hall, Inc., Englewood Cliffs, N.J. Bologna M., P. Grigolini (2003). Physics of Fractal Operators, Spriger-Verlag Choudhury S. (2005). Designing a TMS320F280x Based Digitally Controlled DC-DC Switching Power Supply. Texas Instruments. Application Report Coimbra C.F.M. (2003). Mechanics with variableorder differential operators. Ann Phys. Vol.12, No. 11-12, pp.692-703. Dębowski A., Z. Kossowski, and M. Kowalski (2004). Model of the Megnetolectric Timing Gear Drive of the Combustion Engine. In Proceedings of Modelowanie I Symulacja, Kościelisko (in Polish).

Ferreira N.M. Fonseca, J.A. Tenreiro Machado (2003). Fractional-order hybrid control of robotic manipulators, in Proceedings of The 11th International Conference on Advanced Robotics, Coimbra, Portugal. Kornobis E. (1997). The basis of hosiery laboratory, The Technical University of Lodz Press, Lodz. (in Polish) Lorenzo C.F., and T.T. Hartley (1998). Initialization, Conceptualization, and Application in the Generalized Fractional Calculus. National Aeronautics and Space Administration, Lewis Research Center, Hanover. Lorenzo C.F., and T.T. Hartley (2002). Variable Order and Distributed Order Fractional Operators. Nonlinear Dynamics, vol.29. Masłowski A. (1993). Computer Aided Identification of Parameters and State in Technical Systems. The Technical University of Białystok Press, Białystok (in Polish). Miller K.S. and B. Ross (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, Inc., New York. Oldham K.B. and J. Spanier (1974). The Fractional Calculus. Academic Press, New York – London. Ostalczyk P. (2000). The non-integer difference of the discrete-time function and its application to the control system synthesis, International Journal of Science, vol.31, no.12, pp. 1551-1561. Ostalczyk P. (2001). Discrete-Variable Functions, A Series of Monographs No 1018, Technical University of Lodz Press, Lodz. Ostalczyk P., and T. Rybicki (2006). Variablefractional-order dead-beat control of an electromechanic servo – Part II. 2nd IFAC Workshop on Fractional Differentiation and its Applications. Porto, (submitted for presentation). Oustaloup, A. (1991). La commande CRONE. Éditions Hermès, Paris. Oustaloup, A (1994). La robustesse. Éditions Hermès, Paris Oustaloup A. (1995). La derivation non entiere. Éditions Hermès, Paris Podlubny I. (1999). Fractional Differential Equations, Academic Press, San Diego. Reyes-Melo M.E., J.J. Martinez-Vega C.A. Guerrero-Salazar, U. Ortiz-Mendez (2004). Application of fractional calculus to modelling of relaxation phenomena of organic dielectric materials, in Proceedings of International Conference on Solid Dielectrics, Toulouse, France. Sjöberg M., L.Kari (2002). Non-linear behavior of a rubber isolator system using fractional derivatives., Vehicle Systems Dynamics 37 (3), 217-236. Suarez, J.I., B.M. Vinagre and Y.Q. Chen (2003). Spatial path tracking of an autonomous industrial vehicle using fractional order controllers, in Proceedings of the 11th International Conference on Advanced Robotics, ICAR 2003.