Digital calculations of dynamic electromechanical systems on the ground of state equations in the continuous and discrete time domains

Z E S Z Y T Y N A U K O W E P O L IT E CH NI KI ŚL ĄS K I E J

Seria: G Ó R N I C T W O z. 179 Nr kol. 1039

______ 1989

IN T ER N A T I O N A L CO N FE R E N C E : DY NA M IC S OF MI NING MACHINES D YNAMACH '89

A ndr z ej S K A L N Y Ludger SZKLARSKI

D I G I TA L C A LC UL A T I O N S OF D Y NA MI C E L E C T R O M E C H A N I C A L SYSTEMS ON THE G R O U N D OF S TA T E EQ UA TI O NS IN T H E C O NT IN U O U S AND D IS CR E T E T I M E DOMAI NS

A b s t r a c t . Ba sing on the Lag r an ge equations the electromechanical line ar syst em is d e s c r i b e d using the v e ct or state equations. The finished n um be r of lumped elements is regarded. The models of d i f f e ­ rent c o n f i g u ra ti on s the s e l e ct ed example is discussed; the digital r ep re s e n t a t i o n of state v ar ia b l e s of an ele c tr ic drive is used, wi th the elasti ci t y of m ec ha n i c a l elements of the system taken into account. For the e l e c t r o m e c h a n i c a l systems wi th an a.c. induction driv i ng m otor the d y n a mi ca l m odel de s cribed by the vector d i f fe re n ­ ce e qu ations is writ te n and the numerical example for both d e te r mi ­ n i s t ic and s to ch a s t i c s y s te ms cases is discussed.

1. IN TR O D U C T I O N

The paper deals with the mathematical models of electromechanical systems with lumped parameters.The vector from of equations for different configurations;

of lumped parameters such as mass,electric elements and damping of the moving system.As an illustration of a model described by linear differential equations (in the continuous time domain) is the dynamical system with four points of mo­

ving lumped masses.As a matter of fact this is a d.c.drive(with mechanical part).

The machanical system may be described in the discrete time domain.The model with a.c.induction motor for both deterministic and stochastic loads has been discussed.Selected trancients for such models received from the sinulation tech-

lique are enclosed.

2. D E T E R M I N I S T I C EL EC TR O M E C H A N I C A L S Y S T E M D E SC RI BE D B Y T HE V ECTOR S T A T E E QU AT I ON S IN T HE CO NT I NU OU S T I M E DOMAIN

Mechanical part.

The stationary linear mechanical system is being described by the equations:

104 A, S k s lny, L. S z k l a r s k i

«here: q-vector of generalized coordinates, Q-vector of generalized forces,

and matrices K,P,D are symmetrical positive definite.

Usually this equation is being reduced to the form where the coefficients of components of generalized accelerations equals to one,i.e.

9 + IT1? 9 + if1? 9 * If1? (2)

The equation (2) may be presented by following block system,with zero ini­

tial conditions,as shown in Fig.l.

Fig, 1, Mathematical blok s y st em of the equation of mechan ic al moving port

Basing on above system (Fig.l),and introducing following notation

Xj(t) - 6 (t) - displacements vector' Xjit) * 8 (t) - velocity vector U(t) * Q(t) - input,

we receive the vector state equation

Xj(t) = XjCt)

X2(t) = -K'^P Xj(t) - K ^ D X2(t) + K ' V t ) In more general form this equation will be

X(t) * A X(t) + B U(t)

where the state matrix:

Xj(t)

XjCt)

* X(t) state variables vector

(3)

9 Ï *g

A = and § =

iOi

i<

OLlH1

1 i r 1

inout matrix

Digital Calculations of Dynamic... 105

The matrices K,P,D are determined for the system with a finite number of de­

grees of freedom,with lumped parameters,i.e.massCinertia moment),elasticity,dam­

ping, 3i,ki ,r1 - respectively.

The series structure of mechanical part,with finite number of degrees of free­

dom is shown in Fig.2.

*| W, JW,

Fig. 2. Model of m ec ha n i c a l part with lumped p a r a m e t er ss er i es structure

Such a structure is being characterized by the following forms of matrices

K.P.D

‘ 3, 1

diag

«

P *

kl ■ * 1 0 0 0 . 0

- k l k^+ k₂ -k2 0 0 . 0 0 -k₂ k2+kj -kj G . 0

• ₀

0 0 * kn-l kn-l_

rl "rl ^{0 0 0 . . 0}

-rl ri + r 2 -r_{2 0} Ü . 0

0 - r 2 r2+ r3-r3 ^{0 . 0}

0

0 • 0 -rn-l rn-l

this matrix represents the influen­

ce of elastic linear deformations,

matrix represents the dissipation in the system.

The dynamics of the system is being characterized by the oscillations about the steady state.The state equation of the dynamical system will be written in­

troducing the new state variables and. ^ , i.e.for the displacements^ <^=9^- e2i tp2-S2- e3 , . . , f n.x=0n_r 8n, and io r *** velociti8S “M l " 8

2

8-1»

106 A. Skalny, L. Szklarski

fflx

Fig. 3. The forked st r ucture of mechan i ca l part with lumped parameters

The forked structure for n moving lumped mass points is shown in Fig.3. In thi- s structure n=l»s.

For this structure the matrices K,P,0 are as follows:

P11 p12 p13 pln

J1 J2

p21 p22 0 . . . 0

P = p31 0 p33 •

1 . . . 0

s

p , 0 . . 0 p

pnl Hnn

Pn * *1+*2 + . • + k s

plj= - V l for 5*2,3,. ...n

-k^_j for i = 2,3,...,n and

p22 * kl p33 = k2

And similarly the matrix D :

O i g r i t a l C a l c u l a t i o n s of D y n a m i c , . 107

«hose elements d ^ are being determined from the relations

dn * rx + r2 * . . * rs

djj = -r^_j for j= 2,3,..,n ; dn = -ri_1 for i » 2,3,..,n and

d22~ rl d3,= r2

Electromechanical system.

Let us discuss some driving system,as an example. The mechanical part of this system is shown in Fig.4. is the inertia moment of lumped mass of a separately

Fig. 4. T h e e x a m p l e of a fork ed m e c h a n i c a l part of a d r i v i ng s ystem

excited d.c.motor armature.The remaining three points of moving lumped masses are parts of driven mechanism. The symbols and 8^ denote the steady-state dis placements.Introducing the generalized variables <f>12= 0j- 02 , ^P23=B2" 93 ar,tl

= 8 - 8 . the equations of system dynamics will be as follows T 24 2 4 ’

i(t) = a ^ U V a ^ C t ) * bjU(t)

’¿^t)« a3 <f12(t)t

a4f 12(t)+

+ Vob

f 2 4 ( t ) l a18^24(t )+ a19‘i>24(t)+ a20^23(t )+ a2 1 % (t )+ a2 ^ 1 2 (t)+ a23^12Ct?

108

Skalny, L, Szklarski

The coefficients 3^ of the above equations are some functions of system para­

meters:

R - motor armature circuit resistance, L - motor armature circuit induction, k„,k_- motor constants,

. e r a

\ 2 3 4’ ^nert’i8 nwments, k12’l<23’k24' elasticity constants, r12,r23,r24“' damping constants, ■

ICt) ,U<t),ni0jJ - instantaneous values of armature current,armature voltage and torque resp.

Introducing the state variables:

* j ( t M( t) ; X2Ct)=B1Ct); Xj(t)«^2(t) j k4(t)=f12(t); XjCti^ jf t) ;

x6(t)=^23( t ) : x7( t ^ 2 4 ( t ) : X8(t)=^24( t)

i(t) * A X(t) + 8 U(t) * G mob(t>

V(t) = C X(t)

In this equation U(t) denotes the scalar reference,while m^Ct) is the dis­

turbance .

3. INVESTIGATION OF THE ELECTROMECHANICAL SYSTEM IN THE DISCRETE TIME DOMAIN

Discrete models.

The transition from the model described in continuous time domain to one in the discrete time domain may be achieved in different way. The choice of the method of discretisation should be made with great care,among others,it depends on the tipe of discussed system and its properties and demanded accuracy of nu­

merical calculations.

In the engineering practice,quite reasonable results may be obtained,for li­

near systems by applying the discretisation of differential equations replaing first and second derivatives by finite differences with chosen discretisation tiraeCsampling time). But still better mathematical model will be received by solving the state equations in'a continuous time domain.If the matrices of the equations are known,as well as their eigenvalues and eigenvectors,the matrices of discrete form of state equations may be received.This method may be applied to the linear dynamical system,as well as to the stationary and non-stationary ones.This problem has been developed in following publications (4,5).

The discrete mathematical models may be received by means of different inte­

gral transformations of continuous functions and equations.. In this paper,however the differential transformation has been used. The discrete description of elec-

Digital Calculations of Oynamic...

trie driving system with a.c. cage rotor supplied from the current-source inver­

ter. For the simlicity reasons the dynamics of the driving system may be written with the driven mechanism torque reduced to the motor shaft.It means,that the model of the motor described by the equations in X-Y coordinate system,besides, we may assume that both of stator and rotor windings coincide.

Assuming following relations w>gyn(t)= W K(t),what means that the coordinates X-Y rotate in synchronism with the stator current space vector; «,(t)=t»?(t)/p, where O r(t) - angular velocity of motor rotor, co(t) - angular velocity of mag­

netic field, p - number of pairs of poles.

Besides: Tr= LrRr - time constant of rotor circuit, then the equations des­

cribing the dynamics of the electric drive are as follows:

“ 7

^r Y (t) Lm

- f ~ W

« - + P H ( t ^ C t ) (6)

du> (t) 3pL 3pL_ I

— ■— « isY( t ) — - — mob

dt 2Lr3 rX SY 2Lr0 rY SX 0 00

d 9(t)

CJr(t) dt

The following state variables and components of reference vector were cho- :en:

Xl(t)=H^x(t) - projection of rotor flux space vector to the 0-X axis,

x (t)=Yry(t) * projection of rotor flux space vector on the 0-Y axis, X3 ( t ) = C J r ( t )

x 4 ( t ) = e ( t )

u,(t)= i v(t) - projection of the space vector of stator current to the

1 SX

0-X axis, u 2 ( t ) = c o s y n ( t )

Uj(t)= i^y(t) - projection of stator current space vector to 0-Y axis.

As it was mentioned previously,the case will be discussed when u3(t)=igY=0.

For C-class functions from above described continuous model the discrete model is received by means of Taylor's differential,for total argument k=0,l,2,...

11 0

A. Skalny, L. Szklarski

X .( k + 1 ) =

Hp

(k+l)Tr • * k+1

E

K ^HL_

x , ( k ) --- > X , ( l ) X , { k - l ) + --- U , ( k )

^ 1 1 ( k + l) T 1

k

- r *

1=0

2(l)X2(k-l)

x (k+1) --- X,(k) +

~ fL. .

1 IT +

Hp

(k+l)Tf ‘ k+1

k k

y \ a ) x l(,i) - u2d)x1ck-i)

1=0 1=0

(7)

Xj(k+1) = -

3Hpl V 1

--- ) X,(l)U,(k-l)

L _

r 1=0

H (k+l)0

X.(k+1) = ----X,(k) k+1 }

In these equations X ].(k),X2(k),X.j(k),X4(k),U1(k),U2(k) and M^fk) are discre­

te farms of previously discussed state variables of components of reference vecto -r and load torque. Constant H is of time dimension.

Applying the Cauchy product properties the algebraic twists are received in equation (7).In order to describe this model in state space,the following vectors are defined as follows:

Xj(k)

X2(k)

X.(k)

= X(k) - state variable vector,

Uj(k)

U„(k)

UjCk)

= VU(k) - reference vector

Digital Calculations of Dynamic.. ill

Some additional vector resulting from the approximation of algebraic twists

_K

£

XXj(k) XX2(k)

XX^(k)

= XN(k)

r U;fn x , k - l ) dif UR^k)XR^k) dti UX(k) M

UXj(k)

UX2( k )

^UX^(k)

= UX(k)

where: 'J - number of products of pairs of variables X^(k) and X^(k) ,IT (k) in the equations (7).respectively. In the vectoe form thé discrete model may be described as follows:

X(k+1) = AL(k,H) X(k) + AN(k,H) XN(k) + BL(k,H) VU(k) + BN(k,H)UX(k) In discussion of electromechanical systems often the problem of stochastic dis­

turbances may be encountered.Their nature and point of entering into the system may be different. If the load torque rn^Ct) containes two components: determi­

nistic mg(t) and stochastic one mQS(t),in the discrete model of the drive appear discrete T-transforms.

mob(t) 1 Mob(k)

r a Q (t) r Mo (k)

m (t) T M (k).

os os

In order to discuss the behaviour of such a drive,as a dynamical system,the de -scription of the system basing on the solutions of equations(7) will be especia­

lly suitable.

The vector state equation in this case will be as follows:

(8)

XjCk+l)

X2(k+1)

Xj(k+1)

X4(k+1)

H

(k+ l) T „ H (k+l) T„

’XjCkf

Xj(k)

♦

Xj(k)

x4( k )

n np

u k+l Hp

g k+1

0 0

0 0

XX^(k)

XX2(k)

112

O

3HPL„

2(k+l)Lr0

O

H k+1

O

kTT

r HL

UXj(k)

UX2(k)

UXj(k)

Ck+1)T

O

(k+1)O (k+l)J

0 0 O

Ux(k)

M0(k)

Mo s ( k )

(9)

CX o

O)

T £ S )

rig.

T —^-r i---“r 1— T ~ ~ 1 --~r— “i----1-- T ~ — |— ~~T~ — !

0.00 i .20 Z.*0 3.S0 (.30 ₆.00 7.30 S.40

T £ S )

5. T r a n s i e n t s of s t a t e v a r i a b le s of fo k ed d r i vi ng systeir

D i g i t a l C a l c u l a t i o n s of D y n a m i c . ».« 113

Fig. 6. T r a n s i e n t s of torq ue a nd v e l o c i t y of an a.c. i n d u c t i o n motor with s t o c h a s t i c load

From the structure itself of the equations (7) and (9) the simple algorythms of numerical simulation will appear. The practice resulting from the application of such models conform their usefulness. They deliver very convenient instrument in the synthesis of digital control of electromechanical systems and will be very useful in computer-aided desing of electric drives.

The briet illustration of simulation of some numerical examples are shown diagrams,depicted in Fig 5.It shown the transients of velocities and ascilla- tions in some chosen points (2) of Fig.4,as well as armature current (X9) of driving motor. The digital simulation was carried out basing on the equations (4) and (5). Fig.6 depicts the torque and velocity transients of an a.c.induc­

tion with cage rotor,according to the equation (9).

( Above problems have been solved with the problem RPBP 02.7 )

114

A. Skalny, L. Szklarski

L I T E R A T U R E

[ li P eł c ze w s k i W., K ry nk a M . : Me toda z m i e n n y c h s t a n u w a n a l i z i e d y n a m ik i u k ł a d ó w napędowych. WNT, W a r s z a w a 1984.

f 2 l P u kh o v G.E. D i f f e r e n t i a l T r a n s f o r m a t i o n s of F u n c t i o n s and Equations.

NO, Kiev 198j0,

["31

A.:

w

TaJ Skalny A . : D y n a m i k a u k ł a d ó w e l e k t r o m e c h a n i c z n y c h o w i e l u s t o p n i a c h swobody i es ty m ac ja w czas ie d y s k r e t n y m przy z a k ł ó c e n i a c h s t o c h a s t y c z ­ nych. PWN, Seria Wyd. Pos tę p y N a p ę d u 61., W a r s z a w a 1968.

[

5

("s'] S z k l a r s k i L. , S k al n y A.: T e o r e t y c z n e Z a g a d n i e n i a M a sz y n W y c i ę g o w y c h Cz. I, PWN, S e ri a Wyd. Postępy N a p ę d u El., W a r s z a w a 1975.

f ? ! S z k l a r sk i L . : T s o r e t y c z n e Z a g a d n i e n i a M as z yn W yc ię g o w y c h . Cz. II, PWN, Seria Wyd. P o st ęp y N a p ę d u El. W a r s z a w a 1977.

fal

L.,

A.,

A.:

Re ce n ze nt t Prof, dr hab. inż. S z c z e p a n B o r k o w s k i

C Y F R O W E O B L I C Z E N I A D Y N A M I C Z N Y C H S Y S T E M Ó W E L E K T R O M E C H A N I C Z N Y C H

N A P O D S T AW IE R Ó W N A Ć S T A N U W C I Ą Ł G E 3 I D Y S K R E T N E 3 P R Z E S T R Z E N I C Z A S O W E O

S t r e s z c z e n i e

Na pod s ta wi e ró wn an ia L a g r a n g e ’a o p i s a n o e l e k t r o m e c h a n i c z n y s y st em l i ­ niowy, s t o s u j ę c we k to r o w e rów na ni a stanu. R o z w a ż o n o s k o ń c z o n ę l i c z b ę e l e ­ ment ów s k upionych. P r z e d y s k u t o w a n o m o d el e o r ó ż ny c h k o n f i g u r a c j a c h na w y b r a n y c h p r z yk ła da c h; z a s t o s o w a n o cyfr ow ę r e p r e z e n t a c j ę z m i e n n y c h stanu n a p ę du elektr yc z ne go , b i or ę c pod u wa g ę s p r ę ż y s t o ś ć m e c h a n i c z n y c h e l e m e n ­ t ó w systemu. Dla s y s t em u e l e k t r o m e c h a n i c z n e g o z i n d u k c y j n y m s i l n i k i e m na pręd stały o p i s a n o m o d e l d y n a m i c z n y za p o mo cę w e k t o r o w y c h równań r ó ż n i c z ­ kowych i p r z e d y s k u t o w a n o p r z y k ł a d lic zb o wy dla p r z y p a d k ó w s y s t e m ó w d e ­ t e r m i n i st y cz ny ch i st o ch a s t y c z n y c h .

D i g i t a l C o l c u l a t i o n s of Dynami c. . . 115

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