- Email: [email protected]
Discrete manufacturing optimizing control through product function
Digi tal Comp ter Applications to Process Control, Van © IFAC and orth-Holland P lishing Company (1977)
a ta Lemke, ed.
A5-
11.
I
her.a
TRODU T10 Level 1 is he ro uc ion sv direct process con roces desi n an of control orm a those eXPlore v rocess, he istribu ors, i~ a proach. T e ro uc who e ro uc io in luenced v higher level ecislons. e olicv level ecisions on vehicles 0 ten effects i ribu or ro uc ion si ce is ri u ors plav an im ortant role in au 0 ive e issions and fuel econo y. A dis ri u or calibra ion, as shown in Fi~ure 1, has beco e particularly de an in of the produc ion sys e because engine svste redesign has increased he slo e of spark advance versus dis ri utor spee (RP) at low spee s. These new calibrations are produc esi n decisions whic~ corres ond 0 the hird level of co rol influence. On he process con rol level he eep sl e is seen as a re uc ion i yield because e olerance an re ains . . he a. e while e 0 a which spark a vance e ins .ore si~ ica a ec s i olerance.
anufacturing opera ions in the discrete processin industries have as their prl e ob'ective to produce quali y products which eet s ecific func ional objec ives a he lowes costs. Production anager u develop a syste s approach so ha the i por ant i~plica ions of these objec 've on ro uction are understoo in or er 0 fulfill company objec ives. This paper develops a sys e s approach 0 improvin 'pro uc ivi y by ex loring the role of product func io on he design of a manufacturin process. Process con rol-is used as a method of achievin and main ainin end product functional objectives. Functional performance is he engineerin~ definition of the system. This product is usually just one component of an overall system. For automobiles, the vehicle is the end sYstem and the va~ious componen s include the en~ine, the chassls, etc. n he case of au o~o ive distribu ors, one function is 0 rovi e a rela ionship between spark i i g an en ine spee. I' anufacturing hese produc s he in errela ion hip of ro uc aria les e clearly un ers 00 a d con rolle 0 ee unc io al objec ives.
SPARK ADVANCE VS. DISTRIBUTOR SPEED 15
~ c:
10
o
>
'U
5
0~_~4-+---+------+------f------I
~
/
~
/
~
tAb: /
10 15 20 25 Distributor Speed- rpm x 100 Actual
Slope
/
/
D = Intercept ri u
329
r
a. ce
A5-4
330
lex he To uc a
rol he
Top PI ate Tab
_~"-+~-~----1~
---+-+-+--+----r~1)
Bottom Plate Weight
PRO UCT
o
ES R PTIG
The func ional objec ive of the is o is ribute the suitable for the The spark i in is ributor spee he slope of ne~a ive ue 0 elay in no actual echanical a vance occurs re ion. The nex re ion is known as he portion because advance is over e primary subsys em of the is ribu or. The re~ion is the secon ary advance sys e and he slope is overned by he pri ary and secon arv subsystems. The breakooin s which determine he ifferen re ions are called he ori arv akeoff oint and he secondary takeoff ooint.
g(w,w,R,L,fd,t, 1,S,dC8)) = 0
p
p~:
Pree
~o
v
is ribu or
of
J:;"orce
on
U DU ro he odel is oh ained by solvin a orce balance e uation i .olici lv or arK a vance aJ a given RP. The soring vs em is mo elerl as a linear ra e 01 s an initial ension orce, f1' as ollows:
(2)
( 1) S
.028 -.05
(4)
...-. ......,. ...
AS-4 Step 2)
::
ensi ivi y
nalvsis
~ _~
"":"'"'1~
.,..
331
\..., _ _ J ......
or any e8S wi h he . variance for is ..,iven bv
he non-linear inearized a ou by performin a mo el, com u er fac orial analysis
D
= S
e
w. re Wc 1 Se are he ce he a vance 1 ei her ortions.
- bw
( 7)
e
ere a ee ri ar or sec
::irk ar
(5)
(6)
-.6 6fd
The subsrip secon ary
P
p and
6D
The sensi ivity coefficien scan part numbe r. I components, such as sprin and tab posi ion, are foun to he firs order effec hence they 0 no destro of equa ions (5) an (6).
s
=
(b
wlJere bp ,b s 8re he slo e ,re ec i elv. e
l)
roces.
on ro
s
Ib) p
D
ri arv
(8)
p an
seco.
arv
o e
The easurement of akeoff oin s an slopes on a distribu or cannot be ma e directly ut must be calculated from spark advance versus BP data. This calculation is accomplished bv definin~ an interval where he takeoff occurs, based on he engineerin specifica ion, an fitting a family ot linear regression pairs to the da a. The bes leas s uares re ression is
~~edt~~r ~~er~~~~~ninl~~~~ r~ ~~nln~ ~~se~k~gn~ oin s.
PPO F
CO. ROL P ORL
ri u
r
... ance
ea.
~e.
er·.
~E
332
E
AS-4
Additional i si ht in 0 the con rol sys e stability is achieve by modelin he rocess using finite difference equations. The sa le ins ance, i, corresspon s 0 indivi ual oar s or sampled roups of oarts. A eneral process control odel roposed by Bishop [3J is a olie to the istribu or process. he 0 el variables can be considere wo d i. ensional vectors containing he primary and secondary por ion , therefore, the variables will be treated as vectors and not distinguished between rimary and secondary influences. Input to the control system is the desired soark advance interceot, D, and the output is the actual spark a vance intercept, m. The process is escribed bv the linearized distributor model and P represents all uncontrolled disturbances. For
~~~mPG~bog~~gyi~d:n sP~inih~roe~~t~~~ear~fran~~~
springs caused a large variance in distributor response the process yIeld woul be unaccep able and the control system is considered unstable. In order to reduce the variance of the springs it is desirable to handle the sorings in batch lots from the suppliers or possibly 0 sort the springs. The discrete process model can determise which decision is best by quantifing the variance reduction required for' control.
The process outputs a chan e in spark advance intercept 1, which is summed over time to ~ive the abso i ute spark advance intercept, M • . easurement error, (' , is additive. The controller is a proportional constant which multiplies the error in spark advance intercept to give the new tab position. The difference equation is written as follows:
6rn(i)
m(i+1) - m(i) P(i) + KG(D-E(i) - rn(i»
=
(9)
where is taken to be a constant indeoendent of time and G is normalized to one. sing classical or z-transform techniques, equa ion is solved for M
m(i)
i-1 L
(P(n)-KE(n»(l-K)
i-1-n
(10)
n=o This controller is stable provided he gain, K, is in he range 0 to 2. Por gains reater han one the control system overshoots. n order to correctly include he effec s of he process random behavior, he sa plin lan ust be described. in or er 0 re ove individual oar s which all he oar s. of par s selecte a i. e. I is assu ed he
es ~e
vance
The effect of hese rando vari3bles on the design of the process co troller can be considered. n ti u steady a e res onse occurs when the exoected value of M is zero (the sa e as the desired) and he vari3nce is TT which is a orede ermine value which ives the esired process yiel. .",Then the 10 e, bI , is equal 0 he ideal slope VT equals the aximu variance,V rnax ! however, tor non-ideal slooes,ba Jhe otal varlance is aporoxi~a ely
vT = Vrnax
-Cb -b )w
I
a
t
( 11)
where Wt is he takeoff RPM. T~e ac ual stearlv state resoon e of the control system is determined by takin~ the expected value and variance of equa ion (10) and letting i aporoach infinity: E(rn)
-H
V(rn)
(a
(12)
2 2 2 +K a )/K(2_K)
p
( 13)
E
The solution of eauation (l~) for the ooti. um control ~ain, K, gives the exoected variance in teadv state. The controller gain can influence he orocess ou out variance. ~ gain between one and two results in overshoot and larger variance. A gain less than one results in a slow res onse and larger variance. The ootimum ain is cho en as one considering no easure ent variance. This controller can be implemented orovided: (14)
The reauired rocess variance i rea er the actual variance which is eaual 0 per u a ion variance using a ain of one zero error variance in e ua ion (l~).
s eo 4)
on rol I ple enta ion
han he and
A5-4
ISCRE E _
FAC
I G
P L
ZI G
HRO GH PRO
C F
Since func ional tests are distributed on the plant floor, a decentralized microcomputer control, si ilar to that proposed by Schoeffler [4J , is being implemented. For the information trans ission requirements, a wi e band coax communication system is installed Data on distributor part calibration and results of testing are transmitted between the icro computers and the host co puter. Calibration information can also be transmitted between the orecalibration test and the final calibration test control computers. ACK OWLEDGE E TS The author wishes to acknowledge the helo of D. oyer, who made several useful sug~estions in the approach taken 1 J. err, who contributed to many of the deslgn concepts and to the people involved in the implementation. REFRRE CES 1.
esarovic r M. D., "Multilevel Systems and Concepts ln Process Control"{ Proceedings of the IEEE, 58,1 (January 1970).
2.
Snedecor,G.W., "Statistical. ethods , The Iowa State University Press, Ames,Iowa, USA (1956) .
3.
Bishop, A.B.), "Introduction to Discrete Linear controlsTheory and Application",Academic Press, New York (1975) .
4.
Schoeffler, J.D., and Sherman R.H., "Software Organization for Mini-Computer Systems", IFAC/IFIP 3rd Inter. Conf. , Helsinki, Finland (1970).
IFAC SECRETARIAT
CTIO
333
a ta Lemke, ed.
A5-
11.
I
her.a
TRODU T10 Level 1 is he ro uc ion sv direct process con roces desi n an of control orm a those eXPlore v rocess, he istribu ors, i~ a proach. T e ro uc who e ro uc io in luenced v higher level ecislons. e olicv level ecisions on vehicles 0 ten effects i ribu or ro uc ion si ce is ri u ors plav an im ortant role in au 0 ive e issions and fuel econo y. A dis ri u or calibra ion, as shown in Fi~ure 1, has beco e particularly de an in of the produc ion sys e because engine svste redesign has increased he slo e of spark advance versus dis ri utor spee (RP) at low spee s. These new calibrations are produc esi n decisions whic~ corres ond 0 the hird level of co rol influence. On he process con rol level he eep sl e is seen as a re uc ion i yield because e olerance an re ains . . he a. e while e 0 a which spark a vance e ins .ore si~ ica a ec s i olerance.
anufacturing opera ions in the discrete processin industries have as their prl e ob'ective to produce quali y products which eet s ecific func ional objec ives a he lowes costs. Production anager u develop a syste s approach so ha the i por ant i~plica ions of these objec 've on ro uction are understoo in or er 0 fulfill company objec ives. This paper develops a sys e s approach 0 improvin 'pro uc ivi y by ex loring the role of product func io on he design of a manufacturin process. Process con rol-is used as a method of achievin and main ainin end product functional objectives. Functional performance is he engineerin~ definition of the system. This product is usually just one component of an overall system. For automobiles, the vehicle is the end sYstem and the va~ious componen s include the en~ine, the chassls, etc. n he case of au o~o ive distribu ors, one function is 0 rovi e a rela ionship between spark i i g an en ine spee. I' anufacturing hese produc s he in errela ion hip of ro uc aria les e clearly un ers 00 a d con rolle 0 ee unc io al objec ives.
SPARK ADVANCE VS. DISTRIBUTOR SPEED 15
~ c:
10
o
>
'U
5
0~_~4-+---+------+------f------I
~
/
~
/
~
tAb: /
10 15 20 25 Distributor Speed- rpm x 100 Actual
Slope
/
/
D = Intercept ri u
329
r
a. ce
A5-4
330
lex he To uc a
rol he
Top PI ate Tab
_~"-+~-~----1~
---+-+-+--+----r~1)
Bottom Plate Weight
PRO UCT
o
ES R PTIG
The func ional objec ive of the is o is ribute the suitable for the The spark i in is ributor spee he slope of ne~a ive ue 0 elay in no actual echanical a vance occurs re ion. The nex re ion is known as he portion because advance is over e primary subsys em of the is ribu or. The re~ion is the secon ary advance sys e and he slope is overned by he pri ary and secon arv subsystems. The breakooin s which determine he ifferen re ions are called he ori arv akeoff oint and he secondary takeoff ooint.
g(w,w,R,L,fd,t, 1,S,dC8)) = 0
p
p~:
Pree
~o
v
is ribu or
of
J:;"orce
on
U DU ro he odel is oh ained by solvin a orce balance e uation i .olici lv or arK a vance aJ a given RP. The soring vs em is mo elerl as a linear ra e 01 s an initial ension orce, f1' as ollows:
(2)
( 1) S
.028 -.05
(4)
...-. ......,. ...
AS-4 Step 2)
::
ensi ivi y
nalvsis
~ _~
"":"'"'1~
.,..
331
\..., _ _ J ......
or any e8S wi h he . variance for is ..,iven bv
he non-linear inearized a ou by performin a mo el, com u er fac orial analysis
D
= S
e
w. re Wc 1 Se are he ce he a vance 1 ei her ortions.
- bw
( 7)
e
ere a ee ri ar or sec
::irk ar
(5)
(6)
-.6 6fd
The subsrip secon ary
P
p and
6D
The sensi ivity coefficien scan part numbe r. I components, such as sprin and tab posi ion, are foun to he firs order effec hence they 0 no destro of equa ions (5) an (6).
s
=
(b
wlJere bp ,b s 8re he slo e ,re ec i elv. e
l)
roces.
on ro
s
Ib) p
D
ri arv
(8)
p an
seco.
arv
o e
The easurement of akeoff oin s an slopes on a distribu or cannot be ma e directly ut must be calculated from spark advance versus BP data. This calculation is accomplished bv definin~ an interval where he takeoff occurs, based on he engineerin specifica ion, an fitting a family ot linear regression pairs to the da a. The bes leas s uares re ression is
~~edt~~r ~~er~~~~~ninl~~~~ r~ ~~nln~ ~~se~k~gn~ oin s.
PPO F
CO. ROL P ORL
ri u
r
... ance
ea.
~e.
er·.
~E
332
E
AS-4
Additional i si ht in 0 the con rol sys e stability is achieve by modelin he rocess using finite difference equations. The sa le ins ance, i, corresspon s 0 indivi ual oar s or sampled roups of oarts. A eneral process control odel roposed by Bishop [3J is a olie to the istribu or process. he 0 el variables can be considere wo d i. ensional vectors containing he primary and secondary por ion , therefore, the variables will be treated as vectors and not distinguished between rimary and secondary influences. Input to the control system is the desired soark advance interceot, D, and the output is the actual spark a vance intercept, m. The process is escribed bv the linearized distributor model and P represents all uncontrolled disturbances. For
~~~mPG~bog~~gyi~d:n sP~inih~roe~~t~~~ear~fran~~~
springs caused a large variance in distributor response the process yIeld woul be unaccep able and the control system is considered unstable. In order to reduce the variance of the springs it is desirable to handle the sorings in batch lots from the suppliers or possibly 0 sort the springs. The discrete process model can determise which decision is best by quantifing the variance reduction required for' control.
The process outputs a chan e in spark advance intercept 1, which is summed over time to ~ive the abso i ute spark advance intercept, M • . easurement error, (' , is additive. The controller is a proportional constant which multiplies the error in spark advance intercept to give the new tab position. The difference equation is written as follows:
6rn(i)
m(i+1) - m(i) P(i) + KG(D-E(i) - rn(i»
=
(9)
where is taken to be a constant indeoendent of time and G is normalized to one. sing classical or z-transform techniques, equa ion is solved for M
m(i)
i-1 L
(P(n)-KE(n»(l-K)
i-1-n
(10)
n=o This controller is stable provided he gain, K, is in he range 0 to 2. Por gains reater han one the control system overshoots. n order to correctly include he effec s of he process random behavior, he sa plin lan ust be described. in or er 0 re ove individual oar s which all he oar s. of par s selecte a i. e. I is assu ed he
es ~e
vance
The effect of hese rando vari3bles on the design of the process co troller can be considered. n ti u steady a e res onse occurs when the exoected value of M is zero (the sa e as the desired) and he vari3nce is TT which is a orede ermine value which ives the esired process yiel. .",Then the 10 e, bI , is equal 0 he ideal slope VT equals the aximu variance,V rnax ! however, tor non-ideal slooes,ba Jhe otal varlance is aporoxi~a ely
vT = Vrnax
-Cb -b )w
I
a
t
( 11)
where Wt is he takeoff RPM. T~e ac ual stearlv state resoon e of the control system is determined by takin~ the expected value and variance of equa ion (10) and letting i aporoach infinity: E(rn)
-H
V(rn)
(a
(12)
2 2 2 +K a )/K(2_K)
p
( 13)
E
The solution of eauation (l~) for the ooti. um control ~ain, K, gives the exoected variance in teadv state. The controller gain can influence he orocess ou out variance. ~ gain between one and two results in overshoot and larger variance. A gain less than one results in a slow res onse and larger variance. The ootimum ain is cho en as one considering no easure ent variance. This controller can be implemented orovided: (14)
The reauired rocess variance i rea er the actual variance which is eaual 0 per u a ion variance using a ain of one zero error variance in e ua ion (l~).
s eo 4)
on rol I ple enta ion
han he and
A5-4
ISCRE E _
FAC
I G
P L
ZI G
HRO GH PRO
C F
Since func ional tests are distributed on the plant floor, a decentralized microcomputer control, si ilar to that proposed by Schoeffler [4J , is being implemented. For the information trans ission requirements, a wi e band coax communication system is installed Data on distributor part calibration and results of testing are transmitted between the icro computers and the host co puter. Calibration information can also be transmitted between the orecalibration test and the final calibration test control computers. ACK OWLEDGE E TS The author wishes to acknowledge the helo of D. oyer, who made several useful sug~estions in the approach taken 1 J. err, who contributed to many of the deslgn concepts and to the people involved in the implementation. REFRRE CES 1.
esarovic r M. D., "Multilevel Systems and Concepts ln Process Control"{ Proceedings of the IEEE, 58,1 (January 1970).
2.
Snedecor,G.W., "Statistical. ethods , The Iowa State University Press, Ames,Iowa, USA (1956) .
3.
Bishop, A.B.), "Introduction to Discrete Linear controlsTheory and Application",Academic Press, New York (1975) .
4.
Schoeffler, J.D., and Sherman R.H., "Software Organization for Mini-Computer Systems", IFAC/IFIP 3rd Inter. Conf. , Helsinki, Finland (1970).
IFAC SECRETARIAT
CTIO
333