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STRATEGIES FOR MULTIVAR I A BLE SELF-TUN I N G CONTROL
T imothy Heske th
B . Sc . Hons (E l P. c t r ical E n g i n e e ring)
tt,.Sc. (E l e c t r i cal Eng i n e e r i n g)
A tnes i s presen ted in fu l f i lment o f the requ i rements for the d eg ree o f Do c t o r o f Phi l o sophy in I nd u s trial Management and Engineering a t Massey
U n i versity.
J u l y , 1981
ABSTRACT
The research reported in this thesis develops strategi e s for apply ing self- tuning con trol theory to mu l t ivariable processes. Self-tuning control of sc�lar processes i s now reasonably wel l establ i shed , and att�n tion i s bei ng turned to the mul t iva riable problem, w i th its attendant computational burden to ove rcome, and add it ional considerat ions such as interaction and decoupl i ng. This the s i s sugges ts methods for deal i ng w i th these problems, and implements the controllers on one of the readi ly avai lable desk
top mic rocomputer systems, provid ing a cheap yet effective contro l l er.
The research bui l d s on the work o f C larke and G awthrop [ 1 975 , 1 979] on impl i c i t control lers. The expl i c i t schemes are d erived from contro llers developed by We l l s tead and others [ 1 979 , ( a ) and ( b ) J,
[ 1 979 J u sing and the work on mul tivariable c ontrol lers by Bor i s son latt�rly Koivo [1980]. The ful l mul t i variable controller rel ies on s tandard techn iques for pole placement reported by Wolovi c h [ 1 974 ] .
The proposed contro llers remove interac tions between loops, and other d i s turbances, and ensure that each loop output a t ta in s the set-point requ i red for that loop. In parti cular , the expli c i t pole-placing controller requi re s a pole-placing calcu la tion for each loop , and removes the i n te ra c tions w i th a minimum-variance- l ike the resu l t ing c ontroller being modest in computational need s .
A solut ion is a l so proposed for the ful l mul tivariable pol e
placing problem , which al lows a comparison to be made between the ful l mul tivariable solution and the s impler contro llers proposed.
It is found that the ful l mul tivariable con trol ler demands more computat ion , but that the resul t ing control may be no d ifferent from that a chieved w ith the s impler contro ller.
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The programs which a re d escribed a re wri t ten in Fortran , in the form of subrout ines which are designed for incorporat ion i n to e x i sting control program su i tes. A l ternat ively , they may be run w i th a ded i cated supervi sory program to hand le t iming , input and ou tpu t , d i splay and s torage of info rma tion. The subrou tines a re general in the i r appl i c at ion , and the informat ion they need fo r any pa rti cular process may be established interact ively using a n offl ine program.
S imulation resu l ts a re presented which confirm the robus tness o f the pole placing controller , and ind icate that the proposed techn iques may be used to stabil ise and control a wide variety o f processes; those which a re non-min imum phase, certa in non- l inea r or unstable processe s. On-l ine con t ro l of a comme rcial heat exchanger process i s reported, the p rocess be i ng s imilar to others which have been reported, thus provid i ng a po int o f comparison o f self- tun ing contro l w ith o ther techniques. The process is multivariabl e. Good contro l is achieved , parti cu l a rly in the face of perturbat i ons to the p rocess which resul t i n changes to the parameters o f the mode l describing the proce s s behaviour, cond i tions under which some o ther con t rol lers may n o t b e suitable.
This research has contribu ted techn iques which may be appl i e d successfully t o mul t ivariable self- tuning cont rol. Effi c ient programs have been wri tten which implement the contro l l e rs on a mi crocomputer. Sugge s ti ons for future work include the
development of a program genera tor whic h w i ll al low more compac t code to be developed, d ed i cated to a particular process, and which will execute more quick ly. S tra tegies which t·etter enabl e self- tuning cont rol lers to d eal w i th non-l inear processes a re a lso of interest.
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A CK NOWLEDGEMENTS
I w i s h t o tha nk Pro f e s s o r J . K . S c o t t f o r h i s s u p p o r t , a nd t h e facil i t ies o ff e r e d by the De partm ent o f Ind u s t r i a l Manage m ent and Eng ineer in� t h e Ne w Zea l and Da i ry R e search I ns t i tu te , fo r a c c e s s t o th e i r p l a n t d u r i ng e x p e r i m e n t a l p h a s e s o f t h i s p r o j e c t; D r R . I . C h a p l i n , w h o m a d e p o s s i b l e t h e u s e o f I S I S a nd p r o vid e d supe rvis ion d u r ing the l a t te r s tages o f th i s resea rc� and m os t o f a l l , my wife , Bery l.
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TABLE OF CONTENTS
Page
CHAPTER LITERATURE SURVEY
'1 . 0 In troduct ion . . • . • . . • . . . • . . • . . . . • . • . . . . • . . •
1 . 1 Sel f- tun ing regula tor based on
m inimum variance control . . . • . . . • • . . . • •
1 • 1 • 1 1 • 1 • 2 1 • 1 • 3 1 • 1 • 4
The CARMA model
Optimal pred iction and m inimum variance control . . . . • . • . • • • • . • • • • • •
Self-tuning . . . . Mul tivariable sel f- tun ing
regulators . . . • . • . . . • • • . . • . • • • . •
2 3 4
5 1 . 2 Impl i c i t optima l sel f- tun ing
con tro l lers . . . 6
1 • 2 . 1 1 • 2 . 2 1 • 2 . 3 1 . 2 . 4
Pred iction and control w i th known parame ters • . . . . • . . • . • • . • • • • • . • . • • . • 7 S e l f- tuning . . . . • . . . • • . • • • . . • • • • • . • • 9
C losed loop behaviour • . • . . . • • • . • • 1 2 Mul tivariable sel f-tun ing
controller • • . . . • • • • • • . • . • . . • • • . • . . • 1 3 1 . 3 Impl ic i t pol e assignment sel f-
tun ing control l ers . . . • . . • • • . • • • . • . . • • • • • 1 4 1 .3.1 Pol e assignment based on opt i m a l
self-tuning controller • • • • • • • . • • • • • 1 4 1 . 3 . 2 Deterministic pol e assign ing
controller • . • . • • • . . . . • • • • • • • • • • • • • • 1 5 1 . 4 Expl i c i t self- tuning controllers
w i th pole assignment • • . • • • • • • • . • • • . . • • • • 1 7 1 • 4 . 1
1 • 4 . 2 1 .4 . 3 1 .4 . 4
Determ in i stic pol e-assigning
contro l le r • • • • . • • • • • • • • . • • . • • • • • • • • 1 7 S tochastic pole-assigning regu l a tor 1 8 Po le a ss igning regulator w i th
varying time delays • • • • . • . • • • . • • • • • 2 1 S ervo self-tuners . . . . • . • • • • • • . • • . . • 22 1 . 5 Applicat ions • • • • • • • • • • • • • • • • • • • . . • • • • • • • • 25
1 • 5 • 1
1 • 5 . 2 Implemen ta tion . . . . • • . • • • • • • • . • • • • • • 25 Testing the algorithms • • . • • • • • • . • • • 27 1 . 6 Compar�son of algori thms • • . • • • • • • • • • • • • • • 28
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CHAPTER 2
Page SELF-TUNING ALGORITHMS
2 . 0 I ntroduct ion . • . . • . . . • . . • • • . • • • • . • • 3 1 2 . 1 Pole plac ing mul t ivariable sel f-
tun ing control . . . 3 1 2 . 1 • 1
2. 1 . 2 2 . 1 . 3 2 . 1 • 4 2 . 1 • 5 2 . 1 . 6 2 . 1 • 7 2 . 1 . 8 2 . 1 • 9
Process description . . . • . 3 1
Mult ivariable contro l . . . . • . . . • . . • • 33
Scalar min imum variance regula tor w i th feed forward . . . • • . • 35
Po l e-placing contro l l e r • . • • . . . • . . . 37
Self- tuning on-l ine .. . . • . • . . . • • 4 0
Set point atta inment • • • . • . • • . . . • 4 1
Unmeasured d isturbances . . • . . . • • 4 3
Resta tement o f the mul t ivariable controller . . . • • • • . . . • . . . • 4 4
Non-minimum phase . . . • . . . • • • . . . . • 4 5 2 . 2 Impl ici t mul tivariable contro l l e r .. . • . . . . 46
2 . 2 . 1 2 . 2 . 2 2 . 2 . 3 2 . 2 . 4 2 . 2 . 5 2 . 2 . 6 2.2.7
Self-tuning . . . ••. . . •. . . 4 7
C losed loop . . . • . . . • . . . 48
Set point attainment . . . • 49
Disturbance rej ection . . . 49
Offse t e l imination . . . 4 9
O ther loop inputs as d i s tu rbances. 50
Summary . . . . • . . . • . . • . . . . 50 2 . 3 Ful l mul tivariable pole-plac i ng
control l e r . . . . . . . • . • . 51
Self- tuning case . . . • • . . . . • • . • 52
Se t point a ttain�ent . . . • • . • . . . . • . • 55
CHAPTER 3 METHOD
3 . 0 I ntroduction . . . • 57 3 . 1 Programs . . . 57
3 . 1 • 1
3 . 1 • 2 Expl i c i t pole plac ing a l gori thms . • 58
Programs for impl i c i t self-tun ing. 6 3 3 . 2 Computer systems . • . . . • • • • • • . . • . . • 6 5
3 . 2 . 1 3 . 2 . 2
M i n icomputer implementation . . . • • . 6 5
M i crocomputer implemen ta tion . . . • . . 6 6 3 . 3 Processes controlled . • . . • • • • • . • • • . . . • • 6 8
3 . 3 . 1 3 . 3 . 2
S imulations . • . . . • . . • . . . • . . • . • . • • 6 9
Hea t exchanger . . . • • . . . • . • . . • 70
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CHAPTER 4 Page RESULTS
4 . 0 Introduction • • . . . • . • • . . . . . • . . . • . . . . • . • • . 72 4 . 1 S imulations w i th impl i c i t sel f-tun i ng
controller. . . .. . . 73 4 . 2 S imulations us ing expl i c i t
pole-plac i ng contro l l e r . . . 76 4 . 2 . 1
4 . 2 . 2
4 . 2 . 6 4. 2 . 7
S ingle loop processes . . . • . . • . . . 77
Choice o f observe r and model
polynomials . . . . • . . . . • . . . • . . . . • 9 2
Mul t ivariable processes . . • . . . 1 05
Processes with changing parame ters 1 1 2
Di ffe rences between pole plac ing
contro l l e rs . . . • • • . • • • • • • • . • . • . . . . • 1 2 2
The self- tuning property . . . ... . . 1 29
Ful l pol e-plac ing algori thm . • . . . • • 1 33 4 . 3 C ontrol of heat exchanger . . . • 1 40
4 . 3 . 1 4 . 3 . 2 4 -3 -3 . 4 . 3 . 4 4 -3 -5
Hea t exchanger process . . . • . . . . • 1 40
S imul a ti on of hea t exchange r . . . 1 4 1
S imula t i on resu l ts • . . • . . . • . . . 1 4 3
Resu l ts o f on- l i ne experiments . . • . 1 50
Onl ine impl i c i t c on trol . . . • 1 60
CHAPTER 5 DISCUSSION AND CONCLUSIONS
5.0 In troducti on . . . • . • . • 1 63 5. 1 D iscussi on o f results . . . • • . • • 1 64 5. 2 Me thods employed . . . . • • . . . • • . 1 69 5.3 Relationship to other sel f-tun ingc
control schemes . . . • . . . • 1 70
APPENDIX A Numerical examples of pole-placi ng
controller design . . . • . • • . . . • . 1 73
APPENDIX B Behaviour of heat exchange r under three
term control . . . • • . . . . • . • . . . . • • . • • 1 8 1
REFERENCES 183
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NOTATION
Bo l d fa ce l e t te r s a re u sed to repre s e n t polyno m i a l s , po lynomial matrices or matri ces. A polyno m i a l i s an expressi on i n the back
ward shift operator q-1 • Thu s :
a q-nA nA
nA i s d e f i n e d to be the o rd e r o f the p o l yn o m i a l A. Po lyno m i a l s ope rate on process variable s so tha t :
y i s o n l y d e f i n e d a t d i s c r e t e t i m e i ns tan t s , y( t ) be i ng t h e ( sa m p l e d ) va l u e o f y a t the prese n t t i m e. y ( t- 1 ) , y ( t - 2 ) . • represent values o f y a t prev i ous sam pl i ng instants. The sampl ing i n te�ya l i s not expl i c itly s tated. The dependence of a polynom i al on q w i l l be om i tted for brev i ty unless ambi gu i ty resu l ts. A polynomial matri x may be represen ted equivalently by :
B = B 0 + B - 1 1 q + • • • + B nBq -nB where tne Bi Bre m x p matrices, or by :
l �� � B1 2 :��J
B Bm1 mp
where the Bi j a re scalar polynomials o f order nB.
The s c�
f
a r quan t i ty A ( 1 ) i s the v a l u e o f polyno m i a l A e v a l u a ted w ith q = 1 .P r o c e s s e s m ay b e repre s e n t e d i n A u to R egressive Movi ng Average (ARMA ) form by :
Ay ( t ) = q- kBu ( t ) + q-kDv ( t ) + Ce ( t ) + d
A , B , C a n d D a r e po l y n o m i a l s ( o r po l y n o m i a l m a t r i c e s ) repre sen t ing a s calar (or a mul t i variable) process. The process t i m e d e l ay k i s e x p re s s e d as a w ho l e numbe r of sa mpl i ng i nterval s. y( t) i s the process output , u(t) the contro l l ed input,
v( t) a m easured d isturbance and e ( t) a w h i te noi se sequence. d i s•
an o ffset .
y(t\t-k) is the predicted value of y at time t, given information up to and including time t-k.
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P rin cipa l symb o l s used in this thesis
k
i , j , l , m , n A, B, C, D
A1, A2, B1, C1 A', B', D', A•, B•
y(t)
w(t)
u(t), u ' (t)
v(t)
e(t), e '(t), e"(t)
d , d ' H,G,E
S1 , S2
P , P' , Q, Q', H, S
L, Jl, B1 , B2
�(t), �;y;(t), K
X, 9, x(t)
�· 'f, '1'- Ao
time d e l ay indices
Process polynomial s/po lynomia l m a t rices
Proce s s ou tput Process s8t- point Cont rolled inpu t Measured d i s tu rbanc e Whi te noise sequences O ffse t va l ues
C on t r o l l e r p o l y no m ia l s Hu(t) + G y(T) + Ew(t) = 0
Mu l t iva riable pole p l a c i ng polynomia l ma trix
Po lyno m i als for se rvo c ont rol Gene ra l po lynomia l s
Sca l a r quantities Gene ra l ve c to r s
Polynomials spec i fying p o l es for po l e placement
Obse rve r polynomia l
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