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Implementation of an adaptive controller on a TI TM 990 microcomputer Tooba, Bahram 1982

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IMPLEMENTATION OF AN ADAPTIVE TI  TM 990  CONTROLLER ON A  MICROCOMPUTER  by Bahram Tooba B.Sc.  Purdue U n i v e r s i t y ,  1979  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED  in  SCIENCE  the department of  Electrical We a c c e p t to  this  thesis  the required  THE UNIVERSITY  as  conforming  standard  OF BRITISH COLUMBIA  March ©  Engineering  1982  Bahram T o o b a , 1982  In p r e s e n t i n g  this thesis  requirements British  it  freely available  for  that  f u l f i l m e n t of the  f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y  of  agree  in partial  Columbia,  I agree that f o r reference  permission  scholarly  the Library  shall  and study.  I  f o r extensive  for  that  copying or p u b l i c a t i o n shall  of this  Bahram  Electrical  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  D  DF-fi  a  f?/79i  t  e  March  11  1982  It i s thesis  n o t be a l l o w e d w i t h o u t my  permission.  of  thesis  p u r p o s e s may be g r a n t e d by t h e h e a d o f my  f i n a n c i a l gain  Department  further  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood  make  Engineering  Columbia  Tooba  written  ABSTRACT  The  thesis  identifier  and  microcomputer The  to  implementation  controller  adaptive  on  a  of  Texas  algorithm  [ 2 0 ] , and  of c o m p u t a t i o n  recursive  the  an  adaptive  Instrument  TM 990  system.  Martin-Sanchez amount  tests  least  was  needed  was  selected  because  i n comparison  squares e s t i m a t i o n .  proposed  o f the small  w i t h methods s u c h a s  The c o n t r o l l e r  a s i m p l i f i e d minimum v a r i a n c e c o n t r o l l e r  by  used  is  similar  in self-tuning  regulators. The  thesis  describes  hardware and s o f t w a r e used method.  the  adaptive  algorithm  i n the microcomputer  of  this  The r e s u l t s  show o p e r a t i o n  for  f i r s t - o r d e r and s e c o n d - o r d e r p r o c e s s e s .  and  the  implementation  of the algorithms  TABLE OF  TABLE OF  CONTENTS  CONTENTS  .  L I ST OF  TABLES  L I S T OF  ILLUSTRATIONS  .  ii i v  ACKNOWLEDGEMENTS  . . .v .....  ....vii  I . ADAPTIVE CONTROL 1. 2. 3. 4.  1  Introduction ..... A d a p t i v e P a r a m e t e r E s t i m a t i o n and C o n t r o l Computer S i m u l a t i o n The O u t l i n e o f T h i s T h e s i s  11. IDENTIFICATION AND  CONTROL ALGORITHM  1 ..........4 9 .10  .....  12  1 . Ove r v i ew 12 2'. D i s c r e t e - T i m e S y s t e m s « ....... .... 1 2 3. An A l g o r i t h m f o r M o d e l R e f e r e n c e A d a p t i v e S y s t e m s ..13 3.1. F o r m u l a t i o n . ......16 3.2. H y p e r s t a b i l i t y of t h e C o n t r o l S y s t e m ............20 3.3. R e l a t i o n Between e ( k ) and s ( k ) ...23 3.4. C o n v e r g e n c e o f t h e C o n t r o l B l o c k P a r a m e t e r s ...25 3.5. C o n c l u s i o n s .27 v  III.  HARDWARE AND  SOFTWARE  1. O v e r v i e w 2. The Hardware 2.1. TM 990/101M- 1 M i c r o c o m p u t e r 2.2. RTI-1241-S B o a r d 3. The S o f t w a r e 3.1. The F i r s t - O r d e r S y s t e m s 3.2. The S e c o n d - O r d e r Systems 3.3. A r i t h m e t i c O p e r a t i o n s 3.4. The S a m p l i n g R a t e 3.5. The Computer Programme  28 ......28 28 28 ....29 .......30 .30 34 35 37 39  -iii-  IV.. MICROCOMPUTER EVALUATION OF THE ADAPTIVE IDENTIFIER AND CONTROLLER . . . . . . . . . . . . . . .  .......... 42  1. O v e r v i e w 2. F i r s t - O r d e r P r o c e s s e s 2.1. I d e n t i f i c a t i o n 2.2. C o n t r o l 3. S e c o n d - O r d e r P r o c e s s e s 3.1. I d e n t i f i c a t i o n 3.2. C o n t r o l  .....42 ....43 47 51 58 .58 ..62  V. CONCLUSION  .67  1 . Summmary  APPENDIX A A.l. A. 2. A.3. A. 4. A.5. REFERENCES  67  THE SYSTEM HARDWARE TM 990/101M-1 M i c r o c o m p u t e r RTI-1241-S B o a r d TI 745 T e r m i n a l TM 990/510 C a r d Cage The Power S u p p l y .".  ...  .  70 .......70 71 ...73 ...73 73 75  -iv-  L I S T OF  Table  3.1  TABLES  The s a m p l i n g r a t e s and t h e i r hexadec imal coded d a t a .  corresponding 38  -v-  L I S T OF  FigFig. Fig.  1.1 1.2 2.1  ILLUSTRATIONS  The B l o c k D i a g r a m o f (a) RLS A d a p t i v e I d e n t i f i e r (b) S e l f - T u n i n g R e g u l a t o r ( S T R ) . B a s i c C o n f i g u r a t i o n o f a Model A d a p t i v e s y s t e m (MRAS)  ......5  Reference 8  C o n f i g u r a t i o n of the A d a p t i v e System i n : (a) I d e n t i f i c a t i o n Mode, (b) C o n t r o l Mode.  .......14  Fig.  2.2  G e n e r a l C o n f i g u r a t i o n o f an A s y s m p t o t i c a l l y H y p e r s t a b l e F e e d b a c k Autonomous S y s t e m . ..........19  Fig.  2.3a  The N o n l i n e a r A s y p t o t i c a l l y  Hyperstable  F e e d b a c k Autonomous S y s t e m s , Fig.  2 = 3b  Fig.  3.1  Fig.  3.2  .21  The E q u i v a l e n t Autonomous S y s t e m  .......21  Flowchart f o r the I d e n t i f i c a t i o n of a F i r s t - O r d e r System F l o w c h a r t f o r the C o n t r o l of a F i r s t - O r d e r  ....32  System  33  Fig.  3.3  Flowchart  f o r two's Complement M u l t i p l i c a t i o n . '  Fig.  4.1  Fig.  4.2  The B l o c k D i a g r a m o f t h e A d a p t i v e I d e n t i f i c a t i o n Mode The B l o c k D i a g r a m o f t h e A d a p t i v e C o n t r o l Mode  ...36  System i n 4-4 System  i n the 45  Fig. Fig.  4.3a 4.3b  The E s t i m a t e d P a r a m e t e r s , The P r o c e s s O u t p u t and The P r e d i c r e d O u t p u t  .48 48  Fig. Fig.  4.4a 4.4b  The E s t i m a t e d P a r a m e t e r s The P r o c e s s O u t p u t a n d The P r e d i c r e d O u t p u t  49 49  Fig. Fig.  4.5a 4.5b  The E s t i m a t e d P a r a m e t e r s . The P r o c e s s O u t p u t a n d The P r e d i c r e d O u t p u t  50 50  Fig. Fig.  4.6a 4.6b  The E s t i m a t e d P a r a m e t e r s . The P r o c e s s O u t p u t and The P r e d i c r e d O u t p u t  ...52 52  Fig. Fig.  4.7a 4.7b  The E s t i m a t e d P a r a m e t e r s . The P r o c e s s O u t p u t and The P r e d i c r e d O u t p u t  ..53 53  Fig. Fig.  4.8a 4.8b  The P r o c e s s O u t p u t f o r t h e F i r s t - O r d e r The R e f e r e n c e S i g n a l .  Fig.  4.9a  The P r o c e s s  Output  System.  ...54 54 56  -viFig.  4»9b  The C o n t r o l  Signal.  ..56  Fig. Fig.  4.10a The P r o c e s s O u t p u t ....,57 4.1 Ob The C o n t r o l S i g n a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57  Fig. Fig.  4.11a The E s t i m a t e d P a r a m e t e r s . 4.11b The P r o c e s s O u t p u t a n d The P r e d i c r e d  ..........59 Output 59  Fig. Fig.  4.12a The E s t i m a t e d P a r a m e t e r s . 4.12b The P r o c e s s O u t p u t and The P r e d i c r e d  Output  Fig. Fig.  4.13a The E s t i m a t e d P a r a m e t e r s . 4.13b The P r o c e s s O u t p u t and The P r e d i c r e d  ..61 O u t p u t . . . . . . 61  Fig. Fig.  4.14a The P r o c e s s O u t p u t . 4.14b The C o n t r o l S i g n a l .  Fig. Fig.  4.15a The P r o c e s s O u t p u t f o r t h e S e c o n d - O r d e r 4.15b The R e f e r e n c e S i g n a l  Fig. Fig.  4.16a The P r o c e s s O u t p u t 4.16b The C o n t r o l S i g n a l  60 60  63 ...63 System.  ..65 65  ........66 ....66  -vii-  ACKNQWLEDGEMENT  I w i s h t o thank Dr. M.S. D a v i e s f o r h i s p a t i e n c e , h e l p , and encouragement d u r i n g  the l i f e of t h i s p r o j e c t .  1  I.  1.  Introduction Adaptive  engineers subject  control  for  a  long  control  performance involving varying  of  large  these  interest  been made  in  ;  in  introducing  are  PID  controllers,  and  to  that  undoubtedly With  these  be  are  not  critical, t o be  increasing-  systems  i n the  [27].  PID  many  continually  In  its  spite  of  modest p r o g r e s s  has  into and  is  industry. derivative  90  percent  industry  As  time  good, p e r f o r m a n c e .  integral  controlled  by  the  adjusts  nonlinear. only  in  improving  and  controllers  indeed  controlled  continue an  type be  the  interest  that  u s e d , w i t h ..almost  this can  i s one  to maintain  proportional,  processes  designed loops  of  for  behaviour  analog' c o n t r o l l e r s used  controllers  The  engineering  system  control,  extensively  control  s y s t e m s w i t h unknown and  process  adaptive  action  industrial  need  complex of  for  r e s e a r c h p a p e r s on  1960's,  s y s t e m s ar,e g e n e r a l l y  allowing  electronic  first  the  automatically  Controllers (PID)  from  adaptive  t ke  challenge  early  uncertainties,  strategy  Therefore,  term  i n the  arose  in  a  .time, t h e  p a r a m e t e r s . An  control,  been  increasingly  m o n i t o r s changes  much  has  being published  adaptive  or  ADAPTIVE CONTROL  being  three  evident,  most  satisfactorily processes  controllers  of  with  have  [1],  In  been  control  these t h r e e term c o n t r o l l e r s  used e x t e n s i v e l y demand  i n the  PI  will  future.  f o r e f f i c i e n c y i n the  use  of  2  energy  and  increase  raw in  controllers  materials, the  it  only  terms,  may  be  any  fairly once  of  control  easy  the  which  has  to  tune,  easy  industrial  i n an  of  be a d j u s t e d , and  systematic  The is  adjustment  address  this  small  its  model  mathematical or  system  can  of  implementing why  where it  is  fluctuate. however,  be  A  is  to  a if PID not  switched o f f  used  has  there  well-tuned  include  not  are in  advantage.  feedforward,  number o f  parameters  be  without  done  a  attempt  and  control  control  theory  too  difficult  advanced  to d e r i v e  be  identification.  has  not  t o implement  controllers,  model  f o r the  the c o n t r o l l e r  o b t a i n e d from p h y s i c a l The  process  The  on  i t i s often  parameters  disturbance characteristics.  model can  procedures  Some o f t h e a l g o r i t h m s p r o p o s e d f o r  In d e s i g n i n g  and  advanced  modern  to develop a mathematical  disturbances,  these  i s often  have a l a r g e  computationally  computers.  necessary  which  procedure. Adaptive c o n t r o l l e r s  in explaining  are  type.  i f t h e p r o c e s s dynamics-  which  this  been u s e d more e x t e n s i v e l y . example,  require  problem.  difficulty  a reason  them  settings  action  controllers  observers often  which  controllers,  a l t h o u g h i t might  which  to  these  particularly  and  can  problems  t h r e e or four parameters,  feedback  corresponding  installation  most d e s i r a b l e  controllers  complicated  a  t o t u n e a PI c o n t r o l l e r  slow. T h e r e f o r e the d e r i v a t i v e  More  is  t h a n t h e c o n v e n t i o n a l PID  operating hundred  there  amount o f work t o keep a l l o f  reason  controller very  is  several  substantial for  number  more s o p h i s t i c a t e d  Altough two  however,  and using  appropriate  consideration  drawback w i t h s u c h a p r o c e d u r e i s  3  that  i t  is  attention  time  of  consuming,  personnel  with  identification  and c o n t r o l  be  as  regarded  identification without its  may  skills  need in  the  convenient design  continuous  modelling,  d e s i g n . The a d a p t i v e  and a c o n t r o l  c o n t r o l l e r can  combination techniques  system  of  system  that can  operate  s u p e r v i s i o n . In a d d i t i o n , i f the system parameters and  disturbance  properly system  a  and  statistics  designed  are  adaptive  controller  f o r close to optimal  Recent  increased  concentrated adaptive thereby  algorithms  a viable  for  conventional autopilot  autopilot  on  recursive  controller, designs discussed  adaptive  simplify  justified  control  and  on t h e t a n k e r  least the  recent  engineers  The s u c c e s s  the i n c r e a s e d  of  can  estimation with "self-tuning  a  of  the  adaptive  the c o n v e n t i o n a l i s based variance  regulator".  design  an  to replace  a minimum  r e p l a c e the c o n v e n t i o n a l  i n [ 2 2 ] , which d e s c r i b e s  reference:techniques.  outperformed  these  of adaptive  a l l conditions. This design  squares  so-called  [23],  with  interest in  seems l i k e l y  tests  of  microcomputers,  f o r a p a p e r m a c h i n e . The d e s i g n  in  has  the implementation  minicomputers  has  under a l m o s t  which  in  s t e e r i n g large tankers  installed  PID  that  on  autopilots,  a  tune the  [14] d e s c r i b e s a s u c c e s s f u l a p p l i c a t i o n  controller  autopilots  time,  can c o n t i n u o u s l y  t o PID c o n t r o l l e r s .  controllers  them. R e f e r e n c e adaptive  with  t h e c o s t , and p r o v i d i n g c o n t r o l  alternative  adaptive  varying  performance.  interest  on p r o c e d u r e s  reducing  slowly  Other  PID a u t o p i l o t a r e based  on  model  4 2. A d a p t i v e Parameter E s t i m a t i o n and C o n t r o l The  two  primary  approaches  that have been developed  i)  to parameter  are;  R e c u r s i v e Least Squares (RLS),  i i ) Model Reference  and  Adaptive Systems (MRAS).  Both methods have r e c e i v e d great a t t e n t i o n The  previous  work  done  d e s c r i b e d , although  identification  on  these  in  the  methods  literature.  will  be  briefly  i t i t must be r e a l i z e d t h a t there have been  hundreds of papers p u b l i s h e d on these methods d u r i n g  the  past  20 y e a r s . The  R e c u r s i v e Least Squares method has been developed  successfully suitable  implemented i n the i n d u s t r y , when combined with  control  method. F i g . 1.1 regulator.  and  In  algorithm. shows  this  the method  equation model are determined  [2] block the  is  a  good  diagram  survey of  of  parameters  a of  a  this  self-tuning a difference  by m i n i m i z i n g the c o s t :  K V=  E |y(kT)-y(kT)|  2  with r e s p e c t to these parameters, output  y(kT)  is  the  process  and y ( k T ) i s the p r e d i c t i o n model output. The r e c u r s i v e  a l g o r i t h m can be found are  here  i n [2] or [ 3 ] . The  updated a f t e r each sampling,  and  used  parameter  i n the t u n i n g of the  the c o n t r o l a l g o r i t h m . In the work of Peterka Wittenmark  [4],  and  Wittenmark  [19],  estimates  the  [18], Astrom, and recursive  least  5  y  u  PLANT  + J  ADJUSTABLE  u  y  MODEL  PARAMETER ESTIMATOR  (a)  r u -c  -  CONTROLLER  u  PLANT  i. PARAMETER ESTIMATOR  1 1 1  1  i j  REGULATOR (b)  Fig.  1.1  The B l o c k Diagram o f  (a) RLS A d a p t i v e I d e n t i f i e r (b) S e l f - T u n i n g R e g u l a t o r (STR).  y  6  squares  parameter  variance  controller„  chosen  so  that  estimation In  the  parameters,  calculated  directly.  so t h a t In  and B o r r i s o n and S y d i n g  work.  telecommunications and a computer [10]  Kurz,  [15]  and 6 d i f f e r e n t 12  adaptive  may  be  coincide with  the  parameters  ate  controller  application  that  a plant  it  was  in Kiruna  i n L u n d i n s o u t h e r n Sweden,  and  was  used  with  performed  using  in northern  Sweden  and Schumann e x p e r i m e n t a l l y c o m p a r e  and  parameter  1800  km  e s t i m a t i o n methods,  the  Maximum L i k e l i h o o d (RML)  c o n t r o l schemes.  These  estimation,  e-3 t i mat i o n  a l g o r i t h m s were  method,  p h a s e a n d non-minimum p h a s e ,  The  with time adaptive  of  adaptation  regulator  a l g o r i t h m of  v e r y good e x p e r i m e n t a l  (RLS) c a n be  estimation determined  and t h e  implemented processes.  better  control  and u n s t a b l e  of  when  algorithm  e.g..  minimum  processes,  and  delay. b a s e d on t h e  b r o a d f i e l d of a p p l i c a t i o n s , h o w e v e r of  perform  type of p r o c e s s e s , stable  (RLS)  so c o n s i d e r i n g a t o t a l  the c o n t r o l a l g o r i t h m s  c o u l d be a p p l i e d t o d i f f e r e n t  processes  of  In  control alogorithms,  RLS  these  apart.  Recursive  found t h a t  [14],  of  o n a p r o c e s s -.computer a n d t e s t e d on s i m u l a t e d a n a l o g It  minimal  r e p o r t e d . An i n t e r e s t i n g a s p e c t  was  between  e v a l u a t e two r e c u r s i v e estimation  the  a  structure  parameters  successful  are  [15]  Isermann,  v?ith  t h e work o f C e g r e l l and H e d q v i s t  'self-tuning regulators' second  combined  t h i s case the model estimated  controller  the  is  is  not a l l  the  a proof  type i s not  results  that  from  this  (RLS)  [3],  of  seems the  the parameters  input-output  convegence  available  Another  t o have a  in  spite  problem  with  of  the  model  observations  when  7  feedback explain  exists how  guidelines  in  to  the  overcome  f o r unbiased  estimated  process.  parameters  this  Astrom  problem  parameter  t o become  very  and Wittenmark [ 4 ] and  suggest  some  e s t i m a t i o n which a l l o w t h e close  to  their  optimal  values. The been  Model R e f e r e n c e  introduced  Figure this  1.2  by W h i t a k e r ,  shows  method t h e i n p u t  vector  model  parameters  introduction,  Landau 1973,  [8]  u i s f e d both  being  i s used  been  to  an  book  e-*-0  have  provides  no  by o t h e r s  analytical  estimates, method  was  algorithm  output  The  by Popov  such  from  of  the  laws  methods, P e a r s o n  the experimental by  i s related  results  Martin-Sanchez t o Popov's  1964 t o  techniques Landau  have  uses  of  method  has  there i s  the parameters  are  [20],  been  [13] developed  [ 2 9 ] , Here a g a i n  convergence  MRAS.  [16] f o r the proof  adaptation  as Serdyukov  i t s  doctoral  of  on t h e g r a d i e n t method, a n d t h i s  proposed  which  MRAS.  on d i f f e r e n t  proof  although  and  good e x a m p l e s o f a p p l i c a t i o n s .  of  used  papers  several different  the convergence  been  to the  Since  o f MRAS  of  based  and  t-*- ° ° .  e x c e l l e n t survey  theorem proposed  algorithm  MRAS. I n  an e r r o r e ( t ) . T h i s  types  hyperstability  an  i n 1958.  the process  various  a  based  as many  the  [9] d e s c r i b e s  i n MRAS a n d p r o v i d e s  formulated  [28]  t o the process  used t o generate  force  dedicated  a recent  Kezer  seems t o have  by an a d a p t i v e m e c h a n i s m t o a d j u s t t h e  and  there  dissertations  Yarmon,and  (MRAS),  t h e d i f f e r e n c e between  t h e model o u t p u t  error  method  the b a s i c c o n f i g u r a t i o n of standard  a d j u s t a b l e model w i t h and  Adaptive  good.  A  new  f o r an a d a p t i v e  hypertabil.ity  criterion  DISTURBANCES  y(t)  PLANT  u(t)  e(t)  DISTURBANCES  ADJUSTABLE MODEL  dft)  ADAPTIVE MECHANISM  F i g . 1.2 Basic Configuration of a Model Reference Adaptive System (MRAS)  9  [16]  as  applied  different  from  adjustment carried output in  the  out based  formulation*  1979  Yale  the information  estimation  and  University It  this  method  the  block parameters i s  available  from t h e i n p u t and  be d e s c r i b e d  in  detail  number o f p a p e r s p r e s e n t e d  Workshop on t h e A p p l i c a t i o n o f  includes  MRAS.  In  and c o n t r o l  I I . [24] contains a large  Adaptive Control.  good  Applications  tutorials described  on  both  in this  RLS  book a r e  m o s t l y on t h e s e two m e t h o d s .  3. Computer  Simulation  In o r d e r t o a s s e s s implementation, proposed  were  by  in  subroutines.  simulated  first-order  p r o c e s s and a  were  very  satisfactory  noise  free  [ 3 ] , and  The  the  The  transfer  identical  the  method  Martin-Sanchez  matrix  operations  fuctions  f o r both process.  were  of  the  methods, The  reached  their  (RLS) e s t i m a t i o n  was f o u n d  of i n i t i a l  was p e r f o r m e d .  and o u t p u t  of  The c o m p u t a t i o n  with  t h e two  t o be s l i g h t l y  parameters  were  correct  o f t h e two methods d e t e r i o r a t e d  of n o i s e a t t h e input  a  results  f o r b o t h methods when t h e s y s t e m s  t o the s e l e c t i o n  identification  microcomputer  c o m p u t e r . The programmes  and  second-order  performance  introduction  sensitive  were  for  squares estimation  and t h e e s t i m a t e d p a r a m e t e r s  The  processes.  least  Amdahl/470  Fortran  processes  values.  suitability  and Wittenmark  simulated using  written  performed  their  the- r e c u r s i v e  by A s t r o m  method were  the  [ 7 ] , but i s s i g n i f i c a n t l y  o f t h e p r o c e s s . T h i s method w i l l  the  based  MRAS•  systems  of the i d e n t i f i c a t i o n  Chapter  in  to discrete  when  t i m e was a  more  system little  10  higher  for  (RLS)  operations, the so  case the  especially  of  (RLS)  number of  identification needed about only  25  decided  work  (RLS)  100  that  as  the  number  and  the  p a r a m e t e r s t o be of  on  The  multiply  estimation  results  arithmetic  d i v i d e , was  much more i n  of  identified  from the  because  f o r microcomputer  method t h a t  has  it  been  This  Thesis  the  In  RLS  and the  method  compared  with  method. computer  simulation be  and  implemented  simpler  a l s o because on  it  considered  i s computationally  implementation  not  processes,  increased.  divide operations  t h e M a r t i n - S a n c h e z method w i l l  mainly  the  process,  for Martin-Sanchez  the  of  order  second-order  m u l t i p l y and  operations  Based  our  estimation.  a  was in than  it is a  microprocessor  before.  4.  The  Outline  The  of  aim  of  this  t h e s i s i s to  M a r t i n - S a n c h e z method on found  to perform  fairly  good  the  as  Chapter  II  will  formulation,  the  proof  convergence  of  the  implementation  of  interfacing  of  analyse  software  testing  the of  first-order  the systems  will cover of  be  and  This  evaluate  s y s t e m has  control  on  line  the been with  described.  the  a n a l y s i s of  the  algorithm,  the  h y p e r s t a b i 1 i t y and  the  proof  the  parameters.  the  the  a microprocessor.  identification  results,  i m p l e m e n t and  Chapter  algorithm  different  parts  employed. identifier (with  on  at  of  most  discuss  microprocessor the  Chapter and  III w i l l  the two  system. IV  of  the  and  the  It w i l l  also  cover  the  will  controller  for  parameters)  the and  11  second-order discuss  the  controller.  systems results  (4 parameters and w i l l  maximum).  Chapter  e v a l u a t e the i d e n t i f i e r  V and  will the  12  II.  1.  I D E N T I F I C A T I O N AND CONTROL ALGORITHM  Overview In  will  t h i s chapter, the i d e n t i f i c a t i o n be  described.  algorithm,  the proof  estimates,  The  of the  mathematical  the  proof  of  block parameters  will  be o u t l i n e d .  2. D i s c r e t e - T i m e  Systems  time  process  in  r e p r e s e n t e d by t h e t r a n s f e r  o s + a  a  the  noise free  environment  /  To c o n t r o l a p r o c e s s  convenient  to develop  +o.  in  '°  such as (2.1) d i g i t a l l y  of the d i g i t a l  i t i s  a d i s c r e t e e q u i v a l e n t of G(s) preceded  s a m p l e a n d h o l d b l o c k r e p r e s e n t i n g t h e D/A  result  c a n be  (2.1)  w h e r e m>n.  will  parameter  function  m-f  output  the  time-invariant  G(s)=  a  of  of the c o n t r o l  (SISO) l i n e a r  +....+a. s  s  of  the convergence  A single-input-single-output  algorithm  foundation  hyperstability  and  continuous  and c o n t r o l  controller.  The  conversion at the  discretization  i n a model of t h e p r o c e s s which  t h e z - t r a n s f o r m domain o r , e q u i v a l e n t l y ,  of  G(s)  c a n be r e p r e s e n t e d by an  input-output  d i f f e r e n c e e q u a t i o n of t h e form: y(k) = a y(k-l)+a_ y(k-2) + (  2  b u(k-1)+b u(k-2)+ 0  1  2  +a y(k-n)+ n  +bu(k-m) m  by  (2.2)  13  We w i l l can  study  the i d e n t i f i c a t i o n  be r e p r e s e n t e d  the  algorithm  algorithm  chapter  proposed  be i m p l e m e n t e d  microcomputer This  by a d i f f e r e n c e e q u a t i o n  first  will  and c o n t r o l o f  system  by on  a  to  the  which  such a s (2.2) u s i n g  Martin-Sanchez Texas  t o be d e s c r i b e d  i sdedicated  systems  [20].  This  Instrument  in detail  analysis  TM 990  i n Chapter I I I .  of  the  adaptive  algorithm.  3.  An A l g o r i t h m In  this  f o r Model R e f e r e n c e A d a p t i v e  algorithm,  which uses only  information  process.  The c o n t r o l l e r  of  the  plant.  the  process  control  some  input,  control block  the input  and  or  process  reference.  output  method  can  carry  parameters. control,  The  an  formulation  and  plant  an  error  the  the  to  inverse equal t o  follow  2.1a  algorithm  and  which  needs  2.1a, an i n p u t  identification  block.  and t h e i d e n t i f i c a t i o n  , which w i l l  be u s e d block  2.1b.  initial  Figure  diagram,  2.1a  but the  different.  In t h e  u i s f e d t o both the  The c o m p a r i s o n block  output  by t h e a d a p t i v e  parameters  this  i d e n t i f i c a t i o n and  block  i s quite  the  c a n be  f o r any  of t h e system,  identification  the i d e n t i f i c a t i o n  output,  MRAS  identification  Figures  mode, F i g u r e  output e  the  of the  of the parameters of the p l a n t ,  modes  in  MRAS  of  identification  the  two  a r e shown  resembles  process  out  output  The i d e n t i f i c a t i o n  o u t on l i n e , a n d u n l i k e t h e s t a n d a r d knowledge  i s developed  t o behave a s t h e e x a c t  way t h e c o n t r o l b l o c k  drives the  input  a'priori  adjust  from  attempts  In t h i s  block  carried  an a d a p t i v e  Systems  between  generates  mechanism t o  so as t o  make  e-*-0  14  PLANT  ADJUSTABLE  i  MODEL  t  I  ADAPTIVE MECHANISM  (a) d(k)  -1  u(k)  d(k+i;  CONTROLLER  PLANT  ADAPTIVE MECHANISM  (b) Fig.. 2.1  Configuration of the Adaptive System i n : (a) I d e n t i f i c a t i o n Mode, (b) Control Mode  y(k)  e(k)  15  as  t~*oo.  In the  the  control  mode, F i g u r e  input  to  the  mechanism  to  g e n e r a t e an i n p u t  measure  of the d e v i a t i o n  output.  This  parameters, output  control  error  is  instant  d(k)  of the p l a n t again  uses  t o the p l a n t . output  used  which  be  the d e s i r e d  parameter  will  generate  the  non-zero  as  the  from  the  e isa desired  the c o n t r o l l e r  that  the plant i s  an  input-output  by  scheme  will  y.  An  control The  a t sampling  algorithm  modes  is  s i g n a l u ( k ) from  adaptive  mechanism  i s t h e same d u r i n g  control  of  both the  operation.  be shown t o be  The  asymptotically  by u s e o f t h e Popov h y p e r s t a b i l i t y c r i t e r i o n [ 1 6 ] ,  the c o n t r o l l e r i s designed and  controller  The e r r o r  of the p l a n t  e ( k ) = y (k)-d(k)«,  and  estimation  hyperstable  t-*- co,  output  t h e model p a r a m e t e r s  identification  adaptive  (2.2).  d ( k + l ) and t h e e r r o r to adjust  output i s  the  t o update  k, a n d o f t h e same d i m e n s i o n  required  therefore  will due  remain  parameters,  achieve  equilibrium.  properly  y(k)-*-d(k).  then  the error  The  parameters  e-*-0 of  u n c h a n g e d when e=0, b u t i f t h e e r r o r  to process  plant's  the  which  s o a s t o make e-*»0. I t i s assumed  o f t h e form  Let  If  block,  c o n t r o l l a b l e and c a n be r e p r e s e n t e d  equation  used  2.2b, t h e d e s i r e d  then  perturbations proper  the c o n t r o l l e r parameters.  will  the is  or t h e v a r i a t i o n of the  control action  This, of c o u r s e  as  will  be done  by  be t a k e n t o adjusting  16  3.1.  Formulation Consider  a  process  .characterized  by  the  difference  equation y(k) =  where y ( k ) plant and  at the  (rxl),  u(k)  sampling output  then  addition, It  and  are  the output  instant  B are  real  matrices  i s desired to  find  an  B u s i n g o n l y the  identification  procedure  A(k-1)y(k-1)+  where A ( k - l ) and at  the  predicted  available  Equation the  output  information  sequence  the  k-1;  will the  (rxr).  u  say In  controllable. estimate  output  the  y_ . T h i s  r e f e r e n c e model:  (2.4)  of  d(k)  sampling  instant  the m a t r i c e s  will  A  and  t h e r e f o r e be  the  instant  k,  from  the  k-1.  d e s c r i b e s the  control  u(k)  block,  obtained  d ( k + l ) . Thus e q u a t i o n  [B(k)]-'d(k+1)-  above e q u a t i o n  that  i n p u t u and  i s the c o n t r o l l e r on  i s output  the e s t i m a t e s  at  also  available  u(k)=  In  (2.4)  at  plant  same d i m e n s i o n ,  dimension  i n c l u d e the  instant  output  the  the  input vector  B(k-1)u(k-1)  B(k-1) are  sampling plant  information  will  i n p u t v e c t o r s of i f the  of  identifier  m a t r i c e s A and  B  the  k. F u r t h e r m o r e ,  i t i s assumed t h a t t h e  d(k)=  and  v e c t o r y_ a r e assumed o f  A and  (2.3)  Ay(k-1)+Bu(k-i)  (2.4)  [B(k>]- A(k)y(k) ,  d(k+1) r e p r e s e n t s t h e  but  from  now the  becomes  (2.5)  d e s i r e d output  of  17  the  plant  and,  at  instant  since  governed ensure  by  the the  that  k+1.  u(k)  will  identification same s e t  the  of  process  be  block  the and  equations, ouput  input the  this  to  the  control  input  i s always e q u a l  block  will to  plant are  ideally  the  desired  outputc The control  error block  for will  both be  defined  e(k)=  To  ensure  that  the  the  identification  d(k)  converges A(k)  uses the  update:  These  new  output  parameter  the  (2.6)  a d a p t i v e mechanism t o a d j u s t following  and  as  y(k)-  error  block  to  and  zero  B(k)  is  t-**oo,  as  an  stable  developed.  This  A(k)=  AA(k)+ A(k-1)  (2.7)  B(k)=  AB(k)+ B(k-1)  (2.8)  estimates  of  A and  B can  instant  k.  Let  at  g(k)  be  be  used  the  to  predict  predicted  the  output  plant  at  this  instant  g(k)=  This  i s the  a'posteriori  A(k)y(k-1)+  a'posteriori error  The error  adaptive  s(k)  to  model  defined  s(k)=  B(k)u(k-1)  determine  will  generate  a  new  by:  y(k)-  mechanism  which  (2.9)  g(k)  proposed the  (2.10)  by  Martin-Sanchez  incremental  matrices  uses AA(k)  the and  18  AB(k)  independent  of  (A(0),B(0),y(0),u(0)).'  the  initial  The mechanism  conditions  i s b a s e d on t h e f o l l o w i n g  theorem: THEOREM and  1: The s y s t e m d e f i n e d by e q u a t i o n s  whose g e n e r a l  asymptotically generated  where  a  hyperstable  i f  i n " F i g u r e 2.1  the matrices  A ( k ) and. B ( k ) a r e  A(k-1)+  os(k)y(k-1)  (2.11)  B(k)=  B(k-l)+  0 (k)u(k-1)  (2.12)  p  are  concept  of a b s o l u t e  S  positive  constant  i s a less  I f the c o n t r o l  restrictive  s y s t e m c a n be  a linear  t i m e - i n v a r i a n t p a r t and a n o n l i n e a r  part,  a s shown  in Figure  specifies  2.2,  sufficient  stability  of the combined  satisfied  are the i n e q u a l i t y  the  system.  k,  /c=i  a*  is  a  positive  Hyperstability  conditions  I u,(k)y(k)  to  be  characteristics.  of h y p e r s t a b i l i t y  stability.  coefficients,  into  where  be  A(k)=  d e p e n d i n g on t h e p r o c e s s  The  Popov  will  by:  and  selected  c o n f i g u r a t i o n i s shown  ( 2 . 3 ) t o (2.10)  The  which main  decomposed time7varying Theorem  guarantee criteria  > -«?  constant,  form  to  of the be  (2.13a)  and a l s o t h e i n p u t / o u t p u t  relationship i s : u (k)= ;  F(y(i),i)  u(k)=-u (k) (  for  i < k, a n d  for  k> 0.  (2.13b)  19  \ u(k)  u (k) x  LINEAR TIME-INVARIANT  y(k)  PLANT  NONLINEAR TIME-VARYING BLOCK  F i g . 2.2  General Configuration of an Asymptotically Feedback Autonomous System.  Hyperstable  20 The  first  the  inequality  The if  criterion  system the  c o n d i t i o n i n the  plant  only  hyperstable,  plant  is positive shown  (2.12) can  be  defined  The  by by  familiar  conventional  2.3,  the  Popov  real  on  criterion. k-*-°°  i . e . y(k)-»~0 a s  i s bounded as  the  discrete.  It  k-#-o°, i f t h e  s(k),  Popov. The  path.  The  equations  equivalent  essence  equivalent path  system  f e e d b a c k autonomous s y s t e m  vector  forward  feedback  positive  constraint  is  linear  real.  i n t o an  i n the  unity  i . e . y(k)  form a n o n l i n e a r  identifier  the  is strictly  in Figure  investigated  block  t o the  i s asymptotically hyperstable,  linear  As  is similar  and  feedback  system  a nonlinear  linear  part  the the  with  case  system adaptive  a  time-varying  in this  to  whose s t a t e  to  i s to convert  (2.4)  linear block  reduces  in  to a  gain.  Popov h y p e r s t a b i l i t y  system  in  the  hyperstability satisfied  criterion  discrete is  assumed  form if  [16] [7]  the  f o r s u c h an states  equivalent  that  following  asymptotic  inequality  is  f o r a l l k. (2.13c)  where  s(k)'  updating  is  the  algorithm  inequality,  it will  transpose can  be  be  of. v e c t o r  shown  asymptotically  to  s ( k ) . I f the satisfy  hyperstable.  the  paramter above  21  F i g . 2.3a The nonlinear a s y m p t o t i c a l l y hyperstable feedback autonomous system.  s(k) b—  -s(k)  —  NON-LINEAR TIME-VARYING BLOCK  F i g . ",2.3b- The equivalent autonomous system.  22  3.2.  Hyperstability The  proof  of  o f t h e C o n t r o l System hyperstability  t h e a d a p t i v e mechanism from of  [20] by M a r t i n - S a n c h e z . Popov's  Landau  theorem  (2.14)  by  (2.11) and  i s actually  (2.12)  with is  a particular  case  s y s t e m s i n v e s t i g a t e d by  I.D.  i s as f o l l o w s :  [B-B(k)]u(k-1)  (2.14)  -s(k)',  s(k)'[A(k)-A]y(k-1)+  -l's(k)'s(k)= k=o  system  (2.10)  [A-A(k)]y(k-1)+  is multiplied  -s(k)'s(k)=  This  proof  ( 2 . 3 ) , ( 2 . 9 ) , and  s(k)=  in equations  for discrete  [ 7 ] . The o u t l i n e  From  If  given  f o r the c o n t r o l  s(k)'[B(k)-B]u(k-1)  E' s ( k ) • [ A ( k ) - A ] y ( k - 1 ) +  (2.15)  s ( k ) ' [B(k)-B]u(k-1)  k=o  (2.16) Equation  (2.16) c a n be decomposed  to  make  the  following  two  conditions:  K  (2.17)  l ' s(k) ' [B(k)-B]u(k-1 ) >-X \/k k.-o  (2.18)  I s(k)'[A(k)-A]y(k-1)  >-\* V  2  b  1  where X and X. a r e c o n s t a n t s , a  b  Conditions  (2.17)  and  (2.18)  can  be w r i t t e n i n s c a l a r  form: I I Z s; (k)[a-.(k)-a..]y. (k-1) >-k  2  \/ k  (2.19)  23  1  E  The  above  and  b  f  s (k)[bJk)-bJu (k-l) D  q  conditions  will  be  V  2 k  satisfied  (k) a r e f o r m e d a c c o r d i n g  t o the  k a •. (k)= I s ; ( h ) y ( h - T ) , KJ hz J  M  (2.20)  i f the parameters  (k)  algorithms  i=l n  ;  a  j = 1,n  r  (2.21)  0  b  If  k (k)= E s  (2.21) and  resulting  (h)u (h~1),  p=1,n  q=1,n  (2.22) a r e s u b s t i t u t e d  equations  are a s p e c i a l  (2.22)  i n (2.19) and  case  of  the  (2.20),  more  the  general  equation k, k k, k, E x ( h ) [ E x(h)+c]=1/2[ E x(k)+c] +1/2 E x ( k ) k=o h=o k.o k'O 2  where - ( c / 2 ) > - X ,  c=constant V k  The  of  2  left  hand  therefore, new  2  equations  this  (2.21) and  X.=cpnstant  equation  is  (2.22) c a n be  always  positive;  expressed  in  a  form. a;.(k)= b  If  side  i  (2.23)  2  s : ( k ) y (k-1 )+a.,(k-1 )  i = 1,n  j = 1,n  (2.24)  (k)= s ( k ) u ( k - l )+b ^(k-1)  p=l,n  q=1,n  (2.25)  p  (2.24) and  (2.11) and (2.21),  ;  hyperstabi1i ty.  p  (2.25) a r e c o n v e r t e d  (2.12)  (2.22)  q  are  obtained.  thus c a r r i e s  with  into matrix The  form,  parameter  i t the important  equations  updating property  of of  24  3.3.  R e l a t i o n Between Since  before  the  e ( k ) and s ( k )  a'posteriori  error  s(k)  cannot  be  obtained  t h e p a r a m e t e r s a r e u p d a t e d and t h e p a r a m e t e r s c a n n o t be  updated u n l e s s  s(k) i s a v a i l a b l e ;  the  error  a'priori  a relation  between  s(k)  and  e ( k ) must be e s t a b l i s h e d . T h i s c a n be  done  as f o l l o w s . From  (2.11),  (2.12),  s(k)=  and  (2.14),  [A-A(k-1)-cs(k)y(k-1)']y(k-1) + (2.26)  [B-B(k-1)-*s(k)u(k-l) ]u(k-1) v  From  equations s(k)=  solving  (2.4),  ( 2 . 6 ) , and  (2.26),  e(k)[ay(k-1)'y(k-1)+pu(k-1)'u(k-1 ) ]  e ( k ) / [ 1 + a y ( k - 1 ) ' y ( k - 1 ) + p u ( k - 1 )'u(k-1 ) ]  (2.28)  * ( k ) be d e f i n e d as *(k) = [l+ay(k-1)'y(k-1)+pu(k-1)'u(k-1)] -  (2.28) c a n now  adaptive  rewritten  1  (2.29)  be w r i t t e n as s(k)=  The  (2.27)  (2.27) f o r s ( k ) , s(k)=  Let  (2.3),  (2.30)  *(k) e(k)  algorithm equations  (2.11) and  (2.12) c a n  i n a form, t h a t a v o i d s  t h e need  to c a l c u l a t e  now  be  s(k) in  25  order  3.4.  t o update  the parameters.  A(k) = c*(k)e(k)y(k-1)'+  A(k-1)  (2.31)  B(k)=  B(k-1)  (2.32)  **(k)e(k)u(k-1)»+  Convergence  It  of the Control  Block  i s important t o realize  that  does  not  assure  the  their  true  values.  I t i s easy  which good  the  control  output error  system block given Noda  implies  convergence  laws  technique  The  component  examples  on i n c o r r e c t  at  least  a  local  method  to  the  based uses  <  *(k) i s defined  by  the  gradient  can give  of the  control  of the control convergence  work  of  Ngumo  parameters  i s and  estimation  formulation error  in  Section  s(k) i s given  3.3, t h e i t h by  t ( k ) e ; (k) (2.29)  (2.33)  and e  ( k ) by  e-"(k)=y- ( k ) d ; ( k )  (2.34)  :  Let  for  below:  of the a ' p o s t e r i o r i  s ;(k)=  parameter  s(k)  system  parameters  convergence  on  of  parameters t o  of  b e h a v i o u r . The h y p e r s t a b i l i t y  and i s o u t l i n e d  According  where  to give  based  Martin-Sanchez  [21 ] .  hyperstability of the model  p a r a m e t e r s . The p r o o f o f t h i s by  Parameters  e a n d e ( k - 1 ) be  9=  [a- ,...,a (  t n  ,b ,...,b ]' u  (2.35)  n  6; ( k - 1 )= [ a . ( k - 1 )•>,:.. , a j k - 1 ) ,b-,(k-1 ) , ... , b ( k - 1 ) ]» :  :  L n  (2.36)  26  Let  From  x (k-1 )  be  (k-l)=  [y,(k-1),...,y (k-1),u,(k-1),...,u (k-1)]' n  (2.3),(2.35),  and  y; (k)=  From  (2.4),(2.36),  The  algorithm  identification in  the  (2.37)  x(k-1 ) '6-  (2.38)  t  and  d;(k)=  (2.37)  x(k-l)'G-(k-1)  previously  and  (2.37)  n  control  (2.39)  developed  block  for  p a r a m e t e r s can  adjusting now  be  algorithms  estimation  are  equivalent  method w h i c h m i n i m i z e s  to  (2.40)  a  the  gradient  parameter  following cost function:  J= [e- ( k ) P / 2  This or  is a special  the  adaptive  From  written  f o l l o w i n g form: G;(k)= G - ( k - 1 ) + * ( k ) e ; ( k ) x ( k - 1 )  The  the  cost  case  function  controller  (2.40) and  of  the  used  (2.41)  cost in  function considered  the  usual  minimum  in  [11]  variance  [1].  (2.34)  G;(k)=e-(k-1)+*(k)x(k-1)[y/(k)-d:(k)]  (2.42)  27  From  ( 2 . 4 2 ) , ( 2 . 3 8 ) , and  (2.39)  e[ ( k ) = e ( k - i ) + * ( k ) x ( k - i ) x ( k - i ) ' [e-  -©. (k-1) ]  L  Let  G;(k)  be  the  identification  error:  e -(k)= e ~eL  t  From  (2.43) and  L  (2.44)  (k)  (2.44)  e l ( k ) = [ l - * ( k ) x ( k - l ) x ( k - 1 ) ' ]G;  This  (2.4.3)  is  the  case  (2.45)  (k-1)  considered  in  [21],  which yy  A  j j © ^ (k) | |  will  2  converge to zero  become o r t h o g o n a l . using  3.5.  an  input  signal  this  with  Chapter  introduced,  reference be  S u c h an  as  t->oo, u n l e s s  orthogonality adequate  can  frequency  G^ (k.) and be  x(k)  avoided  by  components.  Conclusions In  was  proves  shown  to  theorem,  this  a  the  provide  a much b e t t e r  simplified  noise  in  controller used  in  formulation  method  formulation a  The  the  imply  the  of  algorithm to a  model  and  could  recursive  control  system a c c o r d i n g that  the  t h a n a PID  similar  to  self-tuning  f o r the  parameter  a  least  method  was  to  Popov  controller  will  c o n t r o l system d r a s t i c a l l y  performance is  control  is similar  form  adaptive  hyperstable  and  is different  simplified  however d o e s not  reduce  controller,  as  method.  establish  adaptive  the  essentially  estimation  identification  identification  a p p r o a c h but  viewed  squares  the  an  or  controller. minimum  regulators, updates.  even This  variance with  a  28  III.  1.  HARDWARE AND  SOFTWARE  Overview The  are  identifier  tested  algorithm  by  and t h e c o n t r o l l e r  developed  in  Chapter II  implementing  the i d e n t i f i c a t i o n  and t h e c o n t r o l  on a m i c r o c o m p u t e r  s y s t e m . The s y s t e m  consists of:  (a)  Texas  Instrument  TM  (b)  Texas  I n s t r u m e n t TM  (c)  Texas  Instrument  990/101M-1 m i c r o c o m p u t e r 990/510  card  SILENT 700  board.  chassis.  (745 m o d e l ) e l e c t r o n i c  data  terminal. (d)  A n a l o g D e v i c e s RTI-1241-S  (e)  GSC  The  components  GOF-2A-1T t r i p l e  were  first  part  parts  and  of t h i s  2. The  the  chapter w i l l  cost  briefly  the  system ease  of  of the system.  The  describe  and the i n t e r f a c i n g .  the software production  on  to i n t e r f a c e ) ,  total  board.  supply.  based  (simplicity  finally,  of the system,  explain  o u t p u t power  selected  requirements,compatibility programming,  ( A / D ) , (D/A) c o n v e r s i o n  The  the  different  second p a r t  will  and i m p l e m e n t a t i o n .  Hardware  2.1. TM  990/101M-1  The  Microcomputer  TMS  9900  microprocessor  TM990/101M-1.  This  16  architecture hardware addressing  and  multiply  the and  bit  CPU  extensive divide.  modes. The c l o c k  rate  is  the  features  heart  is 3  are MHz.  the  a memory t o memory  instruction There  of  a The  set  includes  total  of e i g h t  communication  29  with  t h e T I 745 t e r m i n a l i s c a r r i e d  RS-232-C  serial  interval routine  I/O  interface.  t i m e r s which a r e used  out through There  are three  debugging  is  done  rate.  interface  between  the  t h e TM990/101M-1 m i c r o c o m p u t e r . The TIBUG p r o g r a m two  TI  2708  microprocessor provides  every  seven  automatically system  EPROM's  is  important  and  word  everytime  set  to  also  is  software  service  by u s i n g t h e TIBUG debug  w h i c h p r o v i d e s an i n t e r a c t i v e  uses  two  the  operate  about  bytes)  routines  of  and  300  f e a t u r e s of t h i s microcomputer  (in this  The  the is  Baud.  and  i s l o c a t e d on  RAM.  sets  monitor  user  40 words  microprocessor at  EIA  programmable  t o s e t up t h e i n t e r r u p t  (ISR), and t h e r e f o r e t h e sampling  The  a modified  TIBUG  Baud  Rate  reset.  This  Some o f t h e o t h e r  board  are  listed  in  A p p e n d i x A.  2.2.  RTI-1241-S To  interface  to  the  and  digital  devices  Board the d i g i t a l  continuous to  time  analog  RTI-1241-S  identification  process  simulation, analog  conversion  board  and c o n t r o l  is  provides  necessary.  both  system  to d i g i t a l The  the analog  analog  i n p u t and  output. The efficient  RTI-1241  interfacing  microcomputers. is  series  has of  been the  The s t a n d a r d b o a r d  d e s c r i b e d i n A p p e n d i x A.  designed analog used  for  signals  in the  simple  and  to  990  control  TM  system  30  3.  The  Software  The  software  control systems  of  SISO s y s t e m s  The  two  G,(s)  =  G (s)  = K« /(s +2£o 2  system  identification  3.1.  The The  discrete  required  the c o n t r o l  First-Order transfer  To  a = f  representing  function be  f r e q u e n c y ; and  from  that  of G ( s )  Q,  the  the  and  damping  the  software  c a n be  easily  first-order  system  2  of t h e  r e p r e s e n t e d by  (  exp(-rT),  control for  the  derived.  and  G  (s), in  the d i f f e r e n c e e q u a t i o n  y(k-l)+b,u(k-1)  (3.3)  b, = ( K / r ) [ l - e x p ( - r T ) ] .  i d e n t i f y the parameters  d(k)=  (3.2)  2  Systems  y(k) = a  where:  function  for identification  d e v e l o p e d and  and  form can  second-order  [26].  software be  a  +u ).  2  of the  (s) w i l l  two  (3.1)  ratio  1  ii)  and  here. These  K/(s+r)  u i s t h e undamped n a t u r a l  G  and  transfer  Where  of  identification  are:  2  The  the  be d e s c r i b e d  system,  continuous-time  system  for  will  are: i) a first-order  system. these  two  implementation  a,  and  b^ , t h e a d j u s t a b l e  A(k-1)y(k-1)+B(k-1)u(k-1)  model  (3.4)  31  will  A(k-1)  be u s e d ; where  and B ( k - 1 ) a r e t h e e s t i m a t e s  of  a,  and b . f  From  the d e f i n i t i o n s  i n Chapter I I , the a ' p r i o r i  e(k)=  y(k)-d(k)  (3.5)  e(k)/[1+ay(k-1)'y(k-1)+£u(k-1)'u(K-1)].  (3.6a)  The a ' p o s t e r i o r i  s(k)=  For  the scalar  error i s  systems  parameters  y(k-1)'=y(k-1), therefore  e(k)/[l+ y(k-1) +*u(K-l) ].  s(k)=  The  error i s  (3.6b)  2  2  c  are  adjusted  according  to  the  following  algorithm:  To c o n t r o l  A(k)=  A(k-1)+as(k)y(k-1),  (3.7)  B(k)=  B(k-1)+*s(k)u(k-1).  (3.8)  the process,  the  parameter  the  process  adjustment  will  order  flowchart system  will  remain  be the- c o n t r o l l e r  u(k)=  The  the a ' p o s t e r i o r i  for  error  computation  t h e same, b u t t h e i n p u t t o  u ( k ) , where:  [B(k)]- [d(k+1)-A(k)y(k)]. 1  the  a r e shown  two  modes o f o p e r a t i o n  i n Figures  and  3.1  and 3 . 2 .  (3.9)  f o r the f i r s t  32  ( V  START  _  I  r e a d A/D  channel:  Input  u(k)  r e a d A/D  channel:  Output  y(k)  compute t h e a d j u s t a b l e model A(k-1)y(k-1)+  d(k)=  compute:  the  output  B(k-I)u(k-1)  error  e(k)=  y(k)-d(k)  s(k)=  e(k)/(1+ y(k-1) +  *u(k-l) )  2  2  C  4. u p d a t e t h e model A(k)=  A(k-1)+  B(k)=  B(k-1)+ ps(k)u(k-l)  update  the  I  . 3.1  cs(k)y(k-1)  i n p u t and  y(k-1 )-•—y(k)  Fig,  parameters  the  output  u ( k- 1 )  and  u(k)  * Flowchart  f o r the first  identification  order  system  of  a  33  (  START  i r e a d A/D  channel:  Input  r e a d A/D  channel:  Output  compute: e(k) =  the  d(k) y(k)  error  y(k)-d(k)  s(k)=  e(k)/{1+cy(k-1) +*u(k-l) ) 2  u p d a t e t h e model  2  parameters  A(k)=  A(k-1)+  os(k)y(k-1)  B(k)=  B(k-1)+  js(k)u(k-1)  compute u(k)=  the c o n t r o l l e r  value  u(k)  [B(k)]" [d(k+1)-A(k)y(k)] 1  !! update y ( k-1  the )-*—  i n p u t and y ( k)  and  the  output  u( k-1 ) u (  k)  t  Fig.  3.2  Flowchart  f o r the c o n t r o l  of a  first  order  system.  34  3.2.  The In  the  Second-Order  Systems  t h e c a s e of a s e c o n d - o r d e r p r o c e s s ,  i d e n t i f i c a t i o n and c o n t r o l  extended The  to obtain  transfer  difference  procedure  for  o f t h e f i r s t - o r d e r s y s t e m can  the e q u a t i o n s g i v e n  function  the  in discrete  be  below.  f o r m i s r e p r e s e n t e d by t h e  equation  y ( k ) = a, y ( k - 1 ) + a y ( k - 2 ) + b 2  f  u(k-1 ) + b u ( k - 2 ) .  (3.10)  2  where: a, = 2 e x p ( - £ u T ) c o s (oT) a = 2  -exp(-2£oT)  b, = K { l - e x p ( - £ o T ) [ c o s ( o T ) + U / 0 - £ ) ) s i n ( o T ) ]} 2  2  b = K { e x p ( - £ « T ) [exp(-£«T)-cos(uT) + U / ( 1-£ ) )sin(«T) ] } . 2  2  2  The  •  problem i s to i d e n t i f y the four  Identification:  The a d j u s t a b l e  d(k)=  p a r a m e t e r s a, , a ,b^ ^b^. a  model:  A1(k-1)y(k-1)+A2(k-1)y(k-2)+ B1(k-1)u(k-1)+B2(k-1)u(k-2)  The a ' p r i o r i  (3.11)  error: e(k) = y ( k ) - d ( k )  The a ' p o s t e r i o r i  (3.12)  error:  s(k)= e ( k ) / { 1 + o [ y ( k - l ) + y ( k - 2 ) ] + e [ u ( k - 1 ) + u ( k - 2 ) ] } 2  2  2  2  (3.13) The  p a r a m e t e r s a r e u p d a t e d by t h e a l g o r i t h m  (3.14a-d)  given  in  equations  35  • Control:  A1 (k)= A1 (k-1 ) + cs(k)y(k--1 )  (3. H a )  A2(k) = A 2 ( k - 1 ) + o s ( k ) y ( k ~ 2 )  (3.14b)  B1(k)=  B1 (k-1  (3,14c)  B2(k)=  B2(k-1)+£s(k)u(k-2)  In t h e c o n t r o l  c o m p u t e d . The d e s i r e d o u t p u t  )+*s(k)u(k-1)  (3.1 4 d )  mode t h e c o n t r o l l e r d(k+l)  u(k)  must  be  i s available.  u(k)=[B(k)]" [d(k+1)-A1(k)y(k)-A2(k)y(k~1)-B2(k)u(k-l)] 1  (3.15) The  computation  flowchart only and  similar  difference  Arithmetic The  is  of  The  estimated,  computation.  the  equations  developed  i n the  r e q u i r e s the f o l l o w i n g  performs  the add o p e r a t i o n  in  2's  form.  SUBTRACT:  Same a s ADD.  »  MULTIPLY:  The  is  TM  performed  multiplication the  systems.  operations:  •  i t  system has a  Operations  The m i c r o p r o c e s s o r  complement  but  second-order  s e c t i o n s on t h e m i c r o p r o c e s s o r  arithmetic  ADD:  the  t h e number o f p a r a m e t e r s b e i n g  implementation  previous  for  t o t h e one f o r t h e f i r s t - o r d e r  t h e r e f o r e t h e amount o f  3.3.  •  sequence  9 9 0 microcomputer in  natural  i n 2's complement,  j o b . The f l o w c h a r t  i s shown  has hardware m u l t i p l y ;  binary. a simple  To  perform  the  s u b r o u t i n e would do  i n F i g u r e 3.3 f o r  two  16 b i t  36  START  4_ COMPUTE: C= A xor B  M = /A|  F i g . 3.3  Flowchart  *  |B|  f o r two's Complement  Multiplication.  37  numbers, A,  B.  consecutive  words.  •  DIVIDE:  number  The  result  Similar  is divided  used  to perform  3.4.  The  A very  to  by  a  important  S e r v i c e Routine  number  will  21.33  This  is  programmable  t i m e r has  of the  done  be  down  counted  automatically.  A  achieved.  3.1  by  systems of  microseconds),  21.33  to zero  the  maximum  time  of  some o f t h e  hexadecimal  code  into  microseconds of  3 Hz K=  The  placed  1/3  6  K represented  T h i s number in  hexadecimal  15th  code t o be  to  of  A  by  be  be  timer*  rates  takes  can and  initialization  of  the o p e r a t o r  with  be the the  loading location  a  21.33  f o r example, a s a m p l i n g  i s o b t a i n e d by  bit  reloaded  milliseconds  created  an the  14  t h e d e s i g n a t e d memory  can  rate  taking  15625 i n hex,  is therefore bit  Upon t h e  so t h a t ,  seconds,  0.333/21.33X10" =  number  1-14.  rate  resolution  or T =  be  the of  interval  will  sampling  is selected  sampling  to. set  (every decrement  349  rate  Any  codes.  as an  clock  sampling  TCONST.  is  microseconds.  system,  proper  2  a 32 b i t  implementation  interface  and  lists  the  hexadecimal  the  in  same t e c h n i q u e c a n  software  corresponding the  here  (ISR), which takes advantage  a resolution  Table  stored  o p e r a t i o n ; but  b i t number. The  function  Interrupt  The  number,  Rate  rate.  9901  bit  division.  sampling  TMS  32  mutiply  16  signed  Sampling  is a  i s 3D09. The  shifted  enable  used  by  the  the  t o the  counter left  interrupt.  o p e r a t o r would  and  uses a  '1' i s  Finally thus  be  bits  the 7A13.  SAMPLES  PER  SECONDS  14 BIT  16 BIT  CLOCK REGISTER  CLOCK REGISTER  3  3D09  7A1  5  24A1  4963  10  1250  24A1  20  0928  1251  30  061B  0C37  40  0494  0929  50  03A9  0753  60  030D  06  70  029D  053B  80  024A  0495  90  0209  0413  100  01D5  02AB  TABLE 3 . 1  The  sampling  r a t e s and t h e i r  hexadecimal coded d a t a .  3  1B  correspondig  39  3.5.  The  Computer  The  Programme  identification  and  assembly  l a n g u a g e , and  c o d e was  t r a n s f e r r e d to the microcomputer  EPROM*s  were l o a d e d  creates second  two  files,  file  The  The  to  set the  2)  sampling  This  of e v e r y  r a t e . The At  ISR  larger  than  them t o t h e  old  showed  (2.29),  down t h e  that  0-7  and  the  first-order  and  the  (ISR): This routine will  use  time  the  this  used  rate provided  rate  i s any  a 21.33  interrupt  is  by  number  microseconds repeating  occurs.  a'posteriori  error  s(k).  The  this  number  will  B(k).  This  because  unity.  the  of  the  parameters A(k)  increments  performance  of  small values  these  parameters  (2.31),  c o n v e r g e n c e but for  bits  which  AA(k)  and  and  p,  and  AB(k)  and  adds  in  the  parameters.  A good c h o i c e  enhance the  in Fortran  The  subroutines:  l/s(k),  to update the  computes  EPROM's.  f u r t h e r s l o w e d down by  the  computes  subroutine  equations  be  t o compute t h e  actually  two  The  word.  the  five  this  countdown b e f o r e  subroutine  better  on  contains  both  r a t e can  subroutine  a l w a y s be A  i n t o machine c o d e s .  3 t o 46875 i n t e r r u p t s / s e c o n d w i t h  routine  3)  8-15  for  operator.  counter A  file  I n t e r r u p t S e r v i c e Routine  resolution. the  first  systems c o n s i s t of  system  between  the  programmes  1)  transformed  u s i n g a programme w r i t t e n  contains bits  second-order  the  then  c o n t r o l programmes were w r i t t e n i n  values  and the of  a  (2.32)  in  identifier. o and  also than  p  that when  used  Chapter  Computer  II,  will  simulation  (0.1<[c,£3<1.0) w i l l the  identifier  used w i t h  slow  performs  larger  values  40  ([G,£]>1.0). are  This  s u d d e n , as  simplicity powers of  is especially  i n a s q u a r e wave  i n the two  or  right.  4)  A  subroutine subroutine in  5)  A  will  subroutine, subroutine  value  of  sent  to  the  following  The  large  in  when  mode. The  the  controller control  (A/D)  and  channel  u(k)  is  checked  consequently,  to  -2047  s a t u r a t i o n c o n s t r a i n t s are  because  necessary  b i t r e s o l u t i o n . I f the  the  last  before  being  process.  The  b e c a u s e of  the  applied:  u(k)=  performed  This  i s the  the  u(k)<-2047  a  u(k).  The  if  i s not  the  d(k+l).  c)  system. T h i s  of  to obtain  2047  values,  is  k,  u(k)=  controller  d(k).  instant  u(k)>2047  Bl(k)  the  sampling  if  parameter  call  as the  system  mode,  b)  the  allowing  first  u(k)  12  selected  the  u(k)=  board  For  r e g i s t e r s to  i f -2047<u(k)<2047  above  example.  p r e d i c t e d output  only  the  output  c o n s t r a i n t s are  changes  subroutine.  the  converter  the  a)  conversion high,  the  main programme. At  controller D/A  shifting  active  only  read  for  0.5,0.25,... so  compute  i n the  routine.will the  be  to  called  this  or  input  parameters are  computes  i s to t h i s  call  these  identification  subroutine  c a s e when t h e  reference,  p e r f o r m e d by  which  the  main programme  2,4,...  t o be  left  operating  computation  i.e.  multiplication  This  the  becomes v e r y inhibiting  problem inverse  when of  B1(k)  sampling  small the  is  too  leads  to  c o n v e r g e n c e of  the  and  this  identification i s not  rate  used.  is  being ,.  ;  41  The is  less  program than  400  to i d e n t i f y words  (800  and  bytes),  (1200  bytes)  f o r the second-order  of  system  input,  the  the parameters gradual these  change  systems.  system  these  and  a first-order  less  systems.  than  system  600  words  Intermediate values  output, c o n t r o l l e r ,  e n a b l i n g the operator to i n s p e c t parameters.  programs a l l o w s a sampling  first-order  and  output, predicted  are stored, of  control  of  rate  about  The  the  e x e c u t i o n time f o r  of about 50 Hz  and  100  f o r the  Hz  for  the  second-order  42  IV. MICROCOMPUTER EVALUATION OF IDENTIFIER AND  1.  ADAPTIVE  CONTROLLER  OVERVIEW  In  this  controller  Chapter  the  developed  microcomputer  system.  implementation  in  Chapter  The  are  simulated  by  integrators  and  inverters.  The  sampling  rate,  parameters w i l l Since controller, the  biased  be  investigated.  loop  because of  output,  however,  i n p u t . For  problem  is  t h e p e r f o r m a n c e of  closed  processes.  process  first-order  the  estimated  parameters.  The  second-order  and  more  initial  sensitive  v a l u e s of  parameters close  t o the  do  not  the  i s divided  three d i f f e r e n t  The  part  second  evaluates  the  identified  and  such  factors  values  to  as  f o r the  be  used  process  with  the  The  identified  p a r a m e t e r s may  between  output  the  input  in a l l cases this  was  not  and  processes  are harder  to  selection  of  r a t e and  t o the  In b o t h  into  correct  the  values  the to  true  identify the  estimated  but  oscillate  bias.  two  parts.  and  control  processes the  cases  be  significant  c l o s e to the  sampling  for  converges a  as the  i s evaluated  identification  s y s t e m s , and  on  identifier  parameters.  converge  and  the  true values with a small  T h i s Chapter evaluates  the  of  parameters are very  to the  evaluated  be  designed  processes  and  identifier  be  to  initial  the c o r r e l a t i o n the  III w i l l  effect  t h e c h o i c e of  identifier  the  using o p e r a t i o n a l a m p l i f i e r s  and  the  of  processes  controlled  the  THE  will  The  first  of be  identification  part  first-order investigated.  and  control  of  43  second-order  systems.  identification control  2.  Figure  mode, and  4.1  shows  F i g u r e 4.2  the  shows  system  the  in  system  the  in  its  mode.  F i r s t - O r d e r Processes A  first-order  process  can  be  c h a r a c t e r i z e d by  a  difference  equation y(k)=  where  y_  and  u  a,y(k-1) + b u ( k - 1 )  are  the  identify  this  In  identification  the  sampling The  process  r a t e and  selection The  of  output  these  mode  the p r o c e s s  i s unknown t h i s  the  to  the  satisfactory not will  some  The should  operator  is briefly be  r a t e can  must  values  and  be  the  critical,  the  of  results.  A  the very  their  limits,  a-^-1  initial  noted.that  the  input fast  high and  value signal  sampling  r a t e s the  the b  1  on  .  the  input. If  o n l y based  process  .  here.  based  selected  any  f o r a., and  random  To  b,  provide  described  selected  knowledge a b o u t  i s not  frequency  c h o i c e of be  t h e p a r a m e t e r s a, and  the p r o c e s s  d e s i r a b l e because at very reach  respectively.  r a t e i f d e s i r e d . A good c h o i c e o f  important,  highest  from  input  initial  should  available  although  the  variables  rate  get  the  i s to estimate  information  c h o o s e a new  and  some r e a s o n a b l e  sampling  input  (4.1)  t  on  and  then  sampling  rate,  10  times  to  will  rate,  100  produce  however, i s  p a r a m e t e r s a and  b  b-»-0.  parameters  is  identification  also  important.  computer  program  It will  u(t)  r  u(k)  ADJUSTABLE MODEL  ADAPTIVE MECHANISM  j  ADAPTIVE IDENTIFIER  I Fig. 4.1  The Block Diagram of the Adaptive System i n I d e n t i f i c a t i o n Mode.  d(k) INVERSE  u(k)  MODEL  J  u(t) D/A  y(t) PLANT  4e(k)  *  ADAPTIVE MECHANISM  4.2  y(k) A/D  The Block Diagram o f the Adaptive System i n the C o n t r o l Mode,  46  use  a  set  value  unreasonable are  (hard  value.  Therefore,  any  systems  limit  . This  o f 1.0  system  i s unstable.  limit  when  lower  limit  mode. T h i s  the  is  Numerical  i s a number g r e a t e r analysis  and  have shown t h a t  For  because  the input  p a r a m e t e r s . S u c h an i n p u t  i f  the  A(k)>1.0  B(k) does not have a  than  computer  the  process  the parameters  limit.  in identification  information  mode, b u t h a s a  i s i n the  zero,  hard  control  i . e . |B1|>0.05.  simulation  of  different  s i g n a l t o the process  a v a i l a b l e f o r the algorithm signal could  must  to identify be  a: w h i t e  a s q u a r e wave o r a s i n u s o i d a l i n p u t s c a n a l s o be u s e d .  To  evaluate  the  microcomputer  P2,  a n d P3 w i t h  These  the  assumption  h a v e enough  noise,  sample  an  p a r a m e t e r A ( k ) h a s an u p p e r  The p a r a m e t e r  system  i t encounters  on i t s m a g n i t u d e when t h e s y s t e m limit  algorithms  exceed  the estimated i s a valid  time  a t every  compared a n d a l t e r e d i f t h e y  first-order  the  limit)  identification  system,  three  and c o n t r o l a l g o r i t h m  processes  are  simulated,  on P1,  the following t r a n s f e r functions.  P1  G(s)=l0.0/(s+5.0)  P2  G(s)=0.5/(s+1.0)  P2  G(s)=30.0/(s+20.0)  three  algorithm  the  processes  will  w i l l be e v a l u a t e d  be  identified  f o r the process  P1.  and  the c o n t r o l  47  2.1  Identification The  p r o c e s s P1  can  where t h e c o e f f i c i e n t s parameters. Chapter  Using  be  r e p r e s e n t e d by a d i f f e r e n c e  d e p e n d on the  I I I , p r o c e s s P1  can  the  sampling  rate  tranformation  formulas  be  by  represented  equation and  its  given  the  in  difference  equation:  To  y(k)=  0.37y(k-l)+  1.26u(k-l)  at  5  samples/sec.  y(k)=  0.61y(k~1)+  0.78u(k-l)  at  10  samples/sec.  y(k)=  0.78y(k-l)+  0.44u(k-l)  at  20  samples/sec.  identify  process  P1  above e q u a t i o n s a r e used. shown  i n F i g u r e s 4.3a,  close  to  the  s h o u l d n o t be  increased  even  further  limits  4.5a. of  These  (>50),  indicated  rate.  The  then  the  are  very  be  seen,  sampling  rate  can  sampling  the parameters  The  predicted  rate  selected  the output output  predicted  closely  rate i s  converge  B ( k ) - > 0 . F i g u r e s 4.3b, and  i n the  obtained are  values  t h e p r o c e s s . As  too h i g h . S i n c e as  show t h e p r o c e s s o u t p u t  process  and  i . e . A(k)—M.0,  microprocessor.  to  4.4b,  and  by  the  follows  .the  output.  The this  4.4a,  rates  estimated parameters  change w i t h sampling  selected  4.5b  The  sampling  true parameters  the parameters  their  the  rate  sampling P2  can  f o r P2  be c h a r a c t e r i z e d  y(k)=  0.82y(k-1)+  by  was  5  samples/sec,  the d i f f e r e n c e  0.09u(k-1)  equation  at  48  > 50 LO  F i g . 4.5a  The Estimated Parameters.  20 Samples/Second.  51  The  estimated  they  are  slow  (low  output the  very  The  At  this  output. process  process  therefore,  P3  the  2.2.  was  slower  this  case  used.  The  still  sampling  once a  very  predicted  very  rate  again  close  could  to  perhaps  is  a  faster  process  rate selected equation  0.82y(k-1)+  results  was  than 100  i s given  P1  and  P2,  samples/second.  by:  0.27u(k-1)  f o r the  process  P2  are  shown  in  4.7.  Control performance  process  response the  A  signal  i s n o i s y but  r a t e the d i f f e r e n c e  The and  4.6b  sampling  identification  Figure  i n F i g u r e 4.6a,  identification.  y(k)=  The  input  in Figure  process the  shown  c l o s e t o t h e t r u e v a l u e s . In  frequency)  shown  improve  parameters are  P1  of  this  transient  This  was  of  selected  process  be  done  developed  in  Chapter  signal  to  the  mode h i g h e r response, therefore  this  by  for this  to a step  response  can  goes t h r o u g h  the a d a p t i v e  be  process,  as  the  which shown  impractical.  control Since  than  the  the  shows  present  adaptive  s y s t e m . The  used may  largest  be  very  value  that  controller  a  In t h e  to achieve  the  response.  reference  i n F i g u r e 4.2.  signal the  4.8  in turn generates  r a t e s c o u l d be  however, t h e  Figure  i s evaluated  input, i t i s desirable  faster  into  controller  sampling  test.  introducing III  controller  input  control control  a very  fast  large  and  handled  by  the  LD  Qo UJ >—i  u_ I—1  Y—V  a o  a  i  1  1  1  1  20  F i g . 4.6a  T  40  TIME  1  1  1  60  (SAMPLES)  The E s t i m a t e d Parameters,  5  Samples/Second.  i  I  r 80  i  r  100  53  LO »  I  •—TM  Q3_ -  F i g . 4.7a  The E s t i m a t e d Parameters.  100  Samples/Second.  55  (A/D)  o r (D/A) c o n v e r s i o n b o a r d  taken  to  prevent  data.  As  mentioned  overflow, volts. has  the  the  in  highest  In a d d i t i o n ,  a lower  limit  overflow.  limit  Chapter value  adaptive  to prevent  controller  h a s been  reached.  the system  The  slow  the process  will  rates;  As  be  of  on  the  actual  shown,  the  the p l o t t i n g  that  signal  of  interval achieved will  samples than  on t h i s  signal  were  slight  i s limited  that  this  system  eventually lead  does  in  out a t  satisfactory  improvements  operating  control  signal  through  of the data,  t o ±4 v o l t s ,  control  of  state  P1 was c a r r i e d  goes  is still  number  steady  over  is  Figure  not  ±10  sudden of the volts  action at  satisfactory.  Note  i s saturated f o r a greater represent  4.9b.  A  a  faster  longer  time  r a t e c a n be  but the s a t u r a t i o n o f the c o n t r o l  to i n s t a b i l i t y .  20  changes  the p l o t  the l i m i t  very  at  a  s y s t e m . F i g u r e 4.10 shows t h e c o n t r o l  the  this  i f the steady  reach  controller  50 s a m p l e s / s e c o n d . The r e s u l t although  B1  F i g u r e 4.9 shows t h e r e s p o n s e o f  simplify  i s limited  P1  parameters  only  a l l results  change.  signal  to  process  when t h e r e f e r e n c e i n p u t  control  i s an  slower.  adaptive  is  the  i f the c o n t r o l  dramatically To  there  f o r t h e c o n t r o l l e r , ±10  constant  introduced only  the  samples/second.  time  mode,  the time  no c o n t r o l l e r .  with  any  i s used  signal,  rates  system with  be  may be done by s c a l i n g t h e  III  remain  control  sampling  although  should  overflow. For the process  Since  response  adaptive  different  care  i n t h i s mode o f o p e r a t i o n t h e p a r a m e t e r  d e p e n d s on t h e c o n t r o l  the  This  i s s e t a t 0.05. In t h e c o n t r o l  state  then  i s ±10 v o l t s ,  level  56  F i g . 4.9b  The Control S i g n a l 20 Samples/Second.  57  F i g . 4.10a  F i g . 4.10  The Process Output.  The Control S i g n a l . 50 Samples/Second  58  3.  Second-Order  The process  Processes  c l o s e d loop simulated  transfer  for this  C(s)  3.1.  s + 2  5.28  process  6.5s  rad/sec.  the  can  be  y(k)=  second-order  part i s :  +28.0  and  Q=  formulation represented 0.7ly(k-1) 0.49u(k-1)+  y(k)=  1.33y(k-l) 0.16u(k-l)+  y(k)=  1.66y(k-l) 0.05u(k-l)+  the  limits; that  the  0.61.  Identification  Using  In  of  39.5  R(s) w h e r e : o=  function  identification the  limits  a l l the  constraint  identification same s a m p l i n g estimated  are:  < B1 was  I I I , the  above  a d i f f e r e n c e equation  0.31u(k-2)  at  5  samples/sec.  at  10  samples/sec.  at  20  samples/sec.  -0.53y(k-2)+ 0.12u(k-2) -0.72y(k-2)+ 0.04u(k-2)  p a r a m e t e r s A1  the  result out  indicated are  Chapter  -0.28y(k-2)+  inside  carried  parameters  in  |A1|<2.0 and  will  rates  by  mode t h e  roots are B2  given  have  upper  |A2|<1.0. T h i s w i l l  ensure  unit  and  circle.  A2  In a d d i t i o n ,  i n a minimum p h a s e  with  the  system  in  the  above  shown  in Figures  the  system.  The  operating at  the  equations.  The  4.11a,  4.12a,  and  59  o CM"  in  ]  o  F i g . 4.11a The Estimated Parameters. 5  Samples/Second.  o  F i g . 4.11b  The Process Output and The  Predicted  Output.  60  F i g . 4.12b The Process Output and The Predicted Output.  F i g . 4.13b  The process Output and The Fredicted Output.  62  4.13a,  along  prediction seen,  B2  process  parameters  B1  and  rate i n c r e a s e s . Again  does  not  process  3.2.  the  output  and  m o d e l , F i g u r e s 4.11b, 4.12b, and  the  sampling  with  prevent  the  B2 the  output  4.13b.  As  become  very  of  the  can  be  s m a l l as  the  bias in estimates  reference  model  from  of  B1  and  tracking  the  output.  Control The  is  used  A1  and  process now  A2  simulated  to evaluate  have upper  identification  f o r t h e e v a l u a t i o n of  the  identifier  adaptive c o n t r o l l e r .  The  parameters  the  limits  mode,  but  higher the  than  limits  the  are  i n t h e a r i t h m e t i c o p e r a t i o n s . The  B1  be  must  selected  could  cause a very  when  a  large  value  care, a very  underdamped  sudden c h a n g e o c c u r s  response. sampling and  with  f o r the For  rate  a  limit  second-order  is critical;  a noisy output.  a  A slow  rate,  a h i g h l y underdamped  response  and  response  an  value.  overdamped The  control  parameter  had wave.  an  upper The  controller  action  B 1 was limit  result has  was  limited of  2.0,  is  speeded  rate on  carried  the in  the  out  from  (0<  an  Figure response  <0.1)  overdamped of  the  instability  hand,  results  small  in  limit,  i s set at a large 10  samples/second,  below and  reference  B1  selection  a very  at  prevent  overshoot  cause  limit  the  i n p u t . However, a  other  has  in  parameter  large  the can  when t h e  and  a  cause  the  when B1  t o 0.5  shown up  will  process  fast  for  small value  reference  >1.0)  set  enough t o  limit  system with  i n the  (B1  limits  low  overflow  the  the  A1  input  4.14.  and  is a  The  A2  square  adaptive  significantly  when  63  o CQ' a ID  a  Tsl  >-  I  I  I  3 00  60  I—rv _J  (SAMPLES)  ID  1  o<=> U3-J  I  CD CO - I  F i g . 4.14a The Process Output.  ^=0.25  /3=0.50 10 Samples/Second  i  >—  1  21  i 20  i  1  r 40  TIM  (SAMPLES)  a  CJ ! !  O  CO _J  Fig.-4.14b  The C o n t r o l  Signal.  ]00  64  c o m p a r e d w i t h F i g u r e 4.13b. The are changed step this  so t h a t  response  has  a gain less  As  can  o f t h e p r o c e s s i s now  be  Parameter than  seen,  p r o c e s s does n o t t h e r e s p o n s e , and quantization  described gain  i t becomes a h i g h l y  p r o c e s s the a d a p t i v e c o n t r o l l e r  significantly.  and  simulated  can  has  a lower  unity.  The  result  although  underdamped  shown  B1  the parameters  of t h e p r o c e s s .  In a and  The  i n F i g u r e 4.15.  For  reduce  there i s almost  were  no  overshoot the  process  in Figure  controller  of  t h e two based  the  noise i n  by  selected  4.16.  overshoot,  r e s p o n s e . The  the c o n t r o l i  the  since  i s shown  response are caused  noise.  parameters system.  limit  have a smooth t r a n s i e n t slow  process  limit  processes on  the  65  F i g . 4.15b The Reference S i g n a l .  66  CD  oo"  o ta a  >a — I —  40  TIME  I—CM _ ZD  1  ~1  1 60  i  (SAMPLES)  I  a  <* = 0 . 5 -  OO _ !  F i g . 4.16a  The P r o c e s s Output.  10  a co" a  Samples/Second.  LD  a  ZDR  cr:  CM  UJo  • ° 3  1  r  20  40 TIME  O  CO „  F i g . 4.16b  60  (SAMPLES)  The Control S i g n a l .  1  ]00  67  V.  CONCLUSIONS  1. Summary Adaptive c o n t r o l techniques have been i n the area have been examined. model  reference  adaptive  attention  from r e s e a r c h e r s  proposed  by  simple t o implement algorithm  on  a  The s e l f - t u n i n g r e g u l a t o r s  systems and  Martin-Sanchez  reviewed and. r e s u l t s  have r e c e i v e d the g r e a t e s t  control  was  engineers.  found  A  method  t o be c o m p u t a t i o n a l l y  microprocessor.  The  identification  i s e s s e n t i a l l y a simple form of the r e c u r s i v e  squares e s t i m a t i o n method, and the c o n t r o l l e r was found similar  to  the  minimum '"'var'^lahce'" c o n t r o l l e r  s e l f - t u n i n g r e g u l a t o r s . The asymptotically  method  on  a  TI  to  used  TM 990  has  been . e s t a b l i s h e d  using  the  Popov  microprocessor  assembly  was  language  system and e v a l u a t e d . As was  as  hyperstability this  described;  and  uses  shown,  The  integer  In Chapter IV the a l g o r i t h m was implemented  microcomputer  be  'in the  necessary f o r the implementation of  software was w r i t t e n i n arithmetic.  least  method  hyperstable  theorem. The hardware  and  the  on the result  f o r the system i d e n t i f i c a t i o n are good and the system performed reasonably  well.  The  estimated parameters were v e r y c l o s e to  the t r u e parameters but, at some sample p o i n t s , parameters  were  noisy  the  predicted  although convergence had been reached.  T h i s e f f e c t caused by the use of the i n t e g e r a r i t h m e t i c  i n the  system, d i d not prevent the o v e r a l l performance of the a d a p t i v e identifier The  being s a t i s f a c t o r y . performance  of  the  adaptive  controller,  although  68  a c c e p t a b l e , was achieve  a  not as good as the a d a p t i v e i d e n t i f i e r , and  good performance, some a ' p r i o r i  s t r u c t u r e of the process was  knowledge about the  o f t e n needed. The  most  important  i s that the  controller  shortcoming  of t h i s a d a p t i v e c o n t r o l l e r  always t r i e s  to reach steady s t a t e , the d e s i r e d output,  sample.  This  action  s i g n a l s which Another  are  problem  demands  unrealizable  were  on the gain of  associated  and  with  the the  o p e r a t i o n s of the  in  amplitude  can  cause  one  control  instability.  i n the d i f f i c u l t y  p d i c u s s e d i n Chapter  found to be very important  i n the c o n t r o l mode. The based  large  with t h i s a l g o r i t h m l i e s  s e l e c t i n g the parameters c and parameters  very  to  II.  of  These  when the system i s  s i m p l e s t c h o i c e of these parameters i s process.  In  algorithm,  realization  also  addition the  to  integer  induce  problems  arithmetic  errors  into  the  system.  2.  Future Work The  algorithm  used as a  reliable  algorithm  is  not  the  the s e l e c t i o n of maintain  the  robust  t h i s microcomputer can method,  but  the  An  analytical  these  parameters  economy of computation  procedure  must  be  made  so  which i s a very  p o i n t a r i t h m e t i c o p e r a t i o n board to  for  provide  to  implementation.  such as the Am  more  as  important  The microcomputer system c o u l d be combined with a  unit  control  £ parameters would be i n v a l u a b l e . However,  aspect of t h i s method f o r microcomputer  processing  be  method and needs m o d i f i c a t i o n to  controller.  a and  on  identification a  provide a r e l i a b l e choosing  implemented  convenient  floating  9511  arithmetic  and  reliable  69  implementation. w h i c h use 990  A n o t h e r a l t e r n a t i v e would be  t h e h a r d w a r e m u l t i p l y and  microprocessor  methods, level  however,  may  l a n g u a g e and  bulk  further  implement controller this  self-tuning work.  be  floating very  capability  point  s l o w . The  s t o r a g e w o u l d add  of  the  arithmetic. addition  to the  of a  convenience  TM  Such high of  expansion.  Self-tuning  that  to perform  divide  to write routines  on  regulators  are  microcomputers,  i s usually  very  microcomputer  computationally but  the  minimum  reliable.  It i s therefore  be  for  used  the  r e g u l a t o r s to provide a comparison  awkward  variance suggested  implementation with  to  the  of  present  70  APPENDIX A The  A.I.  TM  System  Hardware  990/101M-1 M i c r o c o m p u t e r  Some o f t h e i m p o r t a n t f e a t u r e s are  listed  •  TMS  9901  multifunctional  the  an  programmable  interval  The  (ACC).  TMS  This  timer.  9902  TMS  facilitates  microprocessor  provides  used  as  an  I/O  ports.  interval  m i c r o s e c o n d s compared  an  interface  21.33  protocol,  between The  timer  with  a  ports,  be u s e d t o s e t up  Controller between  asynchronous communication  communication  serial  interrupt, I/O  Communications  a c c e p t s E I A RS-232-C  the  is  sampling rate.  Asynchronous  component  9902 ACC  interface  T h i s component w i l l  m i c r o p r o c e s s o r and a s e r i a l The  system  component, w h i c h p r o v i d e s  ISR t o p r o v i d e a v a r i a b l e  *  board  below:  The  and  of t h i s microcomputer  TMS  with  the  i t  channel. therefore  terminal  and  the  also  be  resolution,  64  9902 ACC lower  the  can  m i c r o s e c o n d s f o r t h e TMS  9901  PSI . The  TMS  useful  9901  functions  Microcomputer ®  17  the  9901  TMS  9902  ACC  p r o v i d e some o t h e r  in detail  in  TI  TMS  very 9900  Manual. input  are a v a i l a b l e  a r e non-maskable  interrupt  TMS  and  which are described  interrupt  interrupts three  PSI  PSI  990/101M.  Two  and o t h e r s a r e m a s k a b l e . T h e r e a r e  s o u r c e s on b o a r d : The interval  on t h e TM  timer.  The  two s e r i a l 15 m a s k a b l e  I/O  ports,  and  interrupts  are  71  also  automatically  •  prioritized,  Hardware m u l t i p l y and d i v i d e i s a v a i l a b l e  family.  This  microcomputer •  feature  the  program  like  word  of  registers counter  workspace p o i n t e r first  one  of  reasons  available  that  This  because  this  i t allows  (R13,R14,R15)  boards  (ST), and the  the address  u s e f u l when  the user to  990  of  the  words w h i c h c a n be u s e d  feature i s very  16 word w o r k s p a c e , when b r a n c h i n g registers  TM  (PC), status r e g i s t e r  a s e t o f 16 c o n t i g u o u s  to a subroutine,  on t h e  (WP). WP a l w a y s c o n t a i n s  an a c c u m u l a t o r .  three  the  990  was s e l e c t e d .  The h a r d w a r e  are  was  on t h e TM  a  branching  t o u s e a new s e t o f  subroutine,  the  c o n t a i n t h e o l d PC, WP,  last  a n d ST  register. • of  MEMORY- The TM990-101M-1 m i c r o c o m p u t e r RAM  TIBUG  a n d 2K words o f EPROM. I t s h o u l d monitor  software  only  1K word o f EPROM  A.2.  RTI-1241-S The  following »  i s available  be m e n t i o n e d  2K words that  the  1K word o f EPROM, t h e r e f o r e f o r programming.  Board  standard  board  used  in  the  control  system  has  the  features:  The  resolution  jumper-selectable Offset  Binary,  •  There  input  occupies  provides  These SE  bits. forms;  The c o d i n g i)  of the data i s  Natural  Binary, i i )  2's complement.  a r e 16 s i n g l e  256  12  and i n t h r e e  and i i i )  channels.  expanded . t o  is  can or  ended be 128  (SE) o r 8 d i f f e r e n t i a l  doubled DIFF  on  input  board  or  channels  (DIFF) can  be  u s i n g an  72  expansion ®  board-  The s o f t w a r e - p r o g r a m m a b l e  the  gain permits  subranging  over  g a i n s o f 1, 2, 4, 8.  •  Memory-Mapped  this  board  has  microcomputer RTI-1241 of  eight  been  board.  board, words  interface: designed  I t was m e n t i o n e d to  be  The h o s t computer  p e r c e i v e s the analog  when  s t o r e s and r e t r i e v e s  do  other  To  perform  things while  with  interface  d a t a . The computer  the conversion  a  interfaced  i n memory. The c o m m u n i c a t i o n  it  board  that  TM  990  with the  as a block  i s i n t h e same  way  i s therefore free to  is"being  performed.  the (A/D).conversion:  (a)  S e t t h e GAIN,  (b)  S e t t h e MULTIPLEXER  channel,  (c)  Send t h e CONVERSION  COMMAND  There  used  earlier  are only  conversion.  three  (start  instructions  I t t a k e s about  needed  35 m i c r o s e c o n d s  the conversion). to  do  the  t o execute  (A/D)  t h e above  instructions.  To  perform  (a)  Only  t h e (D/A) c o n v e r s i o n :  Send  one  the data  instruction  jumper-selectable. operation  with  t o one o f t h e two DAC's.  is  In t h i s  required. system  2's complement  it  coding.  The is  DAC set  output for  range i s  ±10  volts  73  A,3.  TI 745 T e r m i n a l The  communication  operator This  i s  done t h r o u g h  terminal  can  microcomputer terminal  between  be  board  comes w i t h  using  device,  Rate  for  terminal  A.4.  system  slots  conversion bus, of  A.5.  board.  can connect  be  the microcomputer  large  computer.  The  Baud  either  110 o r 3 0 0 B a u d . T h e  Rate.  Cage card  cage  The b a c k p a n n e l lines  The Power  has four  slots,  board  contains  and  two o f these the  RTI-1241  the address bus, data  t o p e r m i t m e m o r y , I / O , a n d DMA. e x p a n s i o n  Supply  DC v o l t a g e s  voltages  needed ±12  +5 V  power  The 745  modules.  The  These  to  terminal.  E I A RS-232-C c a b l i n g .  as a  can  data  TM 9 9 0  t h e TM 9 9 0 m i c r o c o m p u t e r  and c o n t r o l  CPU  such  TM 9 9 0 / 5 1 0 c a r d  contain  the  and t h e system  the  i s s e t f o r t h e 300 Baud  TM 9 9 0 / 5 1 0 C a r d The  interfaced  a modem w h i c h  t o another  control  t h e TI 745 e l e c t r o n i c  easily  board  this  the microcomputer  f o r the system  V  a r e p r o v i d e d b y a GSC  supply. I t provides:  +5 V ±15  V  ±15  at  5 Amps  at  1 .5 Amps  are :  V  (GOF-2A-1T) t r i p l e  output  the  ±12  V  i s obtained  f r o m ±15  V  using  two  regulators.  75  REFERENCES  [I]  Astrom K.J. " S e l f - T u n i n g R e g u l a t o r s , Design P r i n c i p l e s and A p p l i c a t i o n s , " D e p a r t m e n t o f A u t o m a t i c C o n t r o l , Lund I n s t i t u t e o f T e c h n o l o g y , pp. 1-14. 1980.  [2]  A s t r o m K . 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