We re-examine the division by zero problem which occurs in certainty equivalence based indirect adaptive control algorithms applied to linear systems. By exploiting a parametrization for linear systems induced by the continued fraction description of its transfer function, the division by zero problem obtains a very simple geometric representation that can be used to virtually eliminate the problem in the adaptive algorith

Arent, Krzysztof

Mareels, Iven

Polderman, Jan Willem

English

Abstract- We re-examine the division by zero prob-lem which occurs in certainty equivalence based in-direct adaptive control algorithms applied to linear systems. By exploiting a parametrization for linear systems induced by the continued fraction description of its transfer function, the division by zero problem obtains a very simple geometric representation that can be used to virtually eliminate the problem in the adaptive algorithm

Krzysztof Arentl

Iven Marcels

Jan Willem Polderman

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Adaptive Poleplacement: The Division By Zero Problem

University of Twente Research Information

WA17 11 :40 Proceedings of the 34th Conference on Decision & Control New Orleans, LA - December 1995 Adaptive Poleplacement: The Division By Zero Problem Krzysztof Arentl , Iven Marcels', Jan Willem Polderman3 Abstract- We re-examine the division by zero prob- lem which occurs in certainty equivalence based in- direct adaptive control algorithms applied to linear systems. By exploiting a parametrization for linear systems induced by the continued fraction description of its transfer function, the division by zero problem obtains a very simple geometric representation that can be used to virtually eliminate the problem in the adaptive algorithm. Keywords: Adaptive pole placement control, pole-zero cancellation, continued fraction representation Mathematics subject classification: 93C40 I. INTRODUCTION This paper introduces a novel approach to the pole- zero cancellation problem encountered in adaptive in- direct pole placement control applied to linear finite dimensional discrete time systems [5]. This work ex- ploits ideas of [l] and [ll]. Essentially we propose to exploit a reparametrization of the coefficients of the polynomials that specify the linear system to be controlled based on the contin- ued fraction expansion of the system's transfer func- tion. The important property of this system repre- sentation, from the point of view of adaptive pole placement control, is that verifying controllability, expressed in the new model parameters, is equiva- lent to checking whether some parameters are zero or not. This simple structure of uncontrollable mod- els in the new parameter space is useful to derive a systematic solution of the stabilizability problem in adaptive control. The price paid for this simplicity is that the parametrization is nonlinear. In essence it suffices to use in parallel a number of pa- rameter identification algorithm in conjunction with a switching rule [ll] to decide which parameter iden- tification algorithm produces the estimate for control purposes. Despite the fact that the parametrization is nonlinear we establish that the classical gradient descent based identification algorithms may be used. Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, e-mail: k.arent@math.utwente.nl (corresponding author) Department of Engineering Australian National University, ACT 0200, Australia, e-mail: iven.maree1sOanu.edu.au 3Department 0: Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, e-mail: j .w.poldermanOmath.utwente.nl In this contribution we focus on the system parametrization rather than its application in an adaptive control algorithm. We use tools from the be- havioral approach to system theory [15] to expound our ideas. The paper is organised as follows. In the next section we introduce the system parametrization and explore some of its properties. In Section 3 we describe briefly how this parametrization could be exploited in an adaptive control context. Some conclusion are pro- vided in Section 4. For a more comp1:te description we refer to [6]. 11. SYSTEM REPRESENTATION VIA CONTINUED FRACTION Consider a dynamical system represented as: A(a)y = B((T)u (1) where U is the system input, y is the measurable output, U is the shift operator, for any integer t o ,  (atow)( t )  = w(t + t o ) .  A ( [ ) ,  B ( [ )  E %[[I are polyno- mials: A ( [ )  = tn + . . . + ai[ + uo B([)  = b&" + . . . + b i t  + bo ( 2 )  TI > m The coefficients of A ( [ ) ,  B ( [ )  can be reparameterized as follows: Definition 11.1 Continued fraction parameteriza- tion Consider A ( [ ) ,  B ( [ )  E %[[I as in the system description (2). Let IC E N ,  ri E ?R and Ti([) E %[[I in the form I?;([) = Ed' + y i , d i - l [ d ' - l  + . . . + yi,o di > 0 (3) be such that: A(<)  = Ro(E) B ( t )  = r lR l (0  (4) & ( E )  = ri+I(<pi+l(t) + Ti+2&+2(<)  (5) &(<) = 1 Rk-l(<) = r k ( 6 )  (6) The coefficients yi,j and ri are referred to as the con- tinued fraction parameters for the pair of polynomials A ( 0 ,  BK). 0 0-7803-2685-7/95 $4.00 0 1995 IEEE 498 2 - 0  0 1 - r k ( a )  - - Remark 11.2 If A ( ( )  and B(<) are coprime then process of the continued fraction expansion of A(<)/B'(<),  where I?'([)  = €?(<) /bo .  This is where the notion continued fraction parameterii ,a t' ion comes from. The above algorithmic definition of the contin- ued fraction parameters follows from the Euclidean b ( ( ) , . .  . .  r k ( ( ) ,  r 1 , .  . . .  r k  can be obtained in the algorithm. 0 Remark 11.3 The importance of the continued frac- tion parameters (11.1) for the system (1) stems from the observation that the system is controllable iff all of the r i  continued fraction parameters are non-zero. This fact will be established in the sequel. n e k - 1  A .  Controllability issues - - U r1 - r1(a)  . . .  0 r 2  . . .  Y 1;. .a. ::: .!. j e l  ;" '  - r k ( a )  e k - 1  = 0 (7 )  The system representation (7) may alternatively be written as - . . .  - r l  - r l ( a )  0 . . .  0 r 2  0 0 0 . . .  0 0 . . .  -rl (c) -1 0 . . .  r l + l R l + l ( o )  -Rl(a) - . . .  . . .  . . .  . . .  b ;] = o m  el  u k  ( I )  is a unimodular matrix. Left multiplication of (9) by u k  ( U )  yields an equivalent representation [ 141 , [15]: . . .  ... . . .  . . .  ... r1 -r+) 0 0 . . .  r k - l r k ( a )  - r k ( a ) r k - l ( o )  - r k  0 . . .  r k - 1  - r k - l  ( U )  According to [14], the behavior in terms of left hand side of the above expression that corre- sponds to zeroes in the right hand side. This in con- junction with (6) yields (8) with 1 = k - 2. Along the same lines we can show that (8) holds for (U, y ,  e l , .  ... e k - 2 )  is described by those rows in the all 1 = 0, . . . .  k - 1. It follows that (8) is input/output equivalent to (7) for I = 0, . . . .  IC - 1. The description ( r l ( a ) R l ( a )  + r 2 R z ( u ) ) Y  = r l R l ( U ) U  (11) The result now follows from (5) with i = 0. (8) implies that elimination of all variables ei yields 0 We now investigate the controllability properties of the behaviors described by (1) and (7). We start with the system description (7). Lemma 11.6: Consider system (7). Assume that ri and ri are as in Definition 11.1. Let !?3 := 93 is control- lable iff r i  # 0 V i .  { ( u , y , e l ,  . . . .  e k - 1 )  I s.t. (7) holds }. 499 Proof: Directly from the Hautus test, see also [6] ,  Next we observe that the introduction/elimination of the auxiliary variables has no influence on the con- trollability. Lemma 11.7: Consider the behavior 23 as in Lemma 11.6. For I = 1 , .  . .k - 1, let ~ 4 1 ,  ~ 5 1 .  0 % = { ( ~ , Y , e l , - , e l ) I  % + I , .  . . , ek-1 S . t . ( u , y , e l , .  . . , e k - l )  E '13) Let '130 = { ( U , Y ) P l , . . . , e k - l  s.t. ( u , y , e l , .  . . , e k - ~ )  E '131 !& is controllable iff Bl-1 is controllable, for all 1 = 1 , 2  , . . . ,  k - 1 .  Proof: Consider the proof of Theorem 11.4. It follows that !& is parameterized by (8). Denote by Mi(<) the matrix in (8). Equations (5, 6)) imply that there exists a unimodular matrix Vi(<) such that Hence it follows that rankMl(X) = rankMi-1 (A) + 1 for all X E C The result is now a direct consequence of the gener- 0 alized Hautus test for controllability. We summarize as follows: Theorem 11.8: Consider the dynamical system (1) with continued fraction parameters as given in Defi- nition (11.1). The system is controllable iff rg # 0 V i .  B. Representution issues The link between the coefficients in the polynomials A and B and their continued fraction parameters defines a non trivial parameter transformation. In order to be able to construct an identification algorithm for the continued fraction parameters it is essential to understand this transformation in some detail. First we provide an example, then the general theory. Example 11.9 Consider a pair of polynomials A(<) = t2 + a l t  + a0 B(<) = b l t  + bo (13) where ai, bi E ?J? 3 The continued fraction parameterization of A(<), B(<) is as follows: A(<) m) = r l < + r l Y 2 , 0  = t2 + (Y1,O + Yz,o)5 + (Yl,OY2,0 + r2) for b l  # 0 or A(<) = <% + Y1,1< + Yl,O WE) = T1U for b l  = 0 Let Q in the parameter space {al, QO,  b l ,  bo} represent poIynomials with common factors. Define the corre- sponding set in the parameter space {yl,o, yz,o, TI ,  rz}  as QdP. Q = ( ( Q l , a O , b l , b O )  1 or QCfP = {(Yl,O,Y2,0,7-1,7-2) I 7-1 = 0 0 5  7-2 = 0) QCfP has a simple structure (as proven above in The- orem 11.8). This is the main reason to introduce this parametrization. The second interesting property of the continued frac- tion representation is that the parameters of the poly- nomials (14) are uniquely determined by the contin- ued fraction coefficients of (13) and vice versa. In particular, if b l  # 0 we have bo a1 = -Y1,0 - Y2,o Yl ,O = --a1 + - 61 bo a0 = Yl,OY2,0 + 7-2 Y2,O = -- b l  61 1 T I  TI = b l  bo = -7-1Y2,0 7-2 = (-) - a1(-) + U0 (15) bo 2 bo b l  61 If b l  = 0 we have ai  = ~ i , i  ao = ~ i , o  bo = TO (16) The above calculations (15,16) show clearly that to parameterize the polynomials (13) two sets of contin- ued fraction parameters, one for b l  # 0 and another for b l  = 0 are required. This complicates significantly the identification aspect of the continued fraction pa- 500 4 rameters! 0 The following Lemma 11.10 relates the parametriza- tions in the polynomial framework in general. Lemma 11.10: Let xi fp  = { (rl(<) , . . .  ,rk(t),rl,...,rk)i ri (<) E %[<I is monic , k = 1, . . . , n; A(<) is monic, degA([) = n B ( 0  E w1, degB(t) < 12 } The equations (4,5,6) define a bijective mapping h : PC:p +PC,. Proof: The fact that h(PCifp) c PC, follows from ( 6 ,  5, 4) and from the fact that deg I'i(C) = n. Step 1: h is injective. Suppose that this is not the case. There exist ( T I , .  . . I  Fk, 1'1,. . . , r k )  E PCkfp, (q, . . . r;, , T;, . . . , f ; , )  E PCkfp such that = h ( r i , .  . . , r;,, r;,  . . . , &) Consider the sequences {Ri}f=o and {Ri(<)}$.o cor- responding to (. . . I'i(<), . . . , ri, . . .) and (. . . , Ti(<), . . . ri, . . .). It fo!lows from (6, 5) and from the fact: deg Pi(<) 2 1 V i  that Ri(<), R: (t) are monic Next consider R~ = ri+l (0 + Ri = ri+l(oRi+l(<) + I':+zRi+z(<) It follows from (19),(20) that (. . . , ri(<), . . . , ri, . . .), (. . . ,  I?{(<), . . . , r i , .  . .) satisfy (21) for i = 0. Consider the case i > 0. Observe that because of (18) the pair of equations (21) has only one solution, Hence h is injective. Step 2: h is surjective. Take any (A((),B(<)) E PC, and perform (6, 5, 4) backward such that (18) holds. It is easy to see that (I'l,. . .,I'k, 1'1,. . . , ~ k )  E PCifp and 0 = rikl(t), ~ ~ + ~ ( t )  = R:+z(<)1ri+2 = Y:+~. h(F1,. 1 . 1  r k ,  ' p l y . .  ., T k )  = (A(<), B ( 0 )  for i = 1,. . . , k } and PCn,k,{dl ,...A) := h(PC$,{dl  ,___,  d k }1 The families PC$,tdl,,, , ,dk l and PCn,k,{dl ,..., d k }  de- fine partitions of PCifp and PC, respectively. Proof: The fact that the family PC$,tdl,,,.,dk} is a partition of PCifP is obvious. Because h is bijec- tive (see Lemma II.lO), the family PCn,k,{dl ,._., d k )  is a partition of PC,. 0 This allows the following characterization of the pa- rameter transformation: Theorem 11.12: Consider the transformation h de- fined by (4,5,6) 1. h induces unique bijective trans- f C " i o n s  tn ,k  , { d i  ,.. . , d k }  from P c ~ ~ , { ~ l , ,  , , , d k }  to (A(<), B(E)) E PCn,k,{dl ,..., dk}. 2. tn,k,{d, _.., d k )  is a multilinear map. 3, If di # di for some i or k # k' then Im tn,k,{dl, ..., d k }  f l  Im tn,k',{d: ,.__, d ; ! }  = 8. Proof: 1. Follows from the fact that PC',fp,,tdl,. , , ,dk  is iso- morfic to %" x SfZ"' \ {0} x % and from the bi- jectivity of h ( see Lemma 11.10 ). 2. Follows immediately from the Definition 11.1, es- pecially (4, 5, 6).  3. Followsfrom LemmaII.lO, LemmaII.ll and the first part of Theorem 11.12. 50 1 5 The Lemma 11.11 and Theorem 11.12 imply that in the context of identifying a single system many dis- tinct continued fraction parametrization need to be considered. The natural question is what is the num- ber of such transformations. Lemma 11.13: Define T,,k := # { ( d l , .  . . , d k )  I di E N ,  di  = n}  and 1 , .  . . , n )  where # stands for the number elements of Tn := # { ( d l  , . . . ,  d k )  I di E N ,  g , d ;  = n; le = In the context of LemmaII. l l  and Theorem 11.12 T, reflects the number of all parameter transformations. Observe that if the set { ( d l ,  ..., d k )  I di E N, di = n}  consisted of non-ordered k-tuples then T n , k  would be a partition number [3], [4]. One can evaluate Tn as an explicit algorithm of a. It can be shown that Tn grows exponentially fast with n. k i=l Remark 11.14 Observe that each element in the set P C , , k , { d l ,  , d k }  is parameterized by n variables 7 and k variables T .  Moreover, the corresponding polyno- mials ( A ( [ ) ,  B(()) E P C n , k , { & ,  , d k }  have relative de- gree (deg(A) - deg(B)) d l  and the number of coeffi- cients is 2 n  - d l .  It follows that if le < n - d l  then P C n , k , { d l ,  , d k }  is a measure zero subset of the set PC, . Remark 11.15 If the system (1) has deg(a) = n and deg(B) = m we could restrict ourselves further to the single class of systems described by continued fraction parameters in 7 X , , , - m , { m , ~ ,  , I } .  This way we only sacrifice a measure zero set of systems in the classical parametrization. If only the deg(A) = n of the system (1) is known, we can similarly restrict attention to the n. par ameter transformations t, , - m, I, - m, 1, ,I> 111. IDENTIFICATION O F  SYSTEMS IN CONTINUED FRACTION PARAMETERIZATION In the previous section the continued fraction pa- rameterization for the polynomial pair ( A ( [ ) ,  B ( [ ) )  has been introduced and analysed. Here we con- centrate on the question if the continued fraction parameters are identifiable from input output data. The fact that there is a bijective, be it complicated, mapping between the classical and continued fraction parametrizations of a system suggests that identifica- tion in continued fraction parameter space may be feasible. A .  The gradient algorithm Consider the system (1) where ( A ( [ ) , B ( [ ) )  E p c n , k > { d l ,  ..., dk}. This system can be expressed in the following compact notation Yt = QT(8)dt-1, t E N (22) where T '8 = [7l,dl -1 . . . 7 1 , O  ' ' '?'k,dk -1 . . . y k , O T l  . * . Tk1 6(29) = [un-1(29) . . . ao(29)b,-d1 ( 2 9 )  . . . b 0 ( 2 9 ) ] ~  $t = [Yt - . - Y t - n + l U t - d l  . . . U  t-n+l] The notation: u;(29), bj(29) means that ai or b j  is expressed in terms of the elements from 29. T Denote by 290 the vector of parameters of the true system in continued fraction parameterization and by 8t the vector of estimates of 290 at timefnstant t .  The mismatch between the true model and its estimate can be detected via the one step ahead prediction error e:  et = Yt - Q T ( L ) d t - l  (23) et can be used for improving the estimate of 290. One of the standard methods in adaptive control is to up- date the parameter estimates in the direction of the negative gradient of e: i.e. where f is suitably chosen real, positive valued func- tion and p E %+. It follows from (23) that where (25) The formulation (25) is interesting in the context of analysis of invariant points of (24). Theorem 111.1: The matrix K(29) defined by (26) has full row rank iff ri # 0 for all i 0 Proof: See [6] for details. In general we expect that e: is a non-convex function of I9 with non-trivial local minima. Observe however that in the generic case, when (1) can be parame- terized by elements from ~ C n , d , { d , l , . . . ,  I), the matrix K(I9) is square. If ri # 0 V i  then we conclude, on the 502 ground of Theorein 111.1 that 1<(29) is invertible. This implies that if Fs, t  # 0 V i , t  and dt # 0 Vt  then 29* is an invariant point of (24). if and only if ~ ( 2 9 ' )  E 0! 'This in conjunctrion with (25) implies that every local minimum of e l  is in fact giobai. Every invariant point of (24) specifies the (unique) system (CP + un-&9*)CF-1  + . . ' +  a0(29*))7J = ( b m ( 2 9 * ) P  + . ' .  + bo(d*))u (27) This equation necessarily explains a given data set (y, U ) .  In [l] such a system is called an unfulszfed model. Unfalsified models have been carefully investigated in the context of adaptive pole placement control in [13]. It has been shown there that the pole placement controller based on an unfalsified model (27) behaves as if it was based on the true system. This goes a long way to certify that the above gradient based al- gorithm may be used to construct an adaptive algo- rithm that may overcome the pole-zero cancellation problem. At present it is a conjecture that combining the con- tinued fraction representation, with a gradient based algorithm complemented with a switching rule to guarantee that the ri parameters remain of fixed sign indeed yields a working adaptive pole placement al- gorithm. Simulation evidence points this way [6]. IV. CONCLUSION We introduced a novel parametrization for linear dis- crete time systems that can be used to aid in the solution of the pple-zero cancellation issue in pole placement adapiive control algorithms. Simulation evidence suggests that the identification and adapta- tion issues can be resolved. Further work is ongoing, see also [6]. REFERENCES [l] I<.ARENT, I.M.Y.MAREELs , J.W.POLDERMAN: The pole zero cancellation problem in adaptive control: a solu- tion for minimum phase systems by approximate models. Memorandum No. 1238, Dept. Appl. Math. University o j  Twente, The Netherlands, 1994. R.G.BROCKET Some Geometric Questions in the Theory of Linear Systems. IEEE Trans. Aut. Contr. AC-21, 449- 455. [3] K.L.CHUNG (1968) A course in  probability theory. Har- court, Brace & World, cop. 141 G.H.HARDY AND E.M.WRIGHT (1960): A n  introduction t o  the theory o j  numbers. Oxford U.P. 151 G.C.GOODWIN AND K.S.SIN (1984): Adaptive Filtering Prediction and Control. Prentice-Hall, Inc., Englewood Cliffs, N.J. [6] K. ARENT (1995) Selected Topics in Adaptive Control, PhD Thesis, Twente University. [2] [7] G.KREISSELMEIER AND M.C.SMITH: Stable adaptive reg- ulation of arbitrary n-th order plants. IEEE Trans. Aut. Contr. AC-31, 299-305, 1986. P H .  DE LARMINAT: On the stabilizability condition in in- direct adaptive control. Automutica 20, 793-795, 1984 R.LOZANO-LEAL AND G.C.GOODWIN: A globally conver- gent adaptive pole placement algorithm without a persis- tence of excitation requirement, IEEE Trans. Aut. Contr. AC-30, 795-798, 1985. [lo] R.LOZANO A N D  X-I.ZHAO: Adaptive Pole Placement. Without Excitation Probing Signals IEEE Trans. Aut. Contr. AC-39, 47-58, 1994. [ll] F.M.PAIT, S.A.MORSE: A cyclic switchingstrategy for pa- rameter adaptive control. IEEE Trans. Aut. Contr. AC- [12] J.W.POLDERMAN: Avoiding the Non-Admissible Region of the Parameter Space in Indirect Adaptive Control Al- gorithms. Proceedings of the Eighth International Conjer- ence on Analysis and Optimization of Systems. Juan les Pins, France, 822-829, 1988 [13] J.W.POLDERMAN: Adaptive Control Fd Identification: Conflict or Confluz ? -CWI Tract 67, Center for Mathe- matics and Computer Science, Amsterdam, The Nether- lands, 1988. [14] J.W.POLDERMAN: Proper Elimination of Latent Vari- ables. Proceedings o j  12th World Congress of Automatic Control Sydney, Australia, 1993 Volume 10, 73-76. [15] J.C. WILLEMS (1988): Paradigms and Puzzles in the The- ory of Dynamical Systems. IEEE Trans. Aut. Contr. AC- [8] [9] 39, 1172-1183,1994. 36. 259-294. 503 

Adaptive pole placement: the division by zero problem

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A course in probability theory.

A cyclic switchingstrategy for parameter adaptive control.

Adaptive Control Fd Identification: Conflict or Confluz ? -CWI Tract 67,

Avoiding the Non-Admissible Region of the Parameter Space in Indirect Adaptive Control Algorithms.

Elimination of Latent Variables.

M.C.SMITH: Stable adaptive regulation of arbitrary n-th order plants.

Paradigms and Puzzles in the Theory of Dynamical Systems.

Selected Topics in Adaptive Control,

The pole zero cancellation problem in adaptive control: a solution for minimum phase systems by approximate models.

X-I.ZHAO: Adaptive Pole Placement. Without Excitation Probing Signals

https://research.utwente.nl/en/publications/09f91b16-7b37-495f-b76a-803c962b15d4

Adaptive poleplacement: the division by zero problem

Adaptive poleplacement: the division by zero problem

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