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Physically based parameterizations for designing adaptive algorithms
0005-1098/87 $300+000 PergamonJournalsLtd 1987 Int©rnatmnalFederationof AutomaticControl
Automatwa Vol 23 N o 4 pp 469-477 1987 Pnnted m Great Britain
Physically Based Parameterizations for Designing Adaptive Algorithms* SOURA DASGUPTAf and BRIAN D O ANDERSON,: Special state var|able and transfer functmn descmptmns are developed for systems whose unknown parameters are a wide class of phystcal element values The dependence of the state varmble structures on these parameters is "rank-1" and that of the numerator and denominator polynommls of the transfer functions multdmear Key Words--Adaptive algorithms, physical paramters, rank-l, multflmear
Section 2 specifies the parameterization in question and demonstrates how RLC orcuits fall within ItS ambit Section 3 investigates the corresponding state vanable realizations while Section 4 shows that the resulting minimal transfer functions are the ratios of polynommls having coefficients multihnear in the parameters For example, for a system with two unkown parameters, kl and k2, the transfer function could be
Abstract--Structural propertmes are exammed of systems w~th physical component values as parameters Both state variable reahzattons and transfer functmn descnptmns are investigated The transfer functmns m particular are shown to be the rahos of polynommls wRh coefficients multdlnear m the parameters These structures prove useful m formulating adaptive algorithms 1 INTRODUCTION
A FACTOR critical to the performance of adaptwe algorithms is their underlying parametenzations Parameterlzatlons reflecting greater a pmom knowledge about the unknown system can be expected to lead to improved performance Frequently, adaptive systems (Lmn, 1967, Narendra and Kudva, 1974, Luders and Narendra, 1974, Kreisselmeier, 1977, Anderson, 1977) treat all the transfer function coefficients as the unknown parameters In the process considerable information may be discarded Sometimes, the only unknowns in a system are the values of certain physical elements or parameters As far as transfer functions are concerned, th~s may mean that some of the coeffic|ents are known a przorz, as also are some relatmnships, possibly nonlinear, which exist between them Accordingly, a parameterizatmn ts developed here revolving unknown parameters which have direct physical relevance Such a parameterization has been the basis of adaptive algonthms formulated by Dasgupta et al (1983, 1984, 1986a, b)
W(s, k l, k2) = po(s) + klpl(s) + k2P2(S) + klk2P12(s) qo(s) + klql(s) + k2q2(s) + klk2q12(s)
(11)
where the polynomials Pc )(s) and q~ )(s) are known Section 5 derives conditions under which systems with transfer functions such as (1) conform to the parametenzatlon under study All results m the last two sections apply only to single-input, singleoutput systems 2 THE PARAMETERIZATION
Much of the background material for this section is contained in Anderson and Vongpanltlerd (1973, pp 156-200) To understand how physical element values affect a wide class of systems, our attention will be restricted to electric circuits containing resistors, lnductors and capacitors The extensions to the corresponding chemical and mechanical analogues will of course be immediate Consider a resistor R appearing In an n-port clrcmt Clearly the resistor can be extracted from the rest of the circuit in a manner depicted in Fig 1 Suppose that u l , y l are the port voltage and current (or current and voltage) at the nght hand port of the "circuit without the resistor", and U, Y are the input and output vectors of the terminated
*Recewed 30 April 1986, rewsed 2 December 1986 The original version of th~s paper was presented at any IFAC meeting This paper was recommended for pubhcatmn m rewsed form by Assocmte Edaor G Krelsselmeler the &rectlon of Editor P C Parks 5"Department of Electrical and Computer Engineering, Umvers~ty of Iowa, Iowa City, IA 52242, U S A :~Department of Systems Engineering, Research School of Physical Sciences, Austrahan Nahonal Umverslty Canberra, ACT 2601, Austraha 469
470
S DASGUPTA and B D O ANDERSON and Circuit W)thout Resistor R
FIG I
R
Representationof a ctrcmt w~th a resistor R
|
FIG 2
U :~
Input/Output Block
u~ )t
Independent of k 1
Y
)
Yl
Representationof a orcmt with an unknown parameter kl
_
Input/Output Block Independent of k 1 kn
_
ul(t) = Kyl(t)
VVV
Remark 2 1 The elements which cannot be treated in this manner are mutual lnductors--they allow cross-couphng between energy storage devices-and gyrators Theorem 2 1 below shows how RLC clrcmts conform to this description Before stating the theorem, a definition Is reqmred Consider an m port RLC network having mdimensional input and output vectors U( ) and Y(), respectively, with Y() finite for all finite U ( ) Suppose also that each port is represented m the input vector by either but not both of Its voltage or current and that if a port voltage (current) appears m the input then the correspondmg current (voltage) appears in the output Suppose N of the network components are f~l, ,fiN Then under these condmons ~1, ,f~N are extractable if the conditions of the following defimtlon are met Definition 2 1 Consider the m + N port network o b t a m e d b y e x t r a c t m g f ~ l , ,l) n Let f.7( )and Y( ) be N-dimensional vectors havmg elements ti,( ) and 37,( ) such that (1) ti,( ) is either the voltage or the current at the port created by the removal of fl,, (n) if ti,( ) is the relevant voltage, then 37,( ) Is the correspondmg current and vice versa
FIG 3 Representatmn of systems satisfyingAssumption 2 1
network, with entries of U, Y corresponding to voltages or currents at the left port If a port voltage appears m one of U, Ythe port current must appear m the other With kl then ldent~fied w~th R or R depending whether u~ is a voltage or current, redescrlpUon vm Fig 2 is possible Of course, for th~s description to make sense, the hybrid matrix relating [ U r u ~]r to [ y r y l ] r should exist More generally, when a clrcmt has N unknown physical components, with values k~, , kN, then in many cases an input-output descnptmn of the form m Fig 3 exists Assumption 2 1 formahzes th~s description and constitutes the standing assumptmn for this paper
Assumptlon 2 1 Consider a system with an ndimensional input vector U and m-dimensional output vector Y Suppose it has N unknown physical components with values kl, ,ks Then defining K ~ drag {kl, , kN} there extst two N-vectors yl(t) and u~(t) and an (m + N) x (n + N) dimensional transfer function matrix T(s) such that [yT(s)yI(s)]T = T(s)[UT(s)uI(s)] T
Then for the gwen U( ) and Y(), t)l, ,ON are extractable if there exist a choice of/.7 and Y and a fimte M such that [yr(), ~t( )It = M [ u t ( ) , tTt( )it
(2 1) VVV
Observe that the definmon reqmres the existence of a hybrid description relating the augmented vectors [~-T(), yr( )] and [f.TT(), Ur( )]x to each other As pointed out m Anderson and Vongpamtlerd (1973, pp 171-198), the elements of an RLC network for a gwen input/output set wdl fad to satisfy the condmon for extractablhty if one of the following hold (1) A particular element fl, does not affect the input/output relationship, m which case fZ, can be removed from consideration (11) The problem of finding a state variable descrtpUon is 111posed Since the motivation here ~s to consider parametenzatlons relevant to adaptive control, such a system is of no interest to us Thus, m deahng with RLC circuits, this paper will be restricted to situations where Defimtton 2 1 apphes Theorem 2 1 will now be stated and the proof gwen
Physically based parameterizatmns
Theorem 2 1 Consider an m port LTI lumped RLC clrcmt with m inputs u, and m outputs y, which are the port voltages and currents Suppose every port is represented in the mput vector by e~ther, but not both, of its voltage and current Also, ff a particular voltage appears m the mput the correspondmg current appears in the output and wee versa Suppose the N components of the system are {/~,},u=t with component values {#,},s=t Then a representatton of the form of Assumptmn 2 1 exists with k, = #, or 1/#,, ff fl, are extractable in the sense of Defimtlon 2 1 Proof With appropriate definmons and the extractability property there j = {1,2} such that
exist
M v,
~= {1,2},
471
Remark 2 2 The result tnwally extends to mechanIcal and chemical analogs of RLC clrcmts It also holds for a large number of actwc clrcmts hawng controlled sources and their analogs 3 T H E STATE VARIABLE R E A L I Z A T I O N
In this secUon it will be shown how N-parameter systems satlsfymg Assumptmn 2 1 are reflected in their state variable reahzatmns Before this is done, the followmg defimtlon will be made
Defimtmn 3 1 A state variable rcahzatlon described by the quadruple {F,G,H,J} has a rank-I dependence on N parameters kt, ,kN, If Vie {1, ,N}, there exist a,,b, eR, independent of k,, such that defining ~, as c~thcr 1 0tip
~(s)J
M22JLO(s)A
LM21
m
a, + k,b,
or
61 ~
where Y,,(s) Is either the voltage across fl,, or the current through it, if ~{s) is a voltage tben G~(s) is the corresponding current and vice versa Now the relatmns between the voltages, and currents across resistors, mductors and capamtors arc respectwely gwcn by
V(s) = sLI(s)
Uts) =
0
I: ] oo] I
sl 0 0 11 K~'(s) s
Thus, with
U,(s) =
=L-H,
IS:]+,,= VVV
Observe that if a M I M O system obeys this definmon, then so does any SISO subsystem contamed by it The followmg theorem shows that a system conforming to the reqmrcments of Assumption 2 1 has at least one state variable reahzatlon which has a rank-1 dependence on the N parameters
output description samfylng Assumption 2 1 In other words, if u and y are mput and output vectors, respectively, there exist N input and outputs, all elements of vectors u, and yx, respectwely, such that
slJ
and Y,(s) = Y(s) the result follows
EsI~ jG] pI-,,
Theorem 3 1 Consider a system having an mput
1-I 0 U(s) s
0
, N},
that
Es,-:
Thus, with appropriate ordcnng of fl, and K ~dmg {k,}L, 0
the followmg is true There exist, V t ~ { 1,
with g,,h, vectors In other words, one can claim that
V(s) = ~I(s)
I
a, + k,b,
F,, G,, H,, J,, h, and g, all Independent of k,, such
-
RI(s)
V(s) =
kl
m
VVV
[ s,1 [*,] y,(s)J
=
T(s) u,(s)
472
S DASGUPTA and B D 0 ANDERSON
and
lf l # j then Yl = K-lU 1
e l F = kje sT A, - 1
with K & drag Ikl, ,kNl Assume T(s) is proper Then there exists at least one state variable reahzatmn, w~th input u and output y, whmh has a rank1 dependence on the k, for all but ~solated values of k, P r o o f Since T(s) is proper, matrices A, B, B~, C, Ca, D11, D12, D21 and D2z exist such that x = A t + [B
B1][U T
IVT y T ] T = [ c T
uT] T
k,ks T A - t D22e,eV A 7 1 + 1 - k,e~A,~-lOz,e, es '
whence, with :t, = k,/(1 - k,eT, Ai - 1D22e,), e l f = r,T j, + ot,wj,eT A, - 1
with vj,,wj,,(eT, A[ -~) independent of k, and vs, vector and w j, a scalar If t = j eTF = cqeT A, - 1
cT]Tx
~Dll D12] + LD21 D223[u T uT]T
a
Thus Wl 3
(3 1)
Wt-l,l
As Ul = Ky~, a httle mampulatmn shows that F = V, + ~,
1
~c = {A + B x K [ I - D 2 2 K ] - I C I } x
e~A~ -1
14"t + 1 ' I
+ {B + B 1 K [ I - D 2 2 K ] - l D 2 1 } u WN,I
y = {C + D , 2 K [ I - D 2 2 K ] - ' C , } x + IDll + D l z K [ I - D 2 2 K ] - I D 2 1 } u
Thus
with E, e,T A,-~ and ws, Independent of k, Hence, the result follows VVV Presented below is a non-electrical example where the state varmble reahzatlon has a rank-1 dependence on most element values The dynamics pertinent to the attitude control of the communications technology satelhte, H e r m e s (Dlduch and Balasubramamam, 1982), are as follows
(3 2)
~=Ax+Bu Y= C \
Denote K [ I - DE2K]-~ by F Suppose A, = l - D22K + k,Dz2e,e x,
where (3 3)
where e, is the unit vector with umty m the tth element and zero elsewhere The matrix A, is independent of k, Observe that matrices like F and A, are lnvertlble for all but isolated values of the k, Our analysis shall exclude such points m the k space Also,
A=
B =
I - = K [ A [ -x + k, lAD ' 22ee'~A~T_'
]-~-22e,1]
Consider the jth row of F, A[- 1D22e,eTA[-' ] eTF = k~er A~-1 + k, l _ ~ l - ~ 2 2 e l J
0
1
0
0
-woh/l 1 0
0 0
0 0
Wo - h / l l 1
0
h/I 2 - w 0
- woh/I 2
0
0
0
F I L 1 G 1 cos ot/I l
0
0
0
-FILIGI
sln~t/12
F2L2G2/I2
Physically based parameterlzatlons
c=
having a state variable reahzatlon which has a rank-l dependence on a single parameter ~ has a transfer function whose numerator and denominator are affine in at
['0 °° °0] 0
473
1
Theorem 4 1 An n-dimensional SISO system represented by
U2
Here ~b is the roll, x is the yaw, 11 is the moment of inertia about the roll axis, 12, that about the yaw axis, Wo ~s the orbital rate, h the nominal wheel angular momentum, 0t the offset angle, Lx and L2 the offset and yaw thruster moment arms, respectively, and G1 and G2, the lnpulse bit factors The inputs ux and u2 provide a guide for the level of consumed fuel It is evident that the parameters Ix, 12, Fx, F2, L1, L2, Gx and G2 appear in the state variable representation in a rank-1 fashion Also, although
= F(~)x +
g(ct)u
y = hT(~)X + J(~)u with a system matrix of the form
sl - F(~t) g(~t)] - hT(~t) j(~t) J F s l - Fo go] - h oT ] J o +
=L
k h 2 j [ g l I gz]
(4 1)
t? FsI - A
_c I ol
where Fo, go, ho,Jo, hi, h2, gx and g2 are independent of ct, has a transfer function
has rank 1, the system matrix is obviously not linear in ~ Thus ~ does not quite conform to the definition of rank-1 dependence The parameters wo and h, on the other hand, clearly do not appear m a rank-1 fashmn But, by definition (one is the orbital rate and the other the wheel angular momentum), one can see that they must allow cross-couphng between energy storage elements They thus fall in the same category as mutual lnductors, which as the authors have emphasized, do not fall within the requirements of Assumption 21 4 SISO TRANSFER FUNCTIONS
In this section, ~t will be shown that SISO systems which have rank-1 state variable realizations necessarily have minimal transfer function descriptions which are the ratios of two polynomials multihnear in the unknown parameters At the outset the following definmon of coprlmeness of polynomials in more than one variable will be introduced
W(s) -
a(s) + orb(a) e(s) + ~d(s)
(4 2)
for every ~ The polynomials a(s), b(s), c(s) and d(s) are independent of ~ and obey the following restrictions (a) 6[c(s)] = n, 6[d(s)] < n, 6[a(s)] ~< n and 6[b(s)] <~ n, (b) a(s)d(s) - b(s)c(s) IS factorlzable into two polynomials of degree not exceeding n Conversely, any transfer function of the form (4 2) has a state variable reahzatlon of the form (4 1) provided that conditions (a) and (b) hold
Proof (1) From (41), some calculation shows that W(s) = [a(s)c(s) + ~{a(s)d(s) + g2h2c2(s) - h2?(s)c(s) + g2fl(s)c(s) - fl(s)7(s)}]/{c2(s) + ctc(s)d(s)}
Definmon 41 Consider p,(xx, , x . , x . . 1 , ,x,.), i = 1,2, , r, which are r polynomials in the mdetermmates x~, , x . Then p, are copnme wtth respect to the variables Xl, , x. if there exists no nontnvlal f which is a polynomial in xl, ,x., but ranonal in x. + 1, , ,c.,, such that ff, = p , ,
Vie{l,
,r}
with c(s) = det (sl - Fo) and a, fl, y, d defined by
(4 4)
W(s) = [a(s)/c(s) Ly(s)/c(s)
=
d(s)/c(s)J
F-l,hTol[sl_Fo]_,[g ° I_gI_l
with f polynomials m xl, , x. and rationals in Xn+l~ ,Xm, as well The following theorem shows that a system AJT 23 4-D
(4 3)
hi]
(44)
Because ff'(s) has an nth order reahzatlon, ItS Macmillan degree is not greater than n Thus,
474
S DASGUPTA and B D O ANDERSON
ad - f17 is divisible by c(s) Define b(s) by
where a, b, c, d are polynomials m s and/,2 Suppose
b(s)c(s) = a(s)d(s) - [?(s) - gzc(s)] [fl(s) + h2c(s)]
(4 5)
Then (4 3) has the same form as (4 2) and by (4 5) and (4 4) conditions (a) and (b) are satlsfield (n) Suppose that (a) and (b) hold for (4 2) Let
a(s)d(s) - b(s)e(s) = fl(s)fz(s) = ['~(S) --
g2c(s)]
[fl(S) +
h2c(s)]
with 3[y(s)] < n and 6[fl(s)] < n Then a(s)d(s)L(s)fz(s) IS divisible by e(s) whence
I.is,/c(sq Lfl(s)/c(s)
a(s, k2) + kxb(s, k2) = m(s, k x, k2)p(s, k I , k2) c(s, k2) + kid(s, k2) = m(s, k I , k2)q(s, k 1, k2)
{4 6)
d(s)/c(s).J
has Macmillan degree no greater than n Hence, l,V(s) can be expressed as
hVol[sl_Fo]_1[gohl]+[ 7o h~] gIJ
(l) a(s, k2) , b(s, k2), ¢(s, k 2} and d(s, k~) are copnme with respect to 3 and/'2, (ll) a(s, k2)d(s, k2) -- b(S, k2)((& k2) ~ 0 Then a(s, k2) + klb(s, k2) and c(s, k2) + ktd(s, k2) are coprlme with respect to s, k~ and k 2 Proof According to Hodge and Pedoe (1953, p 36), the ring of polynomials in the variables s, kl, k2 over the field of real numbers is a unique factorlzatlon domain Let
L --g2
so that by reversing the argument in the first part of this theorem a state variable reahzatlon of (42) exists in the form displayed m (41) VVV
Remark 41 Given any W(s) of the form (42), with condmon (a) holding, it may be that (b) cannot be satisfied, because ad - bc has all complex roots and n is odd But then one can multiply both the numerator and denominator in (42) by (s + t/), for any i/ to ensure that the resulting a, b, c and d polynomials satisfy (b) The resulting state variable realization conforming to (41) IS then non mzmmal Thus, a system having a non mlmmal state variable reahzatlon of the form in (41) need not have a minimal realization of the same form Notice also, that, If 6[c(s)] < 6[d(s)], then (a) will hold by replacing ct with 1/~ Thus all transfer functions of the form (42) which are proper at almost all values of a, have state variable descriptions like (41) However, the latter need not be minimal Remark 4 2 If the state variable realization has a rank-1 dependence on kl then cq = kl/(al + klbl) or 1/(al + klbl), whence both the numerator and denominator of the transfer function are linear in kl, even though the degree requirements of (a) may have to be replaced by 3[d(s)]<<.n If a 1 = 0 , kl would need to be replaced by 1/k~ The result of the theorem IS now extended to include the situation where the number of parameters exceeds one To this end Lemmas 4 1 - 4 3 are needed, which derive some results for the coprlmeness of polynommls in more than one variable Lemma 41 Consider a transfer functmn a(s, k2) + klb(s, k2) W(s, k x, k2) = c(s, k2) + kld(s, k2)
with m,p,q polynomials in s, k~ and k 2 and m not trivial Consider the following cases
Case I m t s independent oJ kl immediate that (l) is violated
It is vmually
Case II m t s dependent on kl Then m must be affine in k 1 and p, q are independent of kl Suppose m(s, kl,k2) = r1(s, k2) + klrE(S, k2) Then a(s, k2) = q(s, k2)p(s, k2) b(s, k2) = r2(s, k2)p(s, k2) (4 7)
c(s, k2) = rds, k2)q(s, kz) d(s, k2) = r2(s, k2)q(s, k2)
whence a d - bc = O, which too contradicts the hypothesis here VVV
Remark 4 3 Violation of (11) Implies that W is independent of kx We now show that the transfer function is expresmble as ratio of polynomials multlhnear in k, Moreover, from (47) one can see that If the numerator and denominator of W are not coprlme w r t kl, then a(s, k2) W(s, kx, k2) - - - W(s, O, k2)
c(s, k9
Lemma 4 2 Suppose that the transfer function W(s, kl, ,kN) Is expressible as W(s, kl,
, kN)
= a,(s, k(')) + k,b,(s, k (')) c,(s, k (')) + k,d,(s, k ('))
v,~{1, ,N}
where k (°--4 {kl, ,kN} -- {k,} Suppose a f t , b,c, ~ 0 and a,, b,, c,, d, are copnme with respect to s and k (')
Physmally based parameterlzatlons
475
independent of s, for which
Then
W(s, kl,
,ks) =
e(s, k 1, Q(s, kl,
, k~) ,kN)
where P and Q are multfllnear in k,
Proof The case when N = 2 will be proved The more general case follows along the same lines Suppose al(s, k2) + klbl(s, k2) W(s, kl, k2) = Cl(S' k2) + kldl(S, k2) = a2(s, kl) + k2b2(s, kl) c2(s, kl) + k2d2(s, kl) and suppose that the other hypotheses specialized to N = 2 holds Then [al(s, k2) + klbl(S, k2)][c2(s, kO
+ k2d2(s, kl)] = [a2(s, kl) + k2b2(s, kl)] x [Cl(S, k2) + kldl(s, k2)] By Lemma 4 1, al(s, k2) + klbl(s, k2) and ct(s, k2) + kldl(s, k2) are copnme with respect to s, kl and k2 Thus, al(s, k2) + klbds, k2) d m d e s a2(s, kl) + k2b2(s, kl) Thus, al(s, k2) and bl(s, k2) can be at most linear in k2 Similarly, c~(s, k2) and dds, k2) can be at most hnear In k2 Hence the transfer functmn W(s, k~, k2) can be written as
po(s) + klpl(s) + k2P2(s) + klk2P12(s) qo(S) + klql(s) + k2q2(s) + klk2q12(s) whence the result follows VVV It is of Interest here to consider mlnlmahty with respect to s alone That is, it is to be shown that if the hypothesis of Lemma 4 3 holds, then P and Q have no common factors which are polynomials in s but rational In the k, The following lemma whmh follows from Youla and Gnaw (1979) shows that this is indeed the case
, ks) and Q(s, kl, , ks), polynomials in s, kl, , ks, have no common factors which are polynomials in s, kl, ,ks, then they have no common factors which are polynomials m s but rational m kt, ,ks
P(s, k l ,
, ks)x(s, k l , ,ks) + Q(s, kl,
,ks)y(s, kl,
,ks)= ~b(k,, ,ks)
Thus, dividing both sides by ~ the result is immediate VVV Now Theorem 4 1 and Lemmas 4 1-4 3 together yield the following main result of this section
Theorem 4 2 If the state variable reahzatlon of a SISO linear tlme-lnvanant finite dimensional system has a rank one dependence on N parameters kl, , ks, then it has a mmlmal (with respect to s) (see Definmon 4 1 for copnmeness) transfer function descnptmn whose numerator and denominator polynomials are multlhnear in the k, 5 TRANSFER FUNCTIONS CONFORMING TO THE STANDING ASSUMPTION In this section, condmons under whmh scalar transfer functions with numerator and denominator multlhnear m the unknown parameters, correspond to systems satisfying Assumptmn 2 1 are derived Theorem 5 1 below summarizes these condmons The theorem can be understood by observing that a scalar system satisfying this assumption can be expressed in terms of the following input-output descrlptmn
Y(s) = -a(s).. ~ u t s l , + gT(S) - ~ uI(s) h(s) . . . . Dis) YI(s) = ~(-~tJts) + --7-7, Ul(s) c~sj
(5 1)
Ul(s) = KYI(s) K __4dlag {kl,
,ks}
where a ( ) and c() are scalar polynomials with c( ) the characteristic polynomial of the transfer function relating [U, U1] r to [Y, I"1]r, h( ) and g( ) are N-&menslonal polynomial vectors and D( ) is an N x N polynomial matrix It is not hard to see then, that the transfer function relating U to Y is
Lemma 4 3 If P(s, kl, k2,
a(s) . gX(s) . . . . . . W(s) = - ~ + - - ~ hl_CtS)l -- D(s)K]- lh(s)
(5 2)
Equation (5 2) forms the basis for Theorem 5 1
Proof If P and Q have no common factors which are polynomials m s , kl, ,ks then P and Q are minor copnme as well (see Youla and Gnavl for a
Theorem 5 1 Consider a system with a proper transfer function
definmon) Thus by Youla and Gnavl there exist polynomials x, y and ~,, with ~, nontnvlal and
W(s, k) = P(s, k)/Q(s, k)
(5 3)
476
S DASGUPTA and B D O ANDERSON
where k ~ [kl,
By defimtlon, T(s) and all submatnces thereof have reahzatmns with characteristic polynomml c(s) Thus V r c S scalar polynomials ~,(s) and/~,(s) exist such that
,kN]T (5 4)
P(s,k) = po(s) +~I-I,=s(,~k,)P,(s)
{Irl- 1}
c(s) q,(s) = det ( - O,(s))
(5 5)
and wRh S = {1, ,N} and 6[qo(s)] ~,6[q,(s)] Suppose P(s, k) and Q(s, k) are copnme with respect to s and the elements of k Then the corresponding system samfies Assumptmn 2 1, w~th a proper T(s), fff there exist a scalar polynomml f(s), N&mensmnal vector polynommls g(s) and h(s) and an N x N matnx polynomml D(s), with 6[g(s)], 6[h(s)] and 6[D(s)] all less than or equal to 3[f(s)qo(s)], such that the following hold Yr c S
Irl
c( s )~,(s) = a(s) det O,(s)- gT,(s) Adj D,(s)h,(s) (5 6) Now,
W(s, k) = l~(s, k)/(~(s, k) where
(1) d e t ( - D,) = f(s)qr(s)[qo(s)f(s)] I'1- 1
P(s, k) = a(s) det (c(s)l - D(s)K)
(n) [qo(s)f(s)]l'lpr(s)f(s)= a(s) det D,(s) - gr,(s){AdjD,(s)}h,(s)
+ gr(s)K Adj (c(s)l - D(s)K)h(s)
(5 7)
where Dr, g, and h, are defined as below (a)
D,(s) is a It[ × Irl matnx consisting of the lth rows
and
and ah columns of D(s) V t e r (b) g,(s) and h,(s) are Irl-&mensmnal vectors consistrag, respectively, of the zth elements of g(s) and
h(s) V t ~ r The ordenng of elements in all cases is consistent
Remark 5 1 Since P and Q are coprlme w r t s and kl, , ks, by Lemma 4 3 they are also copnme w r t s alone ProofO) First the "only if" part wdl be proved Suppose Assumptmn 2 1 is satisfied wRh proper T(s) Then (5 1) holds and W(s,k) is gwen by (5 2), with all quantmes obwously defined Thus to complete the proof the p,(s) and q,(s) m (5 4) must be related to the quantmes m (5 2)
0) P ( s , k , ) = d e t l c ( s ) l - D ( s ) { K "
(~(s, k) = c(s) det (c(s)l - D(s)K)
(5 8)
Observe that the coetticaents o f / ~ and (~ can eamly be shown to be multlhnear m the k, Define 2, = l/k, and A = K - 1 Then the polynomials multiplying I I k, m P, (~ can be obtained as follows let
(1) set kj = 0, Vjer, (11) dwlde by Ilk,, (m) set 2, = 0 t~r With K,, A, obviously defined and k, a vector of elements k)' ~¢r, for P the three steps respectwely, translate to
~}l[a(s)+gV(s)I K"
°03[c,s,,-o,,,{o,
N-It)
= det (c(s)Ii, I - D,(s)K,)c (s) [a(s) + gT,(s)K,(c(s)II,I- D,(s)K,)-lh,(s)],
(n)
/~(s, k,)
1--Ik,
N-Irl
- c (s) det (c(s)Ar - D,(s))[a(s) + gT,(s)(c(s)A, -- D,(s))- lh,(s)],
|Er
N-Irl
(m) /~,(s) = c (s) det ( - D,(s))[a(s) - g[(s)D7 l(s)hr(s)] N-Irl
= c (s) [a(s) d e t ( - O,(s)) - gT,(s) AdJ [D,(s)]hr(s)]
(5 9)
Physically based parametenzatmns Similarly, W(s,k) = N-Irl+ 1
~,(s) =
c(s) det ( - D,(s))
a(s)
+
477 gr(s)rK, o
(5 lo) ×[c(s)l-D(s)[o'
00] ;]l":,s,
By (5 5) and (5 6), = a(s)
4,(s) =
ti-M+l
N
c(s) c(s)'"-%(s) = c(s)q,(s)
(511)
and N
p,(s) = c(s) p,(s)
(5 12)
Define ~o(S) = a(s) and ~o(S) = c(s) Thus,
c(s) +
g,r (s)
where r excludes the elements corresponding to The first equatmn is obtained by using the equatmn m Remark 4 3 Thus the proof can be estabhshed easily (n) The "iff" part follows by a trtvial reversal of the above arguments VVV 6
w ( s , k) =
/~(s, k)
with P and O. obwously defined Consider now, the two following cases Case I P and ~ are coprtme with respect to every element o f k Since P, Q, P and ~ are all multdmear m the elements of k, f(s)P(s, k) = ]~(s, k) and f(s)Q(s, k) = (~(s, k),
whence, f (s)po(s) = a(s) f(s)qo(s) = c(s) f (s)q,(s) = q,(s)
Vr c S
f(s)p,(s) = ~,(s)
Vr c S
and
Thus, condmons (0 and (u) are sausfied due to (5 5) and (5 6) Case II ~t and 0 are not coprtme with respect to k,Vte ~ c S Thcn, by cancelhng factors involving k,, it can be seen that
r'(c(s)G1 - D,(s)K,)- ~hY(s)
CONCLUSION
Certain structural aspects of systems whmh have a parametenzatmn mvolwng physical component values as the parameters have been established The systems investigated include RLC c~rcmts and their chemmal and mechamcal analogs REFERENCES Anderson, B D O (1977) An approach to multwanable system ldentfficatlon Automatwa, 13, 401-408 Anderson, B D O and S Vongpamtlerd (1973) Network Analys~s and Synthesis, A Modern Systems Theory Approach Electncal Engmeenng Series, PrenUce-Hall, Englewood Chffs, New Jersey Dasgupta, S, B D O Anderson and R J Kaye (1983) Robust identification of parttally known systems Proc 22nd CDC San Antomo, Texas, 1, 1510-1514, December Dasgupta, S, B D O Anderson and R J Kaye (1984) IdenUficaUon of economically parametenzed system Prepnnts 9th IFAC World Congress, Hungary, X, 96-100, July Dasgupta, S, B D O Anderson and R J Kaye (1986a) Output error ~denttficatmn methods for partmlly known systems Int J Control, 43, 177-191 Dasgupta, S, B D O Anderson and R J Kaye (1986b) AdapUve control of systems wRh unknown phystcal element values Submitted for pubhcatlon Dlduch, C and R Balasubramamam (1982) Integrated atutude control of a satelhte using a mmroprocessor Proc IFAC Syrup Theory and Apphcauon of Dzgttal Control, pp 9-14, New DelM, January Hodge, W V D and D Pedo¢ (1953) Methods of Algebrmc Geometry Cambridge Umverslty Press, Cambridge Krelsselmemr, G (1977) Adaptive observes with exponenual rate of convergence IEEE Tram Aut Control, AC-22, 2-8 Lion, P M (1967) Rap~d identification of linear and nonlinear systems AIAA JI, 5, 1835-1842 Luders, G and K S Narendra (1974) Stable adaptive schemes for identfficatmn ofhnear systems IEEE Trans Aut Control, AC-19, 841-847 Narendra, K S and P Kudva (1974) Stable adapuve schemes for system ~dentfficanon and control--Parts I and II IEEE Tram Syst Man, Cybern SMC-.4, 542-560 Youla, D C and G Gnaw (1979) Notes on n-dimensional systems theory IEEE Trans Ccts Svst CAS-26, 105-111
Automatwa Vol 23 N o 4 pp 469-477 1987 Pnnted m Great Britain
Physically Based Parameterizations for Designing Adaptive Algorithms* SOURA DASGUPTAf and BRIAN D O ANDERSON,: Special state var|able and transfer functmn descmptmns are developed for systems whose unknown parameters are a wide class of phystcal element values The dependence of the state varmble structures on these parameters is "rank-1" and that of the numerator and denominator polynommls of the transfer functions multdmear Key Words--Adaptive algorithms, physical paramters, rank-l, multflmear
Section 2 specifies the parameterization in question and demonstrates how RLC orcuits fall within ItS ambit Section 3 investigates the corresponding state vanable realizations while Section 4 shows that the resulting minimal transfer functions are the ratios of polynommls having coefficients multihnear in the parameters For example, for a system with two unkown parameters, kl and k2, the transfer function could be
Abstract--Structural propertmes are exammed of systems w~th physical component values as parameters Both state variable reahzattons and transfer functmn descnptmns are investigated The transfer functmns m particular are shown to be the rahos of polynommls wRh coefficients multdlnear m the parameters These structures prove useful m formulating adaptive algorithms 1 INTRODUCTION
A FACTOR critical to the performance of adaptwe algorithms is their underlying parametenzations Parameterlzatlons reflecting greater a pmom knowledge about the unknown system can be expected to lead to improved performance Frequently, adaptive systems (Lmn, 1967, Narendra and Kudva, 1974, Luders and Narendra, 1974, Kreisselmeier, 1977, Anderson, 1977) treat all the transfer function coefficients as the unknown parameters In the process considerable information may be discarded Sometimes, the only unknowns in a system are the values of certain physical elements or parameters As far as transfer functions are concerned, th~s may mean that some of the coeffic|ents are known a przorz, as also are some relatmnships, possibly nonlinear, which exist between them Accordingly, a parameterizatmn ts developed here revolving unknown parameters which have direct physical relevance Such a parameterization has been the basis of adaptive algonthms formulated by Dasgupta et al (1983, 1984, 1986a, b)
W(s, k l, k2) = po(s) + klpl(s) + k2P2(S) + klk2P12(s) qo(s) + klql(s) + k2q2(s) + klk2q12(s)
(11)
where the polynomials Pc )(s) and q~ )(s) are known Section 5 derives conditions under which systems with transfer functions such as (1) conform to the parametenzatlon under study All results m the last two sections apply only to single-input, singleoutput systems 2 THE PARAMETERIZATION
Much of the background material for this section is contained in Anderson and Vongpanltlerd (1973, pp 156-200) To understand how physical element values affect a wide class of systems, our attention will be restricted to electric circuits containing resistors, lnductors and capacitors The extensions to the corresponding chemical and mechanical analogues will of course be immediate Consider a resistor R appearing In an n-port clrcmt Clearly the resistor can be extracted from the rest of the circuit in a manner depicted in Fig 1 Suppose that u l , y l are the port voltage and current (or current and voltage) at the nght hand port of the "circuit without the resistor", and U, Y are the input and output vectors of the terminated
*Recewed 30 April 1986, rewsed 2 December 1986 The original version of th~s paper was presented at any IFAC meeting This paper was recommended for pubhcatmn m rewsed form by Assocmte Edaor G Krelsselmeler the &rectlon of Editor P C Parks 5"Department of Electrical and Computer Engineering, Umvers~ty of Iowa, Iowa City, IA 52242, U S A :~Department of Systems Engineering, Research School of Physical Sciences, Austrahan Nahonal Umverslty Canberra, ACT 2601, Austraha 469
470
S DASGUPTA and B D O ANDERSON and Circuit W)thout Resistor R
FIG I
R
Representationof a ctrcmt w~th a resistor R
|
FIG 2
U :~
Input/Output Block
u~ )t
Independent of k 1
Y
)
Yl
Representationof a orcmt with an unknown parameter kl
_
Input/Output Block Independent of k 1 kn
_
ul(t) = Kyl(t)
VVV
Remark 2 1 The elements which cannot be treated in this manner are mutual lnductors--they allow cross-couphng between energy storage devices-and gyrators Theorem 2 1 below shows how RLC clrcmts conform to this description Before stating the theorem, a definition Is reqmred Consider an m port RLC network having mdimensional input and output vectors U( ) and Y(), respectively, with Y() finite for all finite U ( ) Suppose also that each port is represented m the input vector by either but not both of Its voltage or current and that if a port voltage (current) appears m the input then the correspondmg current (voltage) appears in the output Suppose N of the network components are f~l, ,fiN Then under these condmons ~1, ,f~N are extractable if the conditions of the following defimtlon are met Definition 2 1 Consider the m + N port network o b t a m e d b y e x t r a c t m g f ~ l , ,l) n Let f.7( )and Y( ) be N-dimensional vectors havmg elements ti,( ) and 37,( ) such that (1) ti,( ) is either the voltage or the current at the port created by the removal of fl,, (n) if ti,( ) is the relevant voltage, then 37,( ) Is the correspondmg current and vice versa
FIG 3 Representatmn of systems satisfyingAssumption 2 1
network, with entries of U, Y corresponding to voltages or currents at the left port If a port voltage appears m one of U, Ythe port current must appear m the other With kl then ldent~fied w~th R or R depending whether u~ is a voltage or current, redescrlpUon vm Fig 2 is possible Of course, for th~s description to make sense, the hybrid matrix relating [ U r u ~]r to [ y r y l ] r should exist More generally, when a clrcmt has N unknown physical components, with values k~, , kN, then in many cases an input-output descnptmn of the form m Fig 3 exists Assumption 2 1 formahzes th~s description and constitutes the standing assumptmn for this paper
Assumptlon 2 1 Consider a system with an ndimensional input vector U and m-dimensional output vector Y Suppose it has N unknown physical components with values kl, ,ks Then defining K ~ drag {kl, , kN} there extst two N-vectors yl(t) and u~(t) and an (m + N) x (n + N) dimensional transfer function matrix T(s) such that [yT(s)yI(s)]T = T(s)[UT(s)uI(s)] T
Then for the gwen U( ) and Y(), t)l, ,ON are extractable if there exist a choice of/.7 and Y and a fimte M such that [yr(), ~t( )It = M [ u t ( ) , tTt( )it
(2 1) VVV
Observe that the definmon reqmres the existence of a hybrid description relating the augmented vectors [~-T(), yr( )] and [f.TT(), Ur( )]x to each other As pointed out m Anderson and Vongpamtlerd (1973, pp 171-198), the elements of an RLC network for a gwen input/output set wdl fad to satisfy the condmon for extractablhty if one of the following hold (1) A particular element fl, does not affect the input/output relationship, m which case fZ, can be removed from consideration (11) The problem of finding a state variable descrtpUon is 111posed Since the motivation here ~s to consider parametenzatlons relevant to adaptive control, such a system is of no interest to us Thus, m deahng with RLC circuits, this paper will be restricted to situations where Defimtton 2 1 apphes Theorem 2 1 will now be stated and the proof gwen
Physically based parameterizatmns
Theorem 2 1 Consider an m port LTI lumped RLC clrcmt with m inputs u, and m outputs y, which are the port voltages and currents Suppose every port is represented in the mput vector by e~ther, but not both, of its voltage and current Also, ff a particular voltage appears m the mput the correspondmg current appears in the output and wee versa Suppose the N components of the system are {/~,},u=t with component values {#,},s=t Then a representatton of the form of Assumptmn 2 1 exists with k, = #, or 1/#,, ff fl, are extractable in the sense of Defimtlon 2 1 Proof With appropriate definmons and the extractability property there j = {1,2} such that
exist
M v,
~= {1,2},
471
Remark 2 2 The result tnwally extends to mechanIcal and chemical analogs of RLC clrcmts It also holds for a large number of actwc clrcmts hawng controlled sources and their analogs 3 T H E STATE VARIABLE R E A L I Z A T I O N
In this secUon it will be shown how N-parameter systems satlsfymg Assumptmn 2 1 are reflected in their state variable reahzatmns Before this is done, the followmg defimtlon will be made
Defimtmn 3 1 A state variable rcahzatlon described by the quadruple {F,G,H,J} has a rank-I dependence on N parameters kt, ,kN, If Vie {1, ,N}, there exist a,,b, eR, independent of k,, such that defining ~, as c~thcr 1 0tip
~(s)J
M22JLO(s)A
LM21
m
a, + k,b,
or
61 ~
where Y,,(s) Is either the voltage across fl,, or the current through it, if ~{s) is a voltage tben G~(s) is the corresponding current and vice versa Now the relatmns between the voltages, and currents across resistors, mductors and capamtors arc respectwely gwcn by
V(s) = sLI(s)
Uts) =
0
I: ] oo] I
sl 0 0 11 K~'(s) s
Thus, with
U,(s) =
=L-H,
IS:]+,,= VVV
Observe that if a M I M O system obeys this definmon, then so does any SISO subsystem contamed by it The followmg theorem shows that a system conforming to the reqmrcments of Assumption 2 1 has at least one state variable reahzatlon which has a rank-1 dependence on the N parameters
output description samfylng Assumption 2 1 In other words, if u and y are mput and output vectors, respectively, there exist N input and outputs, all elements of vectors u, and yx, respectwely, such that
slJ
and Y,(s) = Y(s) the result follows
EsI~ jG] pI-,,
Theorem 3 1 Consider a system having an mput
1-I 0 U(s) s
0
, N},
that
Es,-:
Thus, with appropriate ordcnng of fl, and K ~dmg {k,}L, 0
the followmg is true There exist, V t ~ { 1,
with g,,h, vectors In other words, one can claim that
V(s) = ~I(s)
I
a, + k,b,
F,, G,, H,, J,, h, and g, all Independent of k,, such
-
RI(s)
V(s) =
kl
m
VVV
[ s,1 [*,] y,(s)J
=
T(s) u,(s)
472
S DASGUPTA and B D 0 ANDERSON
and
lf l # j then Yl = K-lU 1
e l F = kje sT A, - 1
with K & drag Ikl, ,kNl Assume T(s) is proper Then there exists at least one state variable reahzatmn, w~th input u and output y, whmh has a rank1 dependence on the k, for all but ~solated values of k, P r o o f Since T(s) is proper, matrices A, B, B~, C, Ca, D11, D12, D21 and D2z exist such that x = A t + [B
B1][U T
IVT y T ] T = [ c T
uT] T
k,ks T A - t D22e,eV A 7 1 + 1 - k,e~A,~-lOz,e, es '
whence, with :t, = k,/(1 - k,eT, Ai - 1D22e,), e l f = r,T j, + ot,wj,eT A, - 1
with vj,,wj,,(eT, A[ -~) independent of k, and vs, vector and w j, a scalar If t = j eTF = cqeT A, - 1
cT]Tx
~Dll D12] + LD21 D223[u T uT]T
a
Thus Wl 3
(3 1)
Wt-l,l
As Ul = Ky~, a httle mampulatmn shows that F = V, + ~,
1
~c = {A + B x K [ I - D 2 2 K ] - I C I } x
e~A~ -1
14"t + 1 ' I
+ {B + B 1 K [ I - D 2 2 K ] - l D 2 1 } u WN,I
y = {C + D , 2 K [ I - D 2 2 K ] - ' C , } x + IDll + D l z K [ I - D 2 2 K ] - I D 2 1 } u
Thus
with E, e,T A,-~ and ws, Independent of k, Hence, the result follows VVV Presented below is a non-electrical example where the state varmble reahzatlon has a rank-1 dependence on most element values The dynamics pertinent to the attitude control of the communications technology satelhte, H e r m e s (Dlduch and Balasubramamam, 1982), are as follows
(3 2)
~=Ax+Bu Y= C \
Denote K [ I - DE2K]-~ by F Suppose A, = l - D22K + k,Dz2e,e x,
where (3 3)
where e, is the unit vector with umty m the tth element and zero elsewhere The matrix A, is independent of k, Observe that matrices like F and A, are lnvertlble for all but isolated values of the k, Our analysis shall exclude such points m the k space Also,
A=
B =
I - = K [ A [ -x + k, lAD ' 22ee'~A~T_'
]-~-22e,1]
Consider the jth row of F, A[- 1D22e,eTA[-' ] eTF = k~er A~-1 + k, l _ ~ l - ~ 2 2 e l J
0
1
0
0
-woh/l 1 0
0 0
0 0
Wo - h / l l 1
0
h/I 2 - w 0
- woh/I 2
0
0
0
F I L 1 G 1 cos ot/I l
0
0
0
-FILIGI
sln~t/12
F2L2G2/I2
Physically based parameterlzatlons
c=
having a state variable reahzatlon which has a rank-l dependence on a single parameter ~ has a transfer function whose numerator and denominator are affine in at
['0 °° °0] 0
473
1
Theorem 4 1 An n-dimensional SISO system represented by
U2
Here ~b is the roll, x is the yaw, 11 is the moment of inertia about the roll axis, 12, that about the yaw axis, Wo ~s the orbital rate, h the nominal wheel angular momentum, 0t the offset angle, Lx and L2 the offset and yaw thruster moment arms, respectively, and G1 and G2, the lnpulse bit factors The inputs ux and u2 provide a guide for the level of consumed fuel It is evident that the parameters Ix, 12, Fx, F2, L1, L2, Gx and G2 appear in the state variable representation in a rank-1 fashion Also, although
= F(~)x +
g(ct)u
y = hT(~)X + J(~)u with a system matrix of the form
sl - F(~t) g(~t)] - hT(~t) j(~t) J F s l - Fo go] - h oT ] J o +
=L
k h 2 j [ g l I gz]
(4 1)
t? FsI - A
_c I ol
where Fo, go, ho,Jo, hi, h2, gx and g2 are independent of ct, has a transfer function
has rank 1, the system matrix is obviously not linear in ~ Thus ~ does not quite conform to the definition of rank-1 dependence The parameters wo and h, on the other hand, clearly do not appear m a rank-1 fashmn But, by definition (one is the orbital rate and the other the wheel angular momentum), one can see that they must allow cross-couphng between energy storage elements They thus fall in the same category as mutual lnductors, which as the authors have emphasized, do not fall within the requirements of Assumption 21 4 SISO TRANSFER FUNCTIONS
In this section, ~t will be shown that SISO systems which have rank-1 state variable realizations necessarily have minimal transfer function descriptions which are the ratios of two polynomials multihnear in the unknown parameters At the outset the following definmon of coprlmeness of polynomials in more than one variable will be introduced
W(s) -
a(s) + orb(a) e(s) + ~d(s)
(4 2)
for every ~ The polynomials a(s), b(s), c(s) and d(s) are independent of ~ and obey the following restrictions (a) 6[c(s)] = n, 6[d(s)] < n, 6[a(s)] ~< n and 6[b(s)] <~ n, (b) a(s)d(s) - b(s)c(s) IS factorlzable into two polynomials of degree not exceeding n Conversely, any transfer function of the form (4 2) has a state variable reahzatlon of the form (4 1) provided that conditions (a) and (b) hold
Proof (1) From (41), some calculation shows that W(s) = [a(s)c(s) + ~{a(s)d(s) + g2h2c2(s) - h2?(s)c(s) + g2fl(s)c(s) - fl(s)7(s)}]/{c2(s) + ctc(s)d(s)}
Definmon 41 Consider p,(xx, , x . , x . . 1 , ,x,.), i = 1,2, , r, which are r polynomials in the mdetermmates x~, , x . Then p, are copnme wtth respect to the variables Xl, , x. if there exists no nontnvlal f which is a polynomial in xl, ,x., but ranonal in x. + 1, , ,c.,, such that ff, = p , ,
Vie{l,
,r}
with c(s) = det (sl - Fo) and a, fl, y, d defined by
(4 4)
W(s) = [a(s)/c(s) Ly(s)/c(s)
=
d(s)/c(s)J
F-l,hTol[sl_Fo]_,[g ° I_gI_l
with f polynomials m xl, , x. and rationals in Xn+l~ ,Xm, as well The following theorem shows that a system AJT 23 4-D
(4 3)
hi]
(44)
Because ff'(s) has an nth order reahzatlon, ItS Macmillan degree is not greater than n Thus,
474
S DASGUPTA and B D O ANDERSON
ad - f17 is divisible by c(s) Define b(s) by
where a, b, c, d are polynomials m s and/,2 Suppose
b(s)c(s) = a(s)d(s) - [?(s) - gzc(s)] [fl(s) + h2c(s)]
(4 5)
Then (4 3) has the same form as (4 2) and by (4 5) and (4 4) conditions (a) and (b) are satlsfield (n) Suppose that (a) and (b) hold for (4 2) Let
a(s)d(s) - b(s)e(s) = fl(s)fz(s) = ['~(S) --
g2c(s)]
[fl(S) +
h2c(s)]
with 3[y(s)] < n and 6[fl(s)] < n Then a(s)d(s)L(s)fz(s) IS divisible by e(s) whence
I.is,/c(sq Lfl(s)/c(s)
a(s, k2) + kxb(s, k2) = m(s, k x, k2)p(s, k I , k2) c(s, k2) + kid(s, k2) = m(s, k I , k2)q(s, k 1, k2)
{4 6)
d(s)/c(s).J
has Macmillan degree no greater than n Hence, l,V(s) can be expressed as
hVol[sl_Fo]_1[gohl]+[ 7o h~] gIJ
(l) a(s, k2) , b(s, k2), ¢(s, k 2} and d(s, k~) are copnme with respect to 3 and/'2, (ll) a(s, k2)d(s, k2) -- b(S, k2)((& k2) ~ 0 Then a(s, k2) + klb(s, k2) and c(s, k2) + ktd(s, k2) are coprlme with respect to s, k~ and k 2 Proof According to Hodge and Pedoe (1953, p 36), the ring of polynomials in the variables s, kl, k2 over the field of real numbers is a unique factorlzatlon domain Let
L --g2
so that by reversing the argument in the first part of this theorem a state variable reahzatlon of (42) exists in the form displayed m (41) VVV
Remark 41 Given any W(s) of the form (42), with condmon (a) holding, it may be that (b) cannot be satisfied, because ad - bc has all complex roots and n is odd But then one can multiply both the numerator and denominator in (42) by (s + t/), for any i/ to ensure that the resulting a, b, c and d polynomials satisfy (b) The resulting state variable realization conforming to (41) IS then non mzmmal Thus, a system having a non mlmmal state variable reahzatlon of the form in (41) need not have a minimal realization of the same form Notice also, that, If 6[c(s)] < 6[d(s)], then (a) will hold by replacing ct with 1/~ Thus all transfer functions of the form (42) which are proper at almost all values of a, have state variable descriptions like (41) However, the latter need not be minimal Remark 4 2 If the state variable realization has a rank-1 dependence on kl then cq = kl/(al + klbl) or 1/(al + klbl), whence both the numerator and denominator of the transfer function are linear in kl, even though the degree requirements of (a) may have to be replaced by 3[d(s)]<<.n If a 1 = 0 , kl would need to be replaced by 1/k~ The result of the theorem IS now extended to include the situation where the number of parameters exceeds one To this end Lemmas 4 1 - 4 3 are needed, which derive some results for the coprlmeness of polynommls in more than one variable Lemma 41 Consider a transfer functmn a(s, k2) + klb(s, k2) W(s, k x, k2) = c(s, k2) + kld(s, k2)
with m,p,q polynomials in s, k~ and k 2 and m not trivial Consider the following cases
Case I m t s independent oJ kl immediate that (l) is violated
It is vmually
Case II m t s dependent on kl Then m must be affine in k 1 and p, q are independent of kl Suppose m(s, kl,k2) = r1(s, k2) + klrE(S, k2) Then a(s, k2) = q(s, k2)p(s, k2) b(s, k2) = r2(s, k2)p(s, k2) (4 7)
c(s, k2) = rds, k2)q(s, kz) d(s, k2) = r2(s, k2)q(s, k2)
whence a d - bc = O, which too contradicts the hypothesis here VVV
Remark 4 3 Violation of (11) Implies that W is independent of kx We now show that the transfer function is expresmble as ratio of polynomials multlhnear in k, Moreover, from (47) one can see that If the numerator and denominator of W are not coprlme w r t kl, then a(s, k2) W(s, kx, k2) - - - W(s, O, k2)
c(s, k9
Lemma 4 2 Suppose that the transfer function W(s, kl, ,kN) Is expressible as W(s, kl,
, kN)
= a,(s, k(')) + k,b,(s, k (')) c,(s, k (')) + k,d,(s, k ('))
v,~{1, ,N}
where k (°--4 {kl, ,kN} -- {k,} Suppose a f t , b,c, ~ 0 and a,, b,, c,, d, are copnme with respect to s and k (')
Physmally based parameterlzatlons
475
independent of s, for which
Then
W(s, kl,
,ks) =
e(s, k 1, Q(s, kl,
, k~) ,kN)
where P and Q are multfllnear in k,
Proof The case when N = 2 will be proved The more general case follows along the same lines Suppose al(s, k2) + klbl(s, k2) W(s, kl, k2) = Cl(S' k2) + kldl(S, k2) = a2(s, kl) + k2b2(s, kl) c2(s, kl) + k2d2(s, kl) and suppose that the other hypotheses specialized to N = 2 holds Then [al(s, k2) + klbl(S, k2)][c2(s, kO
+ k2d2(s, kl)] = [a2(s, kl) + k2b2(s, kl)] x [Cl(S, k2) + kldl(s, k2)] By Lemma 4 1, al(s, k2) + klbl(s, k2) and ct(s, k2) + kldl(s, k2) are copnme with respect to s, kl and k2 Thus, al(s, k2) + klbds, k2) d m d e s a2(s, kl) + k2b2(s, kl) Thus, al(s, k2) and bl(s, k2) can be at most linear in k2 Similarly, c~(s, k2) and dds, k2) can be at most hnear In k2 Hence the transfer functmn W(s, k~, k2) can be written as
po(s) + klpl(s) + k2P2(s) + klk2P12(s) qo(S) + klql(s) + k2q2(s) + klk2q12(s) whence the result follows VVV It is of Interest here to consider mlnlmahty with respect to s alone That is, it is to be shown that if the hypothesis of Lemma 4 3 holds, then P and Q have no common factors which are polynomials in s but rational In the k, The following lemma whmh follows from Youla and Gnaw (1979) shows that this is indeed the case
, ks) and Q(s, kl, , ks), polynomials in s, kl, , ks, have no common factors which are polynomials in s, kl, ,ks, then they have no common factors which are polynomials m s but rational m kt, ,ks
P(s, k l ,
, ks)x(s, k l , ,ks) + Q(s, kl,
,ks)y(s, kl,
,ks)= ~b(k,, ,ks)
Thus, dividing both sides by ~ the result is immediate VVV Now Theorem 4 1 and Lemmas 4 1-4 3 together yield the following main result of this section
Theorem 4 2 If the state variable reahzatlon of a SISO linear tlme-lnvanant finite dimensional system has a rank one dependence on N parameters kl, , ks, then it has a mmlmal (with respect to s) (see Definmon 4 1 for copnmeness) transfer function descnptmn whose numerator and denominator polynomials are multlhnear in the k, 5 TRANSFER FUNCTIONS CONFORMING TO THE STANDING ASSUMPTION In this section, condmons under whmh scalar transfer functions with numerator and denominator multlhnear m the unknown parameters, correspond to systems satisfying Assumptmn 2 1 are derived Theorem 5 1 below summarizes these condmons The theorem can be understood by observing that a scalar system satisfying this assumption can be expressed in terms of the following input-output descrlptmn
Y(s) = -a(s).. ~ u t s l , + gT(S) - ~ uI(s) h(s) . . . . Dis) YI(s) = ~(-~tJts) + --7-7, Ul(s) c~sj
(5 1)
Ul(s) = KYI(s) K __4dlag {kl,
,ks}
where a ( ) and c() are scalar polynomials with c( ) the characteristic polynomial of the transfer function relating [U, U1] r to [Y, I"1]r, h( ) and g( ) are N-&menslonal polynomial vectors and D( ) is an N x N polynomial matrix It is not hard to see then, that the transfer function relating U to Y is
Lemma 4 3 If P(s, kl, k2,
a(s) . gX(s) . . . . . . W(s) = - ~ + - - ~ hl_CtS)l -- D(s)K]- lh(s)
(5 2)
Equation (5 2) forms the basis for Theorem 5 1
Proof If P and Q have no common factors which are polynomials m s , kl, ,ks then P and Q are minor copnme as well (see Youla and Gnavl for a
Theorem 5 1 Consider a system with a proper transfer function
definmon) Thus by Youla and Gnavl there exist polynomials x, y and ~,, with ~, nontnvlal and
W(s, k) = P(s, k)/Q(s, k)
(5 3)
476
S DASGUPTA and B D O ANDERSON
where k ~ [kl,
By defimtlon, T(s) and all submatnces thereof have reahzatmns with characteristic polynomml c(s) Thus V r c S scalar polynomials ~,(s) and/~,(s) exist such that
,kN]T (5 4)
P(s,k) = po(s) +~I-I,=s(,~k,)P,(s)
{Irl- 1}
c(s) q,(s) = det ( - O,(s))
(5 5)
and wRh S = {1, ,N} and 6[qo(s)] ~,6[q,(s)] Suppose P(s, k) and Q(s, k) are copnme with respect to s and the elements of k Then the corresponding system samfies Assumptmn 2 1, w~th a proper T(s), fff there exist a scalar polynomml f(s), N&mensmnal vector polynommls g(s) and h(s) and an N x N matnx polynomml D(s), with 6[g(s)], 6[h(s)] and 6[D(s)] all less than or equal to 3[f(s)qo(s)], such that the following hold Yr c S
Irl
c( s )~,(s) = a(s) det O,(s)- gT,(s) Adj D,(s)h,(s) (5 6) Now,
W(s, k) = l~(s, k)/(~(s, k) where
(1) d e t ( - D,) = f(s)qr(s)[qo(s)f(s)] I'1- 1
P(s, k) = a(s) det (c(s)l - D(s)K)
(n) [qo(s)f(s)]l'lpr(s)f(s)= a(s) det D,(s) - gr,(s){AdjD,(s)}h,(s)
+ gr(s)K Adj (c(s)l - D(s)K)h(s)
(5 7)
where Dr, g, and h, are defined as below (a)
D,(s) is a It[ × Irl matnx consisting of the lth rows
and
and ah columns of D(s) V t e r (b) g,(s) and h,(s) are Irl-&mensmnal vectors consistrag, respectively, of the zth elements of g(s) and
h(s) V t ~ r The ordenng of elements in all cases is consistent
Remark 5 1 Since P and Q are coprlme w r t s and kl, , ks, by Lemma 4 3 they are also copnme w r t s alone ProofO) First the "only if" part wdl be proved Suppose Assumptmn 2 1 is satisfied wRh proper T(s) Then (5 1) holds and W(s,k) is gwen by (5 2), with all quantmes obwously defined Thus to complete the proof the p,(s) and q,(s) m (5 4) must be related to the quantmes m (5 2)
0) P ( s , k , ) = d e t l c ( s ) l - D ( s ) { K "
(~(s, k) = c(s) det (c(s)l - D(s)K)
(5 8)
Observe that the coetticaents o f / ~ and (~ can eamly be shown to be multlhnear m the k, Define 2, = l/k, and A = K - 1 Then the polynomials multiplying I I k, m P, (~ can be obtained as follows let
(1) set kj = 0, Vjer, (11) dwlde by Ilk,, (m) set 2, = 0 t~r With K,, A, obviously defined and k, a vector of elements k)' ~¢r, for P the three steps respectwely, translate to
~}l[a(s)+gV(s)I K"
°03[c,s,,-o,,,{o,
N-It)
= det (c(s)Ii, I - D,(s)K,)c (s) [a(s) + gT,(s)K,(c(s)II,I- D,(s)K,)-lh,(s)],
(n)
/~(s, k,)
1--Ik,
N-Irl
- c (s) det (c(s)Ar - D,(s))[a(s) + gT,(s)(c(s)A, -- D,(s))- lh,(s)],
|Er
N-Irl
(m) /~,(s) = c (s) det ( - D,(s))[a(s) - g[(s)D7 l(s)hr(s)] N-Irl
= c (s) [a(s) d e t ( - O,(s)) - gT,(s) AdJ [D,(s)]hr(s)]
(5 9)
Physically based parametenzatmns Similarly, W(s,k) = N-Irl+ 1
~,(s) =
c(s) det ( - D,(s))
a(s)
+
477 gr(s)rK, o
(5 lo) ×[c(s)l-D(s)[o'
00] ;]l":,s,
By (5 5) and (5 6), = a(s)
4,(s) =
ti-M+l
N
c(s) c(s)'"-%(s) = c(s)q,(s)
(511)
and N
p,(s) = c(s) p,(s)
(5 12)
Define ~o(S) = a(s) and ~o(S) = c(s) Thus,
c(s) +
g,r (s)
where r excludes the elements corresponding to The first equatmn is obtained by using the equatmn m Remark 4 3 Thus the proof can be estabhshed easily (n) The "iff" part follows by a trtvial reversal of the above arguments VVV 6
w ( s , k) =
/~(s, k)
with P and O. obwously defined Consider now, the two following cases Case I P and ~ are coprtme with respect to every element o f k Since P, Q, P and ~ are all multdmear m the elements of k, f(s)P(s, k) = ]~(s, k) and f(s)Q(s, k) = (~(s, k),
whence, f (s)po(s) = a(s) f(s)qo(s) = c(s) f (s)q,(s) = q,(s)
Vr c S
f(s)p,(s) = ~,(s)
Vr c S
and
Thus, condmons (0 and (u) are sausfied due to (5 5) and (5 6) Case II ~t and 0 are not coprtme with respect to k,Vte ~ c S Thcn, by cancelhng factors involving k,, it can be seen that
r'(c(s)G1 - D,(s)K,)- ~hY(s)
CONCLUSION
Certain structural aspects of systems whmh have a parametenzatmn mvolwng physical component values as the parameters have been established The systems investigated include RLC c~rcmts and their chemmal and mechamcal analogs REFERENCES Anderson, B D O (1977) An approach to multwanable system ldentfficatlon Automatwa, 13, 401-408 Anderson, B D O and S Vongpamtlerd (1973) Network Analys~s and Synthesis, A Modern Systems Theory Approach Electncal Engmeenng Series, PrenUce-Hall, Englewood Chffs, New Jersey Dasgupta, S, B D O Anderson and R J Kaye (1983) Robust identification of parttally known systems Proc 22nd CDC San Antomo, Texas, 1, 1510-1514, December Dasgupta, S, B D O Anderson and R J Kaye (1984) IdenUficaUon of economically parametenzed system Prepnnts 9th IFAC World Congress, Hungary, X, 96-100, July Dasgupta, S, B D O Anderson and R J Kaye (1986a) Output error ~denttficatmn methods for partmlly known systems Int J Control, 43, 177-191 Dasgupta, S, B D O Anderson and R J Kaye (1986b) AdapUve control of systems wRh unknown phystcal element values Submitted for pubhcatlon Dlduch, C and R Balasubramamam (1982) Integrated atutude control of a satelhte using a mmroprocessor Proc IFAC Syrup Theory and Apphcauon of Dzgttal Control, pp 9-14, New DelM, January Hodge, W V D and D Pedo¢ (1953) Methods of Algebrmc Geometry Cambridge Umverslty Press, Cambridge Krelsselmemr, G (1977) Adaptive observes with exponenual rate of convergence IEEE Tram Aut Control, AC-22, 2-8 Lion, P M (1967) Rap~d identification of linear and nonlinear systems AIAA JI, 5, 1835-1842 Luders, G and K S Narendra (1974) Stable adaptive schemes for identfficatmn ofhnear systems IEEE Trans Aut Control, AC-19, 841-847 Narendra, K S and P Kudva (1974) Stable adapuve schemes for system ~dentfficanon and control--Parts I and II IEEE Tram Syst Man, Cybern SMC-.4, 542-560 Youla, D C and G Gnaw (1979) Notes on n-dimensional systems theory IEEE Trans Ccts Svst CAS-26, 105-111