CH 8 RECURSIVE IDENTIFICATION
3. C onvergence
This section is devoted to proving a convergence result on the perturbed extended least squares algorithm proposed in the previous section. The proposed algorithm
CH 8 RECURSIVE IDENTIFICATION
does not fit directly into the framework where known convergence analysis applies, yet our analysis specializes to known techniques. Let us consider the simplest situation of a stable signal model under persistently exciting input signals.
Theorem 3.1: Consider the signal model with assumed order (n,m,/) as in (2.1)-(2.4), possibly overparametrized by some L < min(n,m,/), and assumed stable. Consider the perturbed recursive extended least squares algorithm (2.6)-(2.8). Consider also that the input uk is uniformly persistently exciting as in (2.3). Then there is parameter convergence as k goes to infinity,
ök —> 0 a.s. at the rate of 0([ln(k)*Vk]2) (3.1)
where 0 is the unique parameters associated with (2.1) with all possible pole / zero cancellations at the origin, and arbitrary constant [ i > 1. ***
As a step to prove the Theorem 3.1, let us first introduce the following lemma, which specializes to known results when Dk , vk = 0 in the absence of overparametrization.
Lemma 3.2 Consider that the conditions of Theorem 3.1 apply and that 0 is defined as in the theorem. Then
oo ( 1 ) X [ 0 Tv k - i v k - l T0 ] ^ Kv < k = l (3.2) k (2) f c ä . X l v T P w / t K B i - 1 ) ] < ~ i=l (3.3) (3) n ►Q £ II (3.4)
CH 8 RECURSIVE IDENTIFICATION
Zk 4 yk - ¥ k 0 k - Wk, hk 4 \|/k0 k - öxVk-l, Bk 4 0 - 0k (3.5)
(4) E[bkwk IFk-i] = -YkPkWJw . bk 4 (3-6)
(5) There are some constants pi>0, p2>0, and K < °o such that
k
Sk 4 X[2biZi - b|2 - P lbj2 - p 2zi2] + K ä 0, (3.7)
i=l
Proof:
(1). Suppose the system (2.1) is overparametrized by 0 < s ^ L. Then with (2.3) and (2.2), it is well known that [5] ^ ^ ( j ) 0 for j > s, at the rate of 0 (l/k ) . Therefore Gk(j) —>0 at the rate of 0 (l/k ) for j > s, or
k=l
Of course, for j < s, ök(j) does not approach zero as k —> <», but terms involving Ok(j) for j < s in 0TDk are zero, since the system is overparametrized by s. Combining this result with (3.8) and V < <» in (2.7) leads to (3.2).
(2) Now (tr^BiJI)-1 < Pj so that
X o f c ) < <*> (3.8)
Vi Pj Vi / tr [ B n ] < v ! PiPi-iVi (3.9)
Then from (2.5)
CH 8 RECURSIVE IDENTIFICATION
YiPiPi-lVi = tr[Pi.i] - tr[Pi] (3.10)
Now (3.3) follows from (3.9) by summing up on both sides of (3.10).
(3) Now with the definition for Zk of (3.5)
C(q-’ )zk = CCq'1 ) [ y k - V £ e k - w k ]
= [CCq-1 ) - i] [y k - V j9k ] + yk - Vlßk - C(q_1 )wk
But from (2.4) (2.6), manipulations yield
? k0 = [C (q _1 ) - l] [yk - ylQ v ] + yk - C(q_1 )wk
Thus
C(q_1 )zk = <j>k0 - y£8k = v£0k - 0xvk-i
where the second equality follows from (2.6) and definition of 0k in (3.5). Applying the definition for hk in (3.5) the result (3) is established.
(4) From the definition of bk in (3.6),
bkWk = y£0kWk (3.11)
Also from (2.5),
BkQk = Bk-lök-1 + W k = Bk0k-1 + Vk(yk - Vk^k-l)
0k = 0k-1 + PkVk(0k0 + wk ’ Vk0k-l)
CH 8 RECURSIVE IDENTIFICATION
Substituting (3.12) into (3.11) and taking conditional expectations on both sides gives the result (4). In this manipulation, recall that all quantities in the expression save wk and vk are Fk-i measurable.
(5) Proof is given in Appendix B. AAA
Proof of Theorem 3.1: From (2.5), (3.5)
(Bk-i + YkYk )0k = Bk-lök-l + W k
6k = 0k-1 + Pk-lYk(yk - \|/Tk0k )
9k = §k-l - Pk-lVk(zk + Wk) (3.13)
Now defining Vk = 0 k®k0k and applying (3.13), we have
Vk = 0 KBk-i + VkVk)0k
= Vk-l - 2\l/l0k-l(zk+Wk) + Vkpk-lVk(zk+Wk)2 + 0 kVkVk0k
= Vk-l - 2Vk9k(zk+Wk) - Vkpk-lVk(zk+Wk)2 + 0 kVkVkök
and with definition for bk as in (3.6)
Vk = Vk-i + b { - 2bkZk - 2bkWk - V0Pk-iVk(wk+Zk )2 (3.14)
Performing conditional expectation on both sides and applying (3.6) then E[Vk IFk-i] = Vk-i + E [ b |- 2bkZk IFk-i ] + 2\|/£Pk W * w "
- E [y ^ P k -i Vk(wk + Zk )2 IFk-i ] (3.15)
CH 8 RECURSIVE IDENTIFICATION
a Vfc Sv X ^ b \ X S z \
Xk= tt(Bjj+ tt(Bri)+ p i i2wtt(Bn) +p242 u f f ( B n ) +
i=l i=l
+ X ^ s ^ ( z i + w i ) 2 + X v i [ T O - 5 < y (3-i6)
Then it is easy to verify that
VÜPkVkO;
E[X k IFk. ! ] < X k. 1 + ^ 0 (3.17)
Applying the martingale convergence theorem [ 2,p501], under (3.2) (3.3), we conclude that Xk converges almost surely as k —»<». This implies that,
k k
1 z2
< o o a.s. Ann 7 - _ ± _ . < o o a.s. ( 3 . 1 8)
1™“ ^ j t r ( B i . i )
Also from the definitions of fk , hk and the Schwarz inequality, k
^ ^ t r ( B i .i) < oo a.s. (3.19)
We now demonstrate that under the excitation and stability conditions of the theorem, for some 8 and k,
jfc^ .sup (^maxBkAminBk) ^ ^ ^ 0SUp(trBkAmin®k) ^ 8 < ©° a.s. (3.20)
^ s u p (XmaxBk/ k) < K, a.s. (3.21)
CH 8 RECURSIVE IDENTIFICATION
lack of excitation of certain modes reflects itself in the property that Gk(j) = ^ ( j ) converge to some nonzero random variable for j < s. Then from the result of [5] \j/k is reachable from wk, Uk and vk-i, and since Wk, Uk, vk-i is persistently exciting under (2.2), (2.3) and (2.7), \|/k is persistently exciting. Now under the stability assumption, the technique of [2,p345] applies here to show that (3.20) holds.
With (3.20) holding, it is easy to show that -1 Pk<5l[tr(Bk)] k k V ]P iV i ^ V iV i <3-22> i=l k
V VlVi V
^ t r ( B i)Ri “2Li
tr(Bj) - tr(Bj.i) tr(Bi)Ri m OrBiy
j
tr(Bi)Ri trBkJ
trBox^nx)*1 [ln(trBo)]1 [ln(trBk)]1 ^ < oo (3.23)CH 8 RECURSIVE IDENTIFICATION
Then it is easy to verify that
E[Qk IFk.i] < Qk-i + 2 -¥lPk^ k° ^
Applying the martingale convergence theorem [2,p501], under (3.2) (3.23), we conclude that Qk converges almost surely as k —> <». This implies, recalling the definition of Vk
Thus 0 k§k —> 0, 0k —» 0, from (3.21), at the rate of 0([ln(k)^/k]2) and the result (3.1) is established. Notice that here 0 is such that (3.2) holds, and has the property
Remark The extension of known techniques in the proof of the above theorem cope with the extra signals vk in the signal model and thereby to cope with overparametrization. Clearly, the novel techniques also apply in related situation involving adaptive control (details are omitted here).