A new concept of strong controllability via the Schur complement in adaptive tracking

0.22 MB·English

by  Bernard Bercu

A NEW CONCEPT OF STRONG CONTROLLABILITY VIA THE
SCHUR COMPLEMENT IN ADAPTIVE TRACKING
BERNARD BERCU AND VICTOR VAZQUEZ
Abstract. We propose a new concept of strong controllability associated with
the Schur complement of a suitable limiting matrix. This concept allows us to
extend the previous results associated with multidimensional ARX models. On
the one hand, we carry out a sharp analysis of the almost sure convergence for
8 both least squares and weighted least squares algorithms. On the other hand,
0 we also provide a central limit theorem and a law of iterated logarithm for these
0 two stochastic algorithms. Our asymptotic results are illustrated by numerical
2 simulations.

n
a
J

8 1. Introduction
1
Consider the d-dimensional autoregressive process with adaptive control of order
]
R (p,q), ARX (p,q) for short, given for all n 0 by
d
≥
P
(1.1) A(R)X = B(R)U +ε
. n+1 n n+1
h
t where R stands for the shift-back operator and X ,U and ε arethe system output,
a n n n
m inputanddrivennoise, respectively. ThepolynomialsAandB aregivenforallz C
∈
[ by

1 A(z) = I A z A zp,
d 1 p
v − −···−
1 B(z) = Id +B1z + +Bqzq,
···
9
where A and B are unknown square matrices of order d and I is the identity
9 i j d
2 matrix. Denote by θ the unknown parameter of the model
.
1
θt = (A ,...,A ,B ,...,B ).
0 1 p 1 q
8
Relation (1.1) can be rewritten as
0
v: (1.2) Xn+1 = θtΦn +Un +εn+1
i
X where the regression vector Φ = Xp,Uq t with Xp = (Xt,...,Xt ) and
n n n−1 n n n−p+1
r
a Unq = (Unt,...,Unt−q+1). In all the se(cid:0)quel, we s(cid:1)hall assume the (εn) is a martingale
difference sequence adapted to the filtration F = ( ) where stands for the σ-
n n
F F
algebra of the events occurring up to time n. We also assume that, for all n 0,
E[ε εt ] = Γ a.s. where Γ is a positive definite deterministic covariance≥ma-
n+1 n+1|Fn
trix.
Awideliteratureconcerningtheestimationofθaswellasonthetrackingcontrolis
available, [1], [4], [5], [7], [8], [10], [12], [15]. The purpose of this paper is to establish
sharp asymptotic results for stochastic algorithms associated with the estimation
2000 Mathematics Subject Classification. Primary: 62G05 Secondary: 93C40, 15A09, 60F05,
60F15.
Keywords andphrases. Estimation,adaptivecontrol,Schurcomplement,centrallimittheorem,
law of iterated logarithm.
1
2 BERNARDBERCU ANDVICTOR VAZQUEZ
of θ via the introduction of a new concept of strong controllability. The strong
controllability is closely related to the almost sure convergence of the matrix
n
S = Φ Φt.
n k k
k=0
X
In the particular case q = 0, it was shown in [5] that
S
lim n = L a.s.
n→∞ n
where L is the block diagonal matrix of order dp given by
L = diag(Γ, ,Γ).
···
Under the classical causality assumption, we shall now prove that
S
n
(1.3) lim = Λ a.s.
n→∞ n
where Λ is the symmetric square matrix of order δ = d(p+q)
L Kt
Λ =
K H
(cid:18) (cid:19)
and the matrices H and K will be explicitly calculated. It is well-known [14] that
det(Λ) = det(L)det(S) where S = H KL−1Kt is the Schur complement of L
−
in Λ. Moreover, as L is positive definite, Λ is positive definite if and only if S is
positive definite. Via our new concept of strong controllability, we shall propose a
suitable assumption under which S is positive definite. It will allow us to improve
the previous results [5], [6], [10], [11] by showing a central limit theorem (CLT) and
a law of iterated logarithm (LIL) for both the least squares (LS) and the weighted
least squares (WLS) algorithms associated with the estimation of θ.
The paper is organized as follows. Section2 is devoted to the introduction of our
new concept of strong controllability together with some linear algebra calculations.
Section3 deals with the parameter estimation and the adaptive control. In Section
4, we establish convergence (1.3) and we deduce a CLT as well as a LIL for both
LS and WLS algorithms. Some numerical simulations are provided in Section 5.
A short conclusion is given in Section6. All technical proofs are postponed in the
Appendices.
2. Strong controllability
In all the sequel, we shall make use of the well-known causality assumption on B.
More precisely, we assume that for all z C with z 1
∈ | | ≤
(A ) det(B(z)) = 0.
1
6
In other words, the polynomial det(B(z)) only has zeros with modulus > 1. Conse-
quently, if r > 1 is strictly less than the smallest modulus of the zeros of det(B(z)),
then B(z) is invertible in the ball with center zero and radius r and B−1(z) is a
holomorphic function (see e.g. [9] page 155). For all z C such that z r, we
∈ | | ≤
shall denote
∞
(2.1) P(z) = B−1(z)(A(z) I ) = P zk.
d k
−
k=1
X
STRONG CONTROLLABILITY VIA THE SCHUR COMPLEMENT 3
All the matrices P may be explicitly calculated as functions of the matrices A and
k i
B . For example, P = A , P = B A A , P = (B B2)A +B A A . We
j 1 1 2 1 1 2 3 2 1 1 1 2 3
− − − −
shall often make use of the square matrix of order dq given, if p q, by
≥
P P P P
p p+1 p+q−2 p+q−1
···
P P P P
p−1 p p+1 p+q−2
 ··· 
Π =
··· ··· ··· ··· ···
 Pp−q+2 Pp−1 Pp Pp+1 
 ··· 
 Pp−q+1 Pp−q+2 Pp−1 Pp 
 ··· 
 
while, if p q, by
≤
P P P P
p p+1 p+q−2 p+q−1
··· ···
··· ··· ··· ··· ··· ···
 P P P P 
Π = 1 2 ··· ··· q−1 q .
 0 P1 P2 Pq−2 Pq−1 
···
 
 
··· ··· ··· ··· ··· ···
 0 0 P P 
 ··· 1 ··· p 
 
Definition 2.1. An ARX (p,q) is said to be strongly controllable if B is causal and
d
Π is invertible,
(A ) det(Π) = 0.
2
6
Remark 2.1. The concept of strong controllability is not really restrictive. For ex-
ample, if p = q = 1, assumption (A ) reduces to det(A ) = 0, if p = 2, q = 1 to
2 1
6
det(A B A ) = 0, if p = 1, q = 2 to det(A ) = 0, while if p = q = 2 to
2 1 1 1
− 6 6
A A B A
1 2 1 1
det A2 B1A1 A3 B1A2−+(B12 B2)A1 6= 0.
(cid:18) − − − (cid:19)
For all 1 i q, denote by H be the square matrix of order d
i
≤ ≤
∞
H = P ΓPt .
i k k−i+1
k=i
X
In addition, let H be the symmetric square matrix of order dq
H H H H
1 2 q−1 q
···
Ht H H H
2 1 2 q−1
 ··· 
(2.2) H = .
··· ··· ··· ··· ···
 Hqt−1 ··· H2t H1 H2 
 Hqt Hqt−1 ··· H2t H1 
 
Forall1 i p, let K = P Γ and denote by K the rectangular matrixof dimension
i i
≤ ≤
dq dp given, if p q, by
× ≥
0 K1 K2 Kp−2 Kp−1
··· ···
0 0 K1 Kp−3 Kp−2
 ··· ··· 
K =
··· ··· ··· ··· ··· ··· ···
 0 0 K1 K2 Kp−q+1 
 ··· ··· 
 0 0 K1 Kp−q 
 ··· ··· ··· 
 
4 BERNARDBERCU ANDVICTOR VAZQUEZ
while, if p q, by
≤
0 K1 Kp−2 Kp−1
···
0 0 K1 Kp−2
 ··· 
··· ··· ··· ··· ···
K = 0 0 0 K .
 1 
···
 0 0 0 0 
 
···
 
 
··· ··· ··· ··· ···
 0 0 0 0 
 ··· 
 
Finally, let L be the block diagonal matrix of order dp
Γ 0 0 0
···
0 Γ 0 0
 ··· 
(2.3) L =
··· ··· ··· ··· ···
 0 0 Γ 0 
 ··· 
 0 0 0 Γ 
 ··· 
 
and denote by Λ the symmetric square matrix of order δ = d(p+q)
L Kt
(2.4) Λ = .
K H
(cid:18) (cid:19)
The following lemma is the keystone of all our asymptotic results.
Lemma 2.2. Let S be the Schur complement of L in Λ
(2.5) S = H KL−1Kt.
−
If (A ) and (A ) hold, S and Λ are invertible and
1 2
L−1 +L−1KtS−1KL−1 L−1KtS−1
(2.6) Λ−1 = S−1KL−1 − S−1 .
(cid:18) − (cid:19)
(cid:3)
Proof. The proof is given in AppendixA.
3. Estimation and Adaptive control
First of all, we focus our attention on the estimation of the parameter θ. We shall
make use of the weighted least squares (WLS) algorithm introduced by Bercu and
Duflo [3], [4], which satisfies, for all n 0,
≥
(3.1) θ = θ +a S−1(a)Φ X U θtΦ t
n+1 n n n n n+1 − n − n n
(cid:16) (cid:17)
where the initial value θˆ may be arbitrarily chosen and
b 0 b b
n
S (a) = a Φ Φt +I
n k k k δ
k=0
X
where the identity matrix I with δ = d(p + q) is added in order to avoid useless
δ
invertibility assumption. The choice of the weighted sequence (a ) is crucial. If
n
a = 1
n
we find again the standard LS algorithm, while if γ>0,
n
1 1+γ
a = with s = Φ 2,
n logs n || k||
(cid:16) n(cid:17) Xk=0
we obtain the WLS algorithm of Bercu and Duflo.
STRONG CONTROLLABILITY VIA THE SCHUR COMPLEMENT 5
Next, we are concern with the choice of the adaptive control U . The crucial role
n
played by U is to regulate the dynamic of the process (X ) by forcing X to track
n n n
step by step a predictable reference trajectory x . We shall make use of the adaptive
n
tracking control proposed by Astr¨om and Wittenmark [1] given, for all n 0, by
≥
(3.2) U = x θtΦ .
n n+1 − n n
By substituting (3.2) into (1.2), we obtain the closed-loop system
b
(3.3) X x = π +ε
n+1 n+1 n n+1
−
where the prediction error π = (θ θ )tΦ . In all the sequel, we assume that the
n n n
−
reference trajectory (x ) satisfies
n
n b
(3.4) x 2= o(n) a.s.
k
k k
k=1
X
In addition, we also assume that the driven noise (ε ) satisfies the strong law of
n
large numbers which means that if
n
1
(3.5) Γ = ε εt,
n n k k
k=1
X
then Γ converges a.s. to Γ. That is the case if, for example, (ε ) is a white noise or
n n
if (ε ) has a finite conditional moment of order > 2. Finally, let (C ) be the average
n n
cost matrix sequence defined by
n
1
C = (X x )(X x )t.
n n k − k k − k
k=1
X
The tracking is said to be optimal if C converges a.s. to Γ.
n
4. Main results
Our first result concerns the almost sure properties of the LS algorithm.
Theorem 4.1. Assume that the ARX (p,q) model is strongly controllable and that
d
(ε ) has finite conditional moment of order > 2. Then, for the LS algorithm, we
n
have
S
n
(4.1) lim = Λ a.s.
n→∞ n
where the limiting matrix Λ is given by (2.4). In addition, the tracking is optimal
logn
(4.2) C Γ = a.s.
n n
k − k O n
(cid:18) (cid:19)
We can be more precise in (4.2) by
n
1
(4.3) lim (X x ε )(X x ε )t = δΓ a.s.
n→∞ logn k − k − k k − k − k
k=1
X
Finally, θ converges almost surely to θ
n
logn
(4.4) b k θn −θ k2= O n a.s.
(cid:18) (cid:19)
Our second result is relatebd to the almost sure properties of the WLS algorithm.
6 BERNARDBERCU ANDVICTOR VAZQUEZ
Theorem 4.2. Assume that the ARX (p,q) model is strongly controllable. In addi-
d
tion, suppose that either (ε ) is a white noise or (ε ) has finite conditional moment
n n
of order > 2. Then, for the WLS algorithm, we have
S (a)
(4.5) lim(logn)1+γ n = Λ a.s.
n→∞ n
where the limiting matrix Λ is given by (2.4). In addition, the tracking is optimal
(logn)1+γ
(4.6) C Γ = o a.s.
k n − n k n
(cid:18) (cid:19)
Finally, θ converges almost surely to θ
n
(logn)1+γ
(4.7) b k θn −θ k2= O n a.s.
(cid:18) (cid:19)
(cid:3)
Proof. The proof is given bin AppendixB.
Remark 4.1. One can observe that Theorems 4.1 and 4.2 extend the results of Bercu
[5] and Guo [11] previously established in the AR framework.
Theorem 4.3. Assume that the ARX (p,q) model is strongly controllable and that
d
(ε ) has finite conditional moment of order α > 2. In addition, suppose that (x )
n n
has the same regularity in norm as (ε ) which means that for all 2 < β < α
n
n
(4.8) x β= (n) a.s.
k
k k O
k=1
X
Then, the LS and WLS algorithms share the same central limit theorem
(4.9) √n(θ θ) L (0,Λ−1 Γ)
n
− −→ N ⊗
where the inverse matrix Λ−1 is given by (2.6) and the symbol stands for the
b
matrix Kronecker product. In addition, for any vectors u Rd a⊗nd v Rδ, they
∈ ∈
also share the same law of iterated logarithm
n 1/2 n 1/2
limsup vt(θ θ)u = liminf vt(θ θ)u
n→∞ (cid:18)2loglogn(cid:19) n − − n→∞ (cid:18)2loglogn(cid:19) n −
1/2 1/2
(4.10) b = vtΛ−1v utΓu a.sb.
(cid:16) (cid:17) (cid:16) (cid:17)
In particular,
λ Γ n λ Γ
(4.11) min limsup θˆ θ 2 max a.s.
(cid:18)λmaxΛ(cid:19) ≤ n→∞ (cid:18)2loglogn(cid:19) k n − k ≤ (cid:18)λminΛ(cid:19)
where λ Γ and λ Γ are the minimum and the maximum eigenvalues of Γ.
min max
(cid:3)
Proof. The proof is given in AppendixC.
5. Numerical simulations
The goal of this section is to propose some numerical experiments for illustrating
the asymptotic results of Section 4. In order to keep this section brief, we shall only
focus our attention on a strongly controllable ARX (p,q) model in dimension d = 2
d
with p = 1 and q = 1. Our numerical simulations are based on M = 500 realizations
of sample size N = 1000. For the sake of simplicity, the reference trajectory (x ) is
n
STRONG CONTROLLABILITY VIA THE SCHUR COMPLEMENT 7
chosen to be identically zero and the driven noise (ε ) is a Gaussian (0,1) white
n
N
noise. Consider the ARX (1,1) model
2
Xn+1 = AXn +Un +BUn−1 +εn+1
where
2 0 1 3 0
A = and B = .
0 1 4 0 2
(cid:18) (cid:19) (cid:18) − (cid:19)
Figure 1 shows the almost sure convergence of the LS estimator θ to the four
n
coordinates of θ which are different from zero. One can observe that θ performs
n
very well in the estimation of θ. b
b
2.5 1
2 0.5
1.5 0
1 −0.5
0.5 −1
0 −1.5
−0.5 −2
0 200 400 600 800 1000 0 200 400 600 800 1000
Figure 1. Left: LLN for A, Right: LLN for B.
Figure 2 shows the CLT for the four coordinates of
Z = √NΛ1/2(θ θ)
N N
−
where Λ is the limiting matrix given by (2.4) with L = I , K = 0 and
2
b
4 48 0
H = A2(I B2)−1 = .
2
− 21 0 7
(cid:18) (cid:19)
One can realize that each component of Z has (0,1) distribution as expected.
N
N
0.45 0.5
0.4 0.45
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05 0.05
0 0
−5 0 5 −5 0 5
Figure 2. Left: CLT for A, Right: CLT for B.
8 BERNARDBERCU ANDVICTOR VAZQUEZ
6. Conclusion
Via our new concept of strong controllability, we have extended the analysis of the
almost sure convergence for both LS and WLS algorithms in the multidimensional
ARX framework. It enables us to provide a positive answer to a conjecture in [5] by
establishing a CLT and a LIL for these two stochastic algorithms. In our approach,
the leading matrix associated with the matrix polynomial B, commonly called the
high frequency gain, was supposed to be known and it was chosen as the identity
matrix I . It would be a great challenge for the control community to carry out
d
similar analysis with unknown high frequency gain and to extend it to ARMAX
models.
7. Appendix A.
Proof of Lemma 2.2. Let Σ be the infinite-dimensional diagonal square matrix
Γ 0
··· ··· ···
0 Γ 0
 ··· ··· 
Σ = 0 Γ 0 .
··· ···
 
 ··· ··· ··· ··· ··· 
 
 ··· ··· ··· ··· ··· 
 
Moreover, denote by T the infinite-dimensional rectangular matrix with dq raws and
an infinite number of columns given, if p q, by
≥
P P P P
p p+1 k k+1
··· ···
P P P P
p−1 p k−1 k
 ··· ··· 
T =
··· ··· ··· ··· ··· ···
 Pp−q+2 Pp−q+3 Pk−q+2 Pk−q+3 
 ··· ··· 
 Pp−q+1 Pp−q+2 Pk−q+1 Pk−q+2 
 ··· ··· 
 
while, if p q, by
≤
P P P P
p p+1 k k+1
··· ··· ···
··· ··· ··· ··· ··· ··· ···
 P P P P 
T = 1 2 ··· ··· k−p+1 k−p+2 ··· .
 0 P1 P2 Pk−p Pk−p+1 
··· ···
 
 
··· ··· ··· ··· ··· ··· ···
 0 0 P P 
 ··· ··· 1 2 ··· 
 
After some straightforward, although rather lengthy, linear algebra calculations, it
is possible to deduce from (2.5) that
(A.1) S = TΣTt.
It clearly follows from this suitable decomposition that
(A.2) ker(S) = ker(Tt).
As a matter of fact, assume that v Rdq belongs to ker(Tt). Then, Ttv = 0, Sv = 0
which leads to ker(Tt) ker(S). O∈n the other hand, assume that v Rdq belongs
⊂ ∈
to ker(S). Since Sv = 0, we clearly have vtSv = 0, so vtTΣTtv = 0. However, the
matrix Γ is positive definite. Consequently, Ttv = 0 and ker(S) ker(Tt), which
⊂
implies (A.2). Moreover, it follows from the well-known rank theorem that
(A.3) dq = dim(ker(S))+rank(S).
STRONG CONTROLLABILITY VIA THE SCHUR COMPLEMENT 9
As soon as ker(S) = 0 , dim(ker(S)) = 0 and we obtain from (A.3) that S is of full
{ }
rank dq which means that S is invertible. Furthermore, the left hand side square
matrix of order dq of the infinite-dimensional matrix T is precisely Π. Consequently,
if Π is invertible, Π is of full rank dq, ker(Π) = ker(Πt) = 0 and the left null space
of T reduces to the null vector of Rdq. Hence, if Π is in{ve}rtible, we deduce from
(A.2) together with (A.3) that S is also invertible. Finally, as
(A.4) det(Λ) = det(L)det(S) = det(Γ)pdet(S),
we obtain from (A.4) that Λ is invertible and formula (2.6) follows from [14] page
18, which completes the proof of Lemma 2.2.
Appendix B.
Proof of Theorem 4.1. In order to prove Theorem 4.1, we shall make use of the
same approach than Bercu [5] or Guo and Chen [10]. First of all, we recall that for
all n 0
≥
(B.1) X x = π +ε .
n+1 n+1 n n+1
−
In addition, let
n
s = Φ 2.
n k
|| ||
k=0
X
It follows from (B.1) together with the strong law of large numbers for martingales
(see e.g. Corollary 1.3.25 of [9]) that n = (s ) a.s. Moreover, by Theorem 1 of [2]
n
O
or Lemma 1 of [10], we have
n
(B.2) (1 f ) π 2= (logs ) a.s.
k k n
− k k O
k=1
X
where f = ΦtS−1Φ . Hence, if (ε ) has finite conditional moment of order α > 2,
n n n n n
we can show by the causality assumption (A ) on the matrix polynomial B together
1
with (B.2) that Φ 2= (sβ) a.s. for all 2α−1 < β < 1. In addition, let
g = ΦtS−1 Φ aknd δn =k tr(SO−1n S−1). It is well-known that
n n n−1 n n n−1 − n
(1 f )(1+g ) = 1
n n
−
and (δ ) tends to zero a.s. Consequently, as π 2= (1 f )(1+g ) π 2 and
n n n n n
1+g 2+δ Φ 2, we can deduce fromk(B.2)kthat − k k
n n n
≤ k k
n
(B.3) π 2= o(sβlogs ) a.s.
k k k n n
k=1
X
Therefore, we obtain from (3.4), (B.1) and (B.3) that
n
(B.4) X 2= o(sβlogs )+ (n) a.s.
k k+1 k n n O
k=1
X
Furthermore, we infer from assumption (A ) that
1
(B.5) U = B−1(R)A(R)X B−1(R)ε
n n+1 n+1
−
which implies by (B.4) that
n
(B.6) U 2= o(sβlogs )+ (n) a.s.
k k k n n O
k=1
X
10 BERNARDBERCU ANDVICTOR VAZQUEZ
It remains to put together the two contributions (B.4) and (B.6) to deduce that
s = o(s )+ (n) a.s. leading to s = (n) a.s. Hence, it follows from (B.3) that
n n n
O O
n
(B.7) π 2= o(n) a.s.
k
k k
k=1
X
Consequently, we obtain from (3.4), (B.1) and (B.7) that
n
1
lim X Xt = Γ a.s.
n→∞ n k k
k=1
X
n
and, for all 1 i p 1, X Xt = o(n) a.s. which implies that
≤ ≤ − k k−i
k=0
X
n
1
(B.8) lim Xp(Xp)t = L a.s.
n→∞ n k k
k=1
X
where L is given by (2.3). Furthermore, it follows from relation (1.1) together with
(B.1) and assumption (A ) that, for all n 0,
1
≥
U = B−1(R)A(R)X B−1(R)ε ,
n n+1 n+1
−
= B−1(R)A(R)(π +x )+B−1(R)(A(R) I )ε ,
n n+1 d n+1
−
= V +W .
n n+1
Consequently, as W = P(R)ε , we deduce from (3.4), (B.7) and the strong law of
n n
large numbers for martingales (see e.g. Theorem 4.3.16 of [9]) that, for all 1 i q,
≤ ≤
n
1
lim U Ut = H a.s.
n→∞ n k k−i+1 i
k=1
X
which ensures that
n
1
(B.9) lim Uq(Uq)t = H a.s.
n→∞ n k k
k=1
X
where H is given by (2.2). Via the same lines, we also find that
n
1
(B.10) lim Xp(Uq )t = Kt a.s.
n→∞ n k k−1
k=1
X
Therefore, it follows from the conjunction of (B.8), (B.9) and (B.10) that
S
n
(B.11) lim = Λ a.s.
n→∞ n
wherethelimitingmatrixΛisgivenby(2.4). Hereafter,werecallthattheARX (p,q)
d
model is strongly controllable. Thanks to Lemma 2.2, the matrix Λ is invertible and
Λ−1, given by (2.6), may be explicitly calculated. This is the key point for the
rest of the proof. On the one hand, it follows from (B.11) that n = (λ (S )),
min n
Φ 2= o(n) a.s. which implies that f tends to zero a.s. Hence, by (OB.2), we find
n n
k k
that
n
(B.12) π 2= (logn) a.s.
k
k k O
k=1
X