On the Fixed-Lag Smoothing: How to Exploit the Information Preview Leonid Mirkin

On the Fixed-Lag Smoothing: How to Exploit the Information Preview Leonid Mirkin

On the H ∞ Fixed-Lag Smoothing:
How to Exploit the Information Preview ?
Leonid Mirkin 1
Faculty of Mechanical Engineering, Technion—IIT, Haifa 32000, ISRAEL
Abstract
The problem of the continuous-time H ∞ fixed-lag smoothing over the infinite horizon is studied. The first solution to the problem is
derived in terms of one algebraic Riccati equation of the same dimension as in the filtering case and the mechanism by which the
performance improvements with respect to the H ∞ filtering occur is clarified. It is shown that the H ∞ smoother exploits the information
preview in an “H 2 manner.”
Key words: H ∞ estimation, fixed-lag smoothing, J-spectral factorization, Riccati equation.
1 Introduction
In this paper the so-called fixed-lag smoothing problem is
studied. The problem is to estimate a linear combination of
the system states at time t, t ∈ R+ , on the basis of the measurements available up to time t+h for a given h > 0, which
is referred to as the smoothing lag. This problem corresponds
to the case, where some delay in supplying the estimate can
be tolerated, like in many signal processing and communication applications. It is clear that potentially smoothers
can achieve a better performance than corresponding filters
(h = 0) and the larger is the smoothing lag, the better estimate can be obtained. The questions are how to exploit this
potential and what is the price (e.g., in terms of complexity of the resulting estimator) one has to pay to achieve this
improvement?
The interest to the fixed-lag smoothing can be traced back to
(Wiener, 1949) and in the context of the H 2 (Kalman) theory the problem is currently well understood, see (Anderson
and Moore, 1979) and the references therein. On the other
hand, the H ∞ version of the fixed-lag smoothing problem is
less studied and only few results are available in the literature, all in the discrete time. Arguably, the underlying idea in
the earlier solutions (Grimble, 1991; Theodor and Shaked,
? This research was supported by THE I SRAEL S CIENCE F OUN DATION (grants No. 384/99 and 106/01) and by the Fund for the
Promotion of Research at the Technion. This paper was presented
at the 1st IFAC/IEEE Symposium on System Structure and Control, Prague, Czech Republic, August 2001
1
Phone: +972-4-8293149; Email: [email protected].
Automatica, vol. 39, no. 8, 2003
1994; Zhang et al., 2000) is to treat the delay on equal footing with the rest of the system dynamics by incorporating
the delay into the transfer function or the state-space matrices of the process. Since in the discrete time the delay
dynamics are finite dimensional, this enables one to reduce
the smoothing problem to a standard filtering one that, in
turn, can be solved using known methods. The conceptual
simplicity, however, is somewhat misleading, since the computational burden grows rapidly as the smoothing lag gets
larger and the clear physically meaningful structure of the
delay element does not show up in the final solution. The
latter complicates any further analysis, e.g., the effect of the
smoothing lag on the solution properties cannot be traced
back. To cope with the complexity, Colaneri et al. (1998)
proposed a special modification of the J-spectral factorization procedure aimed to handle the associated discrete delay.
This approach enables one to reduce the computational complexity of the algorithm. Yet the dependence of the resulting
solution on the problem data is not readily recoverable, so
both the structure of the smoother and the dependence of
the achievable performance on h are not clearly seen.
Summarizing, the currently available solutions to the H ∞
fixed-lag smoothing problem are all limited to the discretetime case, are considerably more complicated than the corresponding H 2 solutions, and fall short in accounting for the
effect of the smoothing lag on the achievable performance.
In this paper a novel approach to the solution of the H ∞
fixed-lag smoothing problem is proposed and applied to
solve the continuous-time version of the problem. As in
(Colaneri et al., 1998), the solution is based on the J-spectral
factorization machinery, but the delay is handled in a com-
to appear
The completion operator π h {·} is defined as follows:
pletely different fashion. Following the idea of Meinsma and
Zwart (2000), the transfer matrix to be factorized is modified
so that the parts involving delay are excluded from the factorization procedure completely. This reduces the problem
to the J-spectral factorization of a finite-dimensional transfer matrix and also suggests the structure of the resulting
smoother. The latter consists of an FIR (finite impulse response) part in parallel connection with a finite-dimensional
system having the structure of the H ∞ filter. Furthermore,
the Hamiltonian matrix associated with the modified Jspectral factorization problem turns out to be similar to the
Hamiltonian matrix associated with the filtering J-spectral
factorization. Using this fact, the effect of the smoothing lag
h on the achievable performance level is quantified.
 




B
B
B
A
A
A
 =
 − e−sh 
.
π h e−sh 
−Ah


C 0
Ce
0
C 0



It can be seen that ω = π h {·} ζ defines the FIR mapping
Rt
ω(t) = C t−h eA(t−h−s) Bζ(s)ds.
Finally, a 2n × 2n Hamiltonian matrix H is said to belong
to dom(Ric) if it has no imaginary axis eigenvalues and its
stable (corresponding to the open left half plane
eigenvalues)
co-spectral subspace is complementary to ker 0 I . In other
words, H ∈ dom(Ric) ifthere there existan n ×
n matrix
Y = Y 0 such that that I Y H = As I Y for some
Hurwitz As . The matrix Y above is unique and thus the
function Y = Ric(H) is well defined. The definitions above
are actually dual to the conventionally used (Zhou et al.,
1995), yet they are better suited for the estimation problems
studied in the paper.
The technical contributions of this paper are as follows:
• the first solution to the continuous-time H∞ fixed-lag
smoothing problem is proposed;
• it is shown that the solvability conditions for the smoothing problem are based on one Riccati equation of the same
dimension as in the filtering case;
• it is shown that the sub-optimal H ∞ smoother has structure similar to that of the H 2 optimal smoother and is
compatible with the latter by the complexity;
• the first analysis of the effect of the smoothing lag h on
the achievable smoothing performance is carried out;
• it is demonstrated that the use toward this end the inverse
Riccati solution enables one to account for the effect of
h on the achievable performance in an elegant manner.
2 Problem formulation and solution
Let G1 (s) and G2 (s) be p1 ×m and p2 ×m transfer matrices
given by their state-space realization
G(s) =
Moreover, this paper shows clearly that the mechanism by
which the previewed information is exploited by the H ∞
smoother (from both the smoother structure and the achievable estimation performance points of view) is similar to that
in the H 2 case. This leads one to the qualitative conclusion
that the H ∞ smoother exploits the information preview in
an H 2 manner.
"
G1 (s)
G2 (s)
#

A B




=
 C1 0 
C 2 D2
(1)
and suppose that the following assumptions hold:
(A1): (C2 , A) is detectable;
"
#
A − jωI B
(A2):
has full row rank ∀ω ∈ R;
C2
D2
(A3): D2 D20 = I.
The paper is organized as follows. In Section 2 the smoothing
problem is formulated and solved. Section 3 is devoted to the
analysis of the effect of the smoothing lag on the achievable
performance. A numerical example illustrating the proposed
solution is discussed in Section 4. Some concluding remarks
are provided in Section 5.
The H ∞ fixed-lag smoothing problem is then formulated as
follows:
SPh : Given G as in (1) and the smoothing lag h ≥ 0, determine whether there exists a causal transfer matrix
K, which guarantees
ke−sh G1 − KG2 k∞ < γ
Notation Given a matrix M , kM k and kM kF denote its
spectral and Frobenius matrix norms, respectively, and M 0
denotes the transpose of M . Given a transfer matrix G(s),
its conjugate is defined as G(s)∼ = G0 (−s) and, when G(s)
is stable, kG(s)k2 and kG(s)k∞ denote its H 2 and H ∞
norms, respectively. The chain scattering transformation is
.
denoted as Cr G, Q = (G21 + G22 Q)(G11 + G12 Q)−1 .
for a given γ > 0 and then characterize all such K
when one exists.
Note that when h = 0 the problem SP0 becomes the standard H ∞ filtering problem extensively studied in the literature, see (Hassibi et al., 1999). This case can actually be
2
thought of as the particular case of the general H ∞ problem with the difference that the internal stability is not required. Another special case of SPh solved in the literature (Nagpal and Khargonekar, 1991; Hassibi et al., 1999)
is SP∞ which corresponds to the infinite-horizon version
of the fixed-interval smoothing problem.
J-spectral factorization of Φα , whose (1, 2) and (2, 1)
blocks contain infinite-dimensional elements esh and e−sh ,
respectively. Yet it turns out that this infinite-dimensional
J-spectral factorization problem can be converted to an
equivalent finite-dimensional one. To this end, define the
Hamiltonian matrix
The rest of this section is devoted to the solution of SP h .
First, the case when the system in (1) is stable is addressed
in §2.1. Then, in §2.2, the solution is extended to the general
case. Finally, some remarks are presented in §2.3.
H=
"
A −BB 0
0 −A0
#
−
"
#
BD20 h
C20
i
C2 −D2 B 0
(3)
(with sub-blocks Hij ) and the symplectic matrix function
2.1 Solution: the stable case
Σ(τ ) =
Assume throughout this subsection that the matrix A is Hurwitz, i.e., that the systems in (1) are stable. Instrumental
in our approach is the use of the J-spectral factorization
approach. To this end, for given
p and q denote the sign.
indefinite matrix2 Jp,q = diag Ip , −Iq and introduce the
following definition:



. 
Φ=


0
e−sh G1 I
#"
I
0
0 −γ 2 I
#"
G2
0
e−sh G1 I
#∼
.
= e−Hτ .
A
0
C2
−BB 0
−A0
−D2 B 0
C1 Σ11 (h) C1 Σ12 (h)
BD20 −Σ012 (h)C10
C20
I
0


Σ011 (h)C10 

.

0

−γ 2 I
Moreover, Ωα is a J-spectral factor of Φα iff Ωα =
where Ω is a J-spectral factor of Φ and
Lemma 2 Assume that the system in (1) is stable and let
G2
Σ21 (τ ) Σ22 (τ )
#
Lemma 3 Φα given by (2) admits J-spectral factorization
iff so does
We are now in the position to formulate the following lemma,
which expresses the solution of SPh in terms of a J-spectral
factorization:
"
Σ11 (τ ) Σ12 (τ )
The following result can be formulated:
Definition 1 A (p + q) × (p + q) transfer matrix Φa is said
to admit Jp,q -spectral factorization (or simply J-spectral
factorization) if Φa (jω) has the same inertia as Jp,q for all
ω and there exists a bistable (p+q)×(p+q) transfer matrix
Ωa such that Φa = Ωa Jp,q Ω∼
a.
.
Φα =
"
I
0
∆I

H11 H12 BD20 



.
0 
∆(s) = −π h e−sh 
H
H
C
2  .
 21 22





C1 0
0




(2)
Then SPh is solvable iff Φα admits Jp2 ,p1 -spectral factorization with the factor Ωα such that the lower-right p1 × p1
sub-block of Ω−1
a is bistable. If these conditions hold, then
the set of all solutions to SPh is parametrized as
K(s) = Cr Ωα (s), Q(s)

(4)
Ω,
(5)
∞
PROOF. First, note that
I 0for
any
I 0S∼∈ H the following
equality holds: Φα = S I Φβ S I , where
.
Φβ =
for any Q ∈ H ∞ such that kQk∞ < 1 and K(∞) exists.
"
G2 G∼
2
∼
G2 G∼
2 R
∼ ∼
2
RG2 G∼
2 Ψ + RG2 G2 R − γ I
#
,
.
∼
∼ −1
Ψ = G 1 G∼
G2 G∼
1 − G1 G2 (G2 G2 )
1 , and
PROOF. The result is, in a sense, dual to Theorem 5.1 in
(Meinsma and Zwart, 2000) and can be proved following
similar arguments. Just note that the special structure of (2)
ensures that condition 3) in (Meinsma and Zwart, 2000) is
redundant and condition 2) there is simplified.
.
∼ −1
−S
R = e−sh G1 G∼
2 (G2 G2 )
(the inverse above exists whenever
I 0 (A3) holds). Obviously,
Φα = Ωα JΩ∼
iff
Ω
=
α
α
S I Ωβ for any Ωβ such that
.
Since
S
is
stable,
Φβ = Ωβ JΩ∼
β
Unlike the finite-dimensional case, the result of Lemma 2
is not readily useful. The difficulty is to find the required
Ωα ∈ H ∞ ⇐⇒ Ωβ ∈ H ∞
2
When the dimensions are irrelevant or clear from the context,
the simpler notation J will be used in the sequel.
and hence Ωα is a J-spectral factor of Φα .
3
matrix of Φ−1 , i.e,
The proof is now completed by noticing that the choice
S = ∆ yields the finite-dimensional Φβ , namely Φβ = Φ
(this follows by a tedious yet straightforward algebra).
Ha = H + γ
The idea of extracting the infinite-dimensional bi-stable part
from the J-spectral factorization was proposed by Meinsma
and Zwart (2000) to solve a two-block problem with dead
time. That problem corresponds in our case to h < 0 and
requires an upper triangular extracted factor.
When G is unstable, the direct application of the arguments
in the previous subsection would be considerably complicated. Instead, we use the coprime factorization arguments to
reduce the problem with unstable data to an equivalent problem with stable data. This idea is borrowed from (Meinsma,
1995), though the structure of the estimation problem enables us to end up with simpler factorization.
Lemma 5 Let L2 be any matrix so that A+L2 C2 is Hurwitz
(exists by (A1)). Denote
Theorem 4 The smoothing problem SPh is solvable iff
.
Hγ (h) ∈ dom(Ric) and Y (h) = Ric(Hγ (h)) ≥ 0. Furthermore, if these conditions hold, then all solutions
to SPh
can be parametrized as K = ∆ + Cr Ω, Q , where ∆ is
given by (5),
A
+Y
C2
I
0
C1 Σ11 (h)
0
I
h
i
C10 C1 Σ11 (h) Σ12 (h) .
2.2 Solution: extension to the unstable case
(note that Hγ (0) is the Hamiltonian matrix associated with
the filtering problem for G1 and G2 ). Then the main result
of this subsection is formulated as follows:

Ω=

Σ011 (h)
#
This yields the solvability conditions. The smoother formula
3 by noticing that
I follows
directly from Lemma
0 Ω, Q = ∆ + C Ω, Q . Finally, the realization
Cr ∆
r
I
of Ω is can be derived by standard J-spectral arguments
for finite-dimensional systems, see, e.g., (Meinsma, 1994,
Theorem 2.4).
.
Hγ (τ ) = Σ−1 (τ ) H + γ −2 C100C1 00 Σ(τ )
(h)C20 γ12 M C10
−Σ012 (h)
Hα = Σ−1 (h)Hγ (0)Σ(h) = Hγ (h).
To formulate the solution to SPh define the following matrix
function:
BD20
"
Then the facts that Σ(h) is symplectic and commutes with
H lead to the equality
The fact that the infinite-dimensional factor in Lemma 3 is
lower triangular simplifies the application of Lemma 3 to the
solution of SPh considerably. Indeed, the solvability condition in Lemma 2 involves the bistability of the (2, 2) subblock of the J-spectral factor. Since the (2, 2) sub-blocks of
−1
Ω−1
are equivalent, the solvability of the smoothing
α and Ω
problem is completely reduced to finite-dimensional conditions. This is unlike (Meinsma and Zwart, 2000), where
the extracted factor alters the (2, 2) sub-block that, in turn,
gives rise to an additional “infinite-dimensional” solvability
condition.

−2
"
M 1 N1
M 2 N2
#

A + L 2 C 2 L 2 B + L 2 D2

=

C1
0
0
C2
I
D2


.

˜ = (e−sh M1 + K)M −1 solves the
Then K solves SPh iff K
2
fixed-lag smoothing problem for (stable) N1 and N2 .


,

PROOF. First, it can be easily verified that
.
where M = Σ012 (h) − Y (h)Σ011 (h), and Q ∈ H ∞ satisfies
kQk∞ < γ but otherwise arbitrary.
"
e−sh G1
G2
#
=
"
I e−sh M1
0
M2
#−1 "
e−sh N1
N2
#
and M2 and N2 constitute the left coprime factorization of
G2 in RH ∞ . Then, straightforward substitutions show that
PROOF. As discussed above, the application of Lemma 3
to the conditions of Lemma 2 yields that SP h is solvable iff Φ given by (4) admits J-spectral factorization with
stable (2, 2) block of the inversed J-spectral factor. Yet
this is the finite-dimensional J-spectral factorization problem extensively studied in the literature, see for example
(Meinsma, 1994). Following the arguments in this reference, one can show that the required factorization exists iff
Ha ∈ dom(Ric) and Ric(Ha ) ≥ 0, where Ha is the “A”
.
˜ 2
Gε = e−sh G1 − KG2 = e−sh N1 − KN
˜ ∈ H ∞ whenever
It is now left to show that K ∈ H ∞ ⇔ K
˜ ∈ H ∞ , then K = KM
˜ 2 − e−sh M1 ∈ H ∞
Gε ∈ H ∞ . If K
˜ 2 =
too. On the other hand, if K ∈ H ∞ , then both KM
˜ 2 = e−sh N1 − Gε belong to H ∞ .
e−sh M1 + K and KN
4
˜ follows by the fact that M2 and
Hence, the stability of K
N2 are left coprime.
3 The effect of h on the achievable performance
Lemma 5 can in principle be used to solve SP h . To this end
one needs first to solve the problem with N1 and N2 instead
of G1 and G2 , respectively (note that (A1)–(A3) hold for the
˜
realization of N ’s as well). The resulting smoother, say K,
−sh
˜
is then to be implemented as K = KM2 − e
M1 . It turns
out, however, that after all substitutions the matrix L2 affects
neither the solvability conditions nor the final formulae for
K. Hence, the following result can be formulated:
The solvability conditions for SPh in Theorem 4 are expressed in terms of the Riccati solution Y (h). Yet the use of
the latter for the analysis of the effect of h on the achievable
performance is complicated by the fact that Y (h) is discontinuous as a function of h. On the other hand, it is known
(Scherer, 1990; Gahinet, 1994) that the null space of Y (0)
depends only upon properties of the realization of G2 (s) and
does not depend on γ. Moreover, as will be shown below,
the null space of Y (h) does not depend on the smoothing lag
h either. These facts suggest that the effect of h can easier
be accounted for in terms of the inverse of Y (h).
Theorem 6 Theorem 4 solves SPh even when the system
in (1) is unstable.
To eliminate the singular part of Y (h) from the analysis and
hence simplify the exposition, assume that
2.3 Solution: final remarks
(A4): BD20 = 0;
(A5): (A, B) has no stable uncontrollable modes.
Theorem 4 yields the complete solution to SPh . It is worth
stressing that it requires one Riccati equation of the same
dimension as in the filtering case to be solved. The smoother
itself, however, is infinite-dimensional because of the FIR
term ∆. In fact, the H ∞ smoother reminds the H ∞ predictor
(Mirkin, 2003). The latter also includes a finite-dimensional
part and an FIR part. The difference is that in the prediction case the infinite-dimensional part is in the feedback
connection with the rational part, like in the Smith predictor scheme, while in the smoothing case these components
are in the feedforward connection. Another qualitative difference between the H ∞ predictor and smoother is that the
infinite-dimensional part of the latter does not depend on
the performance level γ and, moreover, coincides with the
infinite-dimensional part of the H2 smoother (γ → ∞).
Thus, one may conclude that the H ∞ smoother exploits the
information preview in an “H 2 fashion.”
Assumption (A4) stays that the measurement noise is independent of the system disturbance, while (A5) just rules out
the problem redundancy. These assumptions thus are quite
reasonable. It is worth stressing, however, that they can easily be omitted by applying a similarity transformation that
extracts the redundant states, see (Mirkin, 2001) for the details.
Assumptions (A1)–(A5) guarantee (Zhou et al., 1995, Chapter 13) that H ∈ dom(Ric) (H is defined by (3)) and, moreover, Ric(H) > 0. Hence, the Hamiltonian matrix
. 0 I 0 I ˜ =
H
I 0 H I 0 =
−A0 −C20 C2
−BB 0
A
#
.
˜ > 0 (since
belongs to dom(Ric) too and Y˜κ = Ric(H)
−1
Y˜κ = Ric(H) ). Note that Y˜κ is the stabilizing solution to
the following Riccati equation:
It is also useful to write explicitly the formula for the socalled central solution 3 (Q = 0):
Corollary 7 Let the conditions of Theorem 4 hold. Then the
“central” smoother is K = ∆ + Kc , where


A − (BD20 + Y (h)C20 )C2 BD20 + Y (h)C20

Kc = 
C1 Σ11 (h)
0
−Y˜κ A − A0 Y˜κ + C20 C2 − Y˜κ BB 0 Y˜κ = 0
(6)
.
so that the matrix Aκ = −(A + BB 0 Y˜κ ) is Hurwitz. Furthermore, the spectrum of Aκ coincides with that of the
“closed-loop state-matrix” of the Kalman filter associated
with G2 (s).
and ∆ is given by (5).
Similarly, the following Hamiltonian matrix is associated
with Y (h)−1 (whenever it exists):
The similarity between the Hamiltonian matrices for the filtering (h = 0) and smoothing (h > 0) cases is intriguing. It
suggests that the dependence of the achievable H ∞ performance on the size the smoothing lag can be quantified and
an additional insight into the mechanism by which the previewed information is exploited in the H ∞ setting can be
provided. This is indeed true and the next section is devoted
to such an analysis.
3
"
. ˜ −1
˜ γ (τ ) =
˜ + γ −2 0 C10 C1 Σ(τ
˜ ),
H
Σ (τ ) H
0 0
. −Hτ
˜
˜ )=
where Σ(τ
e
. The following result, whose proof is
given in §A.1, plays an important role in the sequel:
˜ γ (0) has
Lemma 8 Let γ∞ be the maximal γ for which H
eigenvalues on the jω-axis. Then the following statements
are equivalent:
This is actually the maximum entropy solution.
5
i)
ii)
iii)
As follows from Theorem 11, any performance γ > γ∞
is achievable provided that the smoothing lag h is “large
enough.” This can be clearly seen from, for example, statement iv) of Theorem 11 since kCγ eAκ h Bκ k approaches zero
as h → ∞. This property of the H ∞ smoother is rather
expectable. What is less obvious, is the fact that the “convergence” rate in the H ∞ case is completely determined by
the dynamics of the corresponding Kalman filter (see the
comment after eqn. (6)).
γ > γ∞ ;
the fixed-interval smoothing problem SP∞ is solvable;
˜ γ (h) ∈ dom(Ric) for any h ≥ 0.
H
Remark 9 The state-space solution to the fixed-interval
smoothing problem SP∞ was given by Nagpal and Khargonekar (1991). Yet the solvability condition in Lemma 8 is
different from that in (Nagpal and Khargonekar, 1991) and
more suitable for the further analysis of the effect of h on
the achievable performance.
The last observation can be further developed by noting that
the H 2 solution is obtained from the H ∞ one by taking
γ −2 = 0. Then, using the estimator in Corollary 7 and
calculating the H 2 norm of the resulting error transfer matrix
e−sh G1 −KG2 , one can end up with the following formula:
Remark 10 As a matter of fact, γ∞ can be given in terms
of the H ∞ norm of a stable system. To this end, note that
˜ γ (0) =
H
"
I −Y˜κ
0
I
#"
A0κ
γ −2 C10 C1
−BB 0
−Aκ
#"
I Y˜κ
0 I
#
.
.
Jh = minK∈H ∞ ke−sh G1 − KG2 k22
= J∞ + kC1 eAκ h Bκ k2F ,
Hence, γ∞ = kC1 (sI − Aκ )−1 Bk∞ by (Zhou et al., 1995,
Lemma 4.7).
where J∞ = tr(C1 Wc C10 ) is the achievable performance
in the fixed-interval H2 smoothing problem. Thus, the H 2
fixed-lag smoothing performance approaches the fixedinterval one exponentially and the speed of convergence is
determined by the matrix Aκ , exactly as in the H ∞ case.
This observation supports the claim that the H ∞ smoother
exploits the information preview in an “H 2 fashion” (cf. the
discussion at the end of Section 2).
We shall also need the inverse H ∞ Riccati equation associated with the filtering problem:
−Y˜γ A − A0 Y˜γ + C20 C2 −
0
1
γ 2 C1 C1
− Y˜γ BB 0 Y˜γ = 0. (7)
˜ γ (0)), is
By Lemma 8 the stabilizing solution to (7), Ric(H
well defined whenever γ > γ∞ . Moreover, as limγ→∞ Y˜γ =
Y˜κ and Y˜γ is monotonically non-decreasing function of γ
(Gahinet, 1994), Y˜γ ≤ Y˜κ , for all γ > γ∞ .
There exists, however, a remarkable difference between the
H 2 and H ∞ solutions. As seen from (10), the H 2 performance is improved all the time as the smoothing lag h grows
(though the improvement becomes negligible as h exceeds
several times the dominant time constant of Aκ , see (Anderson and Moore, 1979)). On the other hand, in the H ∞ case
there always exists a finite hγ such that kCγ eAκ h Bκ k < 1
for all h > hγ and every γ > γ∞ . In other words, any performance level γ > γ∞ is achievable with a finite smoothing lag. Moreover, the quantity h∞ , which corresponds to
the limit γ & γ∞ and is the function of the problem data,
can be regarded as the maximal smoothing lag making sense
from the H ∞ performance point of view. Indeed, any further
increase of h does not improve the achievable H ∞ performance at all! Note, however, that the increase of the smoothing lag beyond h∞ might affect some other properties of the
solution, e.g., its H 2 performance (see Section 4).
We are now in the position to state the main result of this
section (see §A.2 for the proof):
Theorem 11 The following statements are equivalent:
i) SPh is solvable;
.
˜ γ (h)) > 0;
ii) γ > γ∞ and Y˜ (h) = Ric(H
iii) γ > γ∞ and Qγ (h) > 0, where Qγ (t) is the solution
to the differential Riccati equation
−Q˙ γ = A0 Qγ + Qγ A − C20 C2 + Qγ BB 0 Qγ ,
(8)
Qγ (0) = Y˜γ , which is monotonically non-decreasing
and limt→∞ Qγ (t) = Y˜κ > 0 for all γ > γ∞ ;
iv) γ > γ∞ and kCγ eAκ h Bκ k < 1, where Bκ and Cγ are
any matrices satisfying
4 Numerical example
Bκ Bκ0 = Y˜κ−1 − Wc ≥ 0,
Cγ0 Cγ = I − (Y˜κ − Y˜γ )Wc −1 (Y˜κ − Y˜γ ) ≥ 0,
Consider SPh for the following simple first-order system:
G1 (s) =
respectively, and Wc ≥ 0 satisfies
Aκ Wc + Wc A0κ + BB 0 = 0.
(10)
h
aq
s−a
0
i
and G2 (s) =
h
aq
s−a
i
1 ,
with a 6=p0 and q > 0. It can be verified that in this case
γ∞ = q/ q 2 + 1 and for any γ > γ∞
(9)
Moreover, kCγ eAκ h Bκ k is monotonically non-increasing function of h.
Y˜γ =
6
1
aq 2
p
1 + q 2 (1 − γ −2 ) − 1 .
1
1
0.414
0.9
γ=1.1γ∞
Filtering
0.2
0.8
0.7
0
0.9
Q (h)
−0.2
γ
min
0.5
γ
γ
∞
0.6
−0.4
0.4
0.8
Smoothing
0.3
γ=γ
∞
−0.6
0.2
−0.8
0.1
0.707
0
−3
10
−2
10
−1
0
10
10
1
10
2
10
0
0.623
1
1.5
q
2
2.5
−1
0
(b) γmin for a = q = 1
(a) Achievable γ for SP0 and SP∞
0.623
1
1.5
2
2.5
h
h
(c) Qγ (h) for a = q = 1
Fig. 1. H ∞ performance
When G(s) is stable (a < 0) Y˜γ > 0 for all γ > γ∞ and
therefore smoothing offers no benefits over filtering. On the
other hand, if a > 0, then Y˜γ > 0 iff γ > 1 > γ∞ . Thus, in
this case smoothing leads to superior estimation performance
and, moreover, the lower is the system noise intensity (i.e.,
the smaller is q), the larger is the potential advantage of
smoothing over filtering, see Fig. 1(a).
.
Assume hereafter that a > 0 and denote η = √ 12
q +1
of G1 − esh KG2 until the latter reaches its minimal value
as Qγ converges to Y˜κ . This agree well with the known
fact (Hassibi et al., 1999, Ch. 10) that as the smoothing lag
h → ∞ both H 2 and H ∞ approaches result in the same estimator, which actually minimizes the estimation error energy
for every input. The reader is referred to Figs. 2(a) and 2(b)
to compare the H ∞ and H 2 solutions and see how the both
approach the same error transfer function as h increases, yet
from different directions.
< 1.
Using condition iv) of Theorem 11 it can be shown that the
minimal achievable H ∞ performance γmin for SPh is
γmin =
(
1
2
(1 −pη)eah/η + (1 + η)e
γ∞ = 1 − η 2
−ah/η
5 Concluding remarks
if h ≤ h∞
otherwise,
In this paper the first solution to the continuous-time H∞
fixed-lag smoothing problem has been derived. The derivation has been based on the reduction of the associated
infinite-dimensional J-spectral factorization problem to an
equivalent finite-dimensional one. The resulting smoother
consists of two components connected in parallel: a finitedimensional estimator reminiscent of the H ∞ filter and an
FIR block of the length of the smoothing lag. The solvability conditions are based on one algebraic Riccati equation
of the same dimension as the Riccati equation associated
with the filtering problem. It has also been shown that any
performance level γ > γ∞ can be achieved if the smoothing
lag is “large enough” (γ∞ stands for the H ∞ performance
level of the infinite-horizon version of the fixed-interval
smoothing problem). Moreover, the effect of the smoothing
lag on the achievable performance has been quantified.
. 1
1+η
η ln 1−η
. The plot γmin vs. h is depicted in
where h∞ = 2a
Fig. 1(b) for the case of a = q = 1. It is seen that γ∞ is
achievable with h = h∞ ≈ 0.623 and any further increase
of h does not improve the H ∞ estimation performance.
The increase of h, however, does affect the Riccati solution
Y (h) and, consequently, the central estimator in Corollary 7.
This is clearly seen from Fig. 1(c), which shows the solutions Qγ (h) = 1/Y (h) to the differential Riccati equation
(8) for two different values of γ: γ = γ∞ (solid line) and
γ = 1.1γ∞ (dashed line). According to condition iii) of
Theorem 11, h∞ corresponds to the zero crossing of Qγ (h)
and the latter then continues to change until it practically
converges to Y˜κ at h ≈ 2.5 (here γ = γ∞ is assumed).
Note that the the implementation issues for the H ∞
smoother have not been addressed in the paper. The potential problem might be in implementing the FIR block, which
is built upon a Hamiltonian matrix and requires the matrix
exponentials to be computed. Yet the computation of the
matrix exponents for matrices having positive eigenvalues,
especially for a large smoothing lag, might be a difficult
numerical problem. Hence, alternative realizations for the
H ∞ fixed-lag smoother that involve only matrix exponentials of Hurwitz matrices is required. This is the subject of
future research.
It is then interesting to see how the variation of Qγ (h)
in h > h∞ affects the estimation error. To this end, consider Fig. 2(a), which depicts the magnitudes plots of G1 −
esh KG2 under the central smoother K achieving γmin for
several values of h ∈ [0, 2.5]. It is seen that in the interval [0, h∞ ] the error transfer function is virtually all-pass
and its H ∞ norm decreases as h increases. The further increase of h, which keeps the peak of the error unchanged,
has completely different effect on the estimation error and
shows up in the decrease of the error bandwidth. In other
words, the increase of h beyond h∞ reduces the H 2 norm
7
5
0
0
Magnitude (dB)
Magnitude (dB)
5
−5
−10
−15
−5
−10
−15
h=0
h=0.31
h=0.62
h=0.86
h=2.5
−20
h=0
h=0.31
h=0.62
h=0.86
h=2.5
0
10
Frequency, ω
−20
1
10
0
1
10
10
Frequency, ω
(b) H 2 design
(a) H ∞ design
Fig. 2. Optimal |G1 (jω) − ejωh K(jω)G2 (jω)| for a = q = 1
smoothing in an H ∞ -setting. IEEE Trans. Automat. Control 36 (2), 151–166.
Scherer, C., 1990. H ∞ -control by state-feedback and fast
algorithms for the computation of optimal H ∞ -norms.
IEEE Trans. Automat. Control 35 (10), 1090–1099.
Theodor, Y., Shaked, U., 1994. Game theory approach
to H ∞ -optimal discrete-time fixed-point and fixed-lag
smoothing. IEEE Trans. Automat. Control 39 (9), 1944–
1948.
Wiener, N., 1949. Extrapolation, Interpolation and Smoothing of Stationary Time Series. The MIT Press, Cambridge,
MA.
Zhang, H., Xie, L., Soh, Y. C., 2000. H ∞ deconvolution filtering, prediction, and smoothing: A Krein space
poloynominal approach. IEEE Trans. Signal Processing
48 (3), 888–892.
Zhou, K., Doyle, J. C., Glover, K., 1995. Robust and Optimal
Control. Prentice-Hall, Englewood Cliffs, NJ.
Acknowledgments
The author wishes to thank U. Shaked for catalyzing this
research and P. Bolzern and P. Colanery for stimulating discussions. He is also indebted to G. Meinsma, the J-spectral
factorization guru, for his counterexamples and help in the
proof of the main result.
References
Anderson, B. D. O., Moore, J. B., 1979. Optimal Filtering.
Prentice-Hall, Englewood Cliffs, NJ.
Colaneri, P., Maroni, M., Shaked, U., 1998. H ∞ prediction
and smoothing for discrete-time systems: A J-spectral
factorization approach. In: Proc. 37th IEEE Conf. Decision and Control. Tampa, FL, pp. 2836–2842.
Gahinet, P., 1994. On the game Riccati equations arising in
H ∞ control problems. SIAM J. Control Optim. 32 (3),
635–647.
Grimble, M. J., 1991. H ∞ fixed-lag smoothing filter for
scalar systems. IEEE Trans. Signal Processing 39 (9),
1955–1963.
Hassibi, B., Sayed, A. H., Kailath, T., 1999. Indefinite
Quadratic Estimation and Control: A Unified Approach
to H 2 and H ∞ Theories. SIAM, Philadelphia.
Meinsma, G., 1994. J-spectral factorization and equalizing
vectors—the extended remix. Tech. Rep. EE9404, Dept.
of Electrical and Computer Eng., The University of Newcastle.
Meinsma, G., 1995. Unstable and nonproper weights in H ∞
control. Automatica 31 (11), 1655–1658.
Meinsma, G., Zwart, H., 2000. On H ∞ control for dead-time
systems. IEEE Trans. Automat. Control 45 (2), 272–285.
Mirkin, L., July 2001. Continuous-time fixed-lag smoothing
in an H ∞ setting. Tech. Rep. ETR–2001–01, Faculty of
Mechanical Eng., Technion—IIT.
Mirkin, L., 2003. On the extraction of dead-time controllers
and estimators from delay-free parametrizations. IEEE
Trans. Automat. Control 48 (4), 543–553.
Nagpal, K. M., Khargonekar, P. P., 1991. Filtering and
Appendix
A.1 Proof of Lemma 8
i) ⇐⇒ ii)
Standard completing to square arguments (Hassibi et al.,
1999, §10.4.2) yield that the optimal performance level for
SP∞ , γFI , can be characterized as follows:
2
γFI
= supω∈[0,∞) σ
¯ Ψ(jω) ,
where

˜ 11
H

.
∼ −1
 ˜
Ψ = G1 I − G ∼
G2 G∼
2 (G2 G2 )
1 =  H21
0

˜ 12 C 0
H
1

˜ 22 0  .
H

C1 0
Clearly, γ > γFI iff γ 2 I − Ψ(jω) > 0 for all ω. Since Ψ(s)
is strictly proper, the latter condition is equivalent to the requirement that (γ 2 I − Ψ)−1 has no poles on the imaginary
8
and Y˜ (h) = Yα−1 (h)Yβ (h). On the other hand, it is the
standard result from the Riccati theory that Y˜ (h) as in the
last formula satisfies Y˜ (h) = Qγ (h), where Qγ (t) is the
solution to to the differential Riccati equation (8). Thus, in
order to show the existence of Y˜ (h) it is sufficient to show
that (8) has no escape points on the interval [0, h]. To this
end the following result is required:
axis. The equality γ∞ = γFI then follows by standard arguments associated with the computation of the H ∞ norm
(see, e.g., (Zhou et al., 1995, Lemma 4.7)) and the fact that
˜ has no imaginary axis eigenvalues.
H
i) ⇐⇒ iii)
˜ γ (0) has imagIt follows from the reasoning above that H
˜ γ (0) 6∈
inary axis eigenvalues ∀γ ∈ [0, γ∞ ]. Therefore, H
dom(Ric) for any γ ≤ γ∞ and one only needs to prove
˜ γ (h) ∈ dom(Ric) whenever γ > γ∞ . The first step
that H
toward this end is to prove the latter for h = 0:
Claim 13 Qγ (t) is monotonically non-decreasing function
of t in the sense that Qγ (t1 ) ≥ Qγ (t2 ) whenever t1 ≥ t2 .
Moreover, Qγ (t) is bounded and limt→∞ Qγ (t) = Y˜κ > 0.
˜ γ (0) ∈ dom(Ric) for all γ > γ∞ .
Claim 12 H
PROOF. Differentiating (8) by t one gets:
˜ γ (0) has no jω-axis eigenvalues, there
PROOF. Since H
i
h
exists a full row rank matrix Y˜α Y˜β such that
h
h
i
˜ γ (0) = AL Y˜α Y˜β
Y˜α Y˜β H
¨ γ = −(A + BB 0 Qγ )0 Q˙ γ − Q˙ γ (A + BB 0 Qγ ).
Q
˜ γ (0)) it is readily seen
Using the fact that Qγ (0) = Ric(H
1
0
˙
that Qγ (t)|t=0 = γ 2 C1 C1 ≥ 0. Then, denoting by ΦQ (t, 0)
the state transition matrix associated with −(A + BB 0 Qγ ),
one gets:
i
for some Hurwitz AL . Now, to prove the Claim it is only left
to show that Y˜α is nonsingular. But the latter is guaranteed
by the fact that (A, B) has no stable uncontrollable modes,
see (Zhou et al., 1995, Theorem 13.7).
Q˙ γ (t) =
Now, since Y˜γ is monotonically non-decreasing function of
γ (Gahinet, 1994), Y˜γ ≤ Y˜κ for every γ > γ∞ . Define the
function
.
Q∆ (t) = Y˜κ − Qγ (t),
(A.3)
for some Hurwitz AL (h). In particular, when h = 0
i
h
i
h
˜ H
˜ γ (h)Σ(h)
˜ −1
˜ γ (0) = I Y˜γ Σ(h)
I Y˜γ H
h
i
= AL (0) I Y˜γ ,
which is monotonically non-increasing. It can then be verified that Q∆ satisfies
Q˙ ∆ = A0κ Q∆ + Q∆ Aκ + Q∆ BB 0 Q∆ ,
.
with Q∆ (0) = Y˜κ − Y˜γ ≥ 0, where Aκ = −(A + BB 0 Y˜κ ) is
Hurwitz and Q˙ ∆ (t) ≤ 0 for all t. It follows from this equation that if there exists a vector η 6= 0 such that Q∆ (t)η = 0,
then Q˙ ∆ (t)η = 0 too. Then, the continuity of Q∆ (t) means
that no eigenvalues of Q∆ can cross 0. Since Q∆ is symmetric, it has only real eigenvalues and thus its inertia is
unchanged. This, in turn, implies that Q∆ (t) ≥ 0 for all t.
Therefore, as Q∆ is monotonically non-increasing, its limit
as t → ∞ exists. Finally, since Aκ is Hurwitz, the only
equilibrium of Q∆ in the positive semi-definite region is
Q∆ = 0.
.
˜ γ (0)). Hence, if H
˜ γ (0) ∈ dom(Ric),
where Y˜γ = Ric(H
˜
then Hγ (h) ∈ dom(Ric) too iff
(A.1)
and in that case
˜ 11 (h) + Y˜γ Σ
˜ 21 (h)
Y˜ (h) = Σ
−1
˜ 22 (h) .
˜ 12 (h) + Y˜γ Σ
Σ
Consider the following first-order system:
i
h
i
h
˜
Y˙ α Y˙ β = − Yα Yβ H
(A.2)
As follows from Claim 13, Y˜ (h) exists for all h ≥ 0 that, in
˜ γ (h) ∈ dom(Ric). This completes the
turn, implies that H
proof of Lemma 8.
i
h
i
Yα (0) Yβ (0) = I Y˜κ .
Clearly, (A.1) is equivalent to the non-singularity of Yα (h)
with the initial condition
≥ 0,
from which the monotonicity statement of the Claim follows
immediately.
˜ γ (h) ∈ dom(Ric) iff there exists a matrix
Now, note that H
˜
Y (h) satisfying
h
i
h
i
˜ γ (h) = AL (h) I Y˜ (h)
I Y˜ (h) H
˜ 11 (h) + Y˜γ Σ
˜ 21 (h) =
det Σ
6 0
0
0
1
γ 2 ΦQ (t, 0)C1 C1 ΦQ (t, 0)
h
9
.
where Θ = Q∆ (0)1/2 . Because Q∆ is bounded (by
Claim 13), ΘW (t)Θ < I for all t. Using the explicit
formula for Q∆ , the condition Q(t) > 0 can be rewritten as
A.2 Proof of Theorem 11
It follows from Lemma 8 that γ∞ is the lower bound for the
achievable smoothing performance for every h. It therefore
will be assumed throughout the rest of the proof that γ >
γ∞ .
0
Y˜κ − eAκ t Θ I − ΘW (t)Θ
or, equivalently,
i) ⇐⇒ ii)
"
Y˜κ
−1
0
e Aκ t Θ
#
ΘeAκ t > 0
> 0.
By Theorem 4 SPh is solvable iff Hγ (h) ∈ dom(Ric) and
Y (h) = Y˜ −1 (h) > 0 (the latter follows from assumptions
(A4) and (A5)). If Hγ (h) ∈ dom(Ric), then the solvability
is clearly equivalent to the positive definiteness of Y˜ (h). On
the other hand, for every γ > γ∞ Hγ (h) 6∈ dom(Ric) iff
Y˜ (h) is singular and hence condition ii) does not hold. This
completes the proof.
Then, as Y˜κ > 0, the latter holds iff the Schur complement
of Y˜κ is positive definite of, i.e., iff
ii) ⇐⇒ iii)
Yet since W (t) = Wc − eAκ t Wc eAκ t , the inequality above
can be reformulated as
0
Θ eAκ t Y˜κ−1 eAκ t + W (t) Θ < I.
0
As follows from the proof of Lemma 8, Qγ (h) = Y˜ (h) for
every h ≥ 0. Then the equivalence of ii) and iii) follows by
Claim 13 on p. 9.
0
ΘeAκ t Bκ Bκ0 eAκ t Θ < I − ΘWc Θ.
Now, noticing that Cγ0 Cγ = Θ(I − ΘWc Θ)−1 Θ the latter
inequality can be equivalently rewritten as Cγ Υ(t)Cγ0 < I,
0
.
where Υ(t) = eAκ t Bκ Bκ0 eAκ t . The reasoning above yields
that
Q(h) > 0 ⇐⇒ kCγ eAκ h Bκ k < 1.
Finally, the monotonicity of the norm in the last inequality
follows from the fact that
˙ = eAκ t Aκ (Y˜κ−1 − Wc ) + (Y˜κ−1 − Wc )A0κ eA0κ t
Υ
0
= −eAκ t Y˜ −1 C 0 C2 Y˜ −1 eAκ t ≤ 0
iii) ⇐⇒ iv)
Consider equation (A.2) and denote
h
i
Zα Zβ
"
#
i I Y˜
h
κ
.
= Yα Yβ
,
0 −I
so that Q∆ = Zα−1 Zβ , where Q∆ (t) is defined by (A.3).
Since
# "
#
# "
"
A0κ
0
I Y˜κ ˜ I Y˜κ
=
H
,
0 −I
BB 0 −Aκ
0 −I
κ
Z˙ α Z˙ β
i
h
= Zα Zβ
i
"
−A0κ
0
−BB 0 Aκ
#
h
i h
i
with the initial conditions Zα (0) Zβ (0) = I Q∆ (0) .
Solving this equation one gets:
Zβ (t) = Q∆ (0)eAκ t ,
2
κ
(the latter equality follows by (6) and (9)). This completes
the proof of Theorem 11.
(A.2) can be rewritten as
h
ΘeAκ t I − ΘW (t)Θ
0
Zα (t) = I − Q∆ (0)W (t) e−Aκ t ,
0
. Rt
where W (t) = 0 eAκ s BB 0 eAκ s ds ≥ 0 (note, that Wc =
limt→∞ W (t)), and then:
0
Q∆ (t) = eAκ t I − Q∆ (0)W (t) −1 Q∆ (0)eAκ t
0
= eAκ t Θ I − ΘW (t)Θ −1 ΘeAκ t .
10