Filters for linear continuous-time singular systems.

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Filters for Linear Continuous-time Singular Systems
Jun-e Feng and James Lam and Shengyuan Xu
Abstract—This paper is concerned with designing filters for
linear continuous-time singular systems. Two types of filters are
developed; one is an infinite-time filter and the other is a finite-
time one. For the former, the convergence of the filtering error
to zero is asymptotic with time; but for the latter, it is finite
time. Sufficient conditions for the solvability of these design
problems are obtained in terms of rank conditions. It is noted
that the designed filters are in linear state-space form and no
information on the unknown input is required. A numerical
simulation is used to illustrate the proposed method.
I. INTRODUCTION
Mathematically speaking, a continuous-time singular sys-
tem model is described by a set of coupled differential
and algebraic equations, which include dynamic information
of a real plant as well as algebraic constraints. Singular
systems are also referred to as descriptor systems, implicit
systems, generalized state-space, differential-algebraic sys-
tems or semi-state systems. They are often encountered in
various applications such as power systems, social economic
systems, circuits and robotics. Therefore, considerable atten-
tion has been drawn to singular systems, see, [2], [12], and
references therein.
The problem of filter design has been a major area in
optimal control and fault diagnosis. There are mainly two
kinds of commonly used filters: Kalman filter and H∞filter,
see [4], [9], [10], [11], [14], for instance. A Kalman filter
generally ensures an optimal estimation of the state variables
in the sense that the covariance of the estimation error
is minimized, while an H∞ filter guarantees the transfer
function from the disturbance to the filtering error output
satisfies a prescribed H∞-norm bound constraint. Theses
methods have been extended to singular systems. For exam-
ple, Kalman filtering problems were tackled in [1] and [6] for
discrete-time stochastic systems. In [7] and [8], H∞filters
were designed, respectively for discrete- and continuous-time
singular systems. Both full- and reduced-order filters were
studied in [12] for continuous- and discrete-time singular
systems. While in [13], the problem of H∞ filtering was
investigated for uncertain descriptor systems with discrete
and distributed delays.
In this paper, we are concerned with the filter design
for linear continuous-time singular systems. Two types of
filters are developed; one converges asymptotically and the
This work was partially supported by RGC Grant HKU 7031/06P
J. Feng is with the School of Mathematics, Shandong University, Jinan,
250100 fengjune@sdu.edu.cn
J. Lam is with the Department of Mechanical Engineering, University of
Hong Kong, Pokfulam Road, Hong Kong james.lam@hku.hk
S. Xu is with the Department of Automation, Nanjing University of
Science and Technology, Nanjing, 210094 syxu02@yahoo.com.cn
other in finite time. In the latter case, the filter structure
comprises two infinite-time filters and a finite delay element.
It is noted that the filters designed are in state-space form
and no information on the unknown input is required.
A brief outline of this paper is as follows. In Section II,
we first make an equivalent transformation of the original
systems. A basic result is presented for the existence of
filters for linear continuous-time singular systems on infinite
time interval. In Section III, two sufficient conditions which
guarantee the existence of the infinite-time filter and the
finite-time filter are obtained in terms of rank relationships.
Systematic design method for filters is also given in this part.
Section IV provides an illustrative example and Section V
concludes the paper.
Notation. Throughout this paper, Rndenotes the n-
dimensional Euclidean space. The Moore-Penrose inverse of
a matrix M is denoted by M†. I is the identity matrix with
compatible dimensions. C represents the complex plane. ℜ(·)
means the real part of a complex number. Notations rank(·),
det(·), λj(·) stand for the rank, the determinant, and the
jth eigenvalue of a matrix, respectively. Matrices, if their
dimensions are not explicitly stated, are assumed to have
compatible dimensions for algebraic operations.
II. PRELIMINARIES
Consider the following linear descriptor system:



E ˙ x(t)
y(t)
z(t)
=
=
=
Ax(t) + Bw(t),
Cx(t) + Dw(t),
Lx(t),
(1)
where x(t) ∈ Rnis the state; y(t) ∈ Rmis the measurable
output; z(t) ∈ Rqis the signal to be estimated; w(t) ∈ Rp
is the disturbance input. The matrix E ∈ Rn×nmay be
singular; we shall assume that rankE = r ≤ n. A, B, C,
D and L are known real constant matrices with appropriate
dimensions. And descriptor system (1) is assumed to be
regular and impulse-free [2]. Without loss of generality, we
assume that L is of full row rank. Denote
˜E =
?
E
0
0
0
˜C =?
?
,
˜A =
?
A
0
?,
B
0
˜L =?
?
,
˜ x(t) =
?
x(t)
w(t)
?
,
CDL
0
?.
Now we consider the following filter for the estimate of z(t)
on infinite horizon:
?˙ζ(t)
where ζ ∈ Rqand ¯ z ∈ Rqare the state and output of the
filter, respectively.ˆA ∈ Rq×q,ˆB ∈ Rq×mandˆD ∈ Rq×m
=
=
ˆAζ(t) +ˆBy(t),
ζ(t) +ˆDy(t),
¯ z(t)
(2)
Joint 48th IEEE Conference on Decision and Control and
28th Chinese Control Conference
Shanghai, P.R. China, December 16-18, 2009
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are unknown matrices of appropriate dimensions, which are
to be determined such that ¯ z asymptotically estimates z.
Remark 1 Although system (1) is a singular system, filter
(2) is a standard qth order state-space system, where q is the
dimension of the signal to be estimated.
Lemma 1: The qth order filter (2) asymptotically esti-
mates z; that is, ¯ z − z → 0 as t → ∞, if there exists a
matrix M such that the following conditions hold:
(c1) ˆAM˜E +ˆB˜C = M˜A,
(c2) M˜E =˜L −ˆD˜C,
(c3) ˆA is Hurwitz.
Proof. Let ξ(t) = ζ(t) − M˜E
?
?
x(t)
0
?
, then from (c1) and
(c2), we have
˙ξ(t)=
˙ζ(t) − M˜E
?
˙ x(t)
0
?
=
=
=
ˆAζ(t) +ˆBy(t) − M˜A˜ x(t)
ˆAξ(t) + (ˆAM˜E +ˆB˜C − M˜A)˜ x(t)
ˆAξ(t).
From (c2), we obtain
¯ z(t) − z(t)=
=
=
=
ζ(t) +ˆDy(t) −˜L˜ x(t)
(ξ(t) + M˜E˜ x(t)) +ˆD˜C˜ x(t) −˜L˜ x(t)
ξ(t) + (M˜E +ˆD˜C −˜L)˜ x(t)
ξ(t).
Using (c3), we obtain lim
(2) estimates z(t) asymptotically.
t→∞(¯ z(t)−z(t)) = 0 and hence filter
?
III. MAIN RESULTS
We first present a sufficient condition which ensures the
existence of infinite-time filter (2) for system (1). Some of
the techniques employed in the proof have been adapted from
[3].
Theorem 1: There exists a filter of form (2) for system
(1), if for any λ ∈ C,ℜ(λ) ≥ 0 the following condition
holds:
rank







0
0
0
0
C
L
0
0
E
D
0
0
0
0
C
L
D
0
−A−B





= rank

C
L
0
D
0
0
0
0
0
0
C
E
D
0
−A−B

(3)
= rank

0
0
C
0
0
D
−B
C
L
0
E
D
0
0
0
λE − A


.
Proof. Based on Lemma 1, the proof is separated into two
steps.
Step (1). We prove the first equality in condition (3) is
equivalent to the existence of a solution to nonlinear matrix
equations (c1) and (c2). Denoting¯E =˜E − I, notice that
rank









0
0
0
0
C
L
0
0
E
D
0
0
0
0
C
L
D
0
−A−B
˜C
˜L
0
0
˜E
0
I
0
0
0
˜C
0
˜L
0
˜E



= rank

0
0
˜C
˜L
−˜A
I
0
0
0
0
˜C˜A
˜C
˜L˜A
˜L
¯E˜A



= rank

0
0
I
0
0
0
0
0
I
0
0
0
0
0
I






0
0
˜C
˜L
0
0
˜E
˜C
˜L
−˜A



?
I
˜A
0
I
?
= rank




,
and
rank









C
L
0
D
0
0
0
0
0
0
C
E
D
0
−A
˜C
−B
0
0
˜C
˜E
I
0
0
0
˜C
0
0
˜E



= rank

˜L
0
−˜A
0
I
0
0
˜C˜A
˜C
˜L
¯E˜A



= rank

0
0
I
0
0
0
0
I






˜C
˜L
0
0
0
˜C
˜E−˜A



?
I
˜A
0
I
?
=rank



.
On the other hand, from conditions (c1) and (c2) of Lemma
1, we obtain
ˆA˜L =˜L˜A − [ˆD T M]


˜C˜A
˜C
¯E˜A

,
(4)
where T =ˆB −ˆAˆD. Since L is of full row rank,˜L is also
of full row rank. Therefore, there exists a matrix G such that
?˜LT
[ H1
E1 ] =
GT?Tis nonsingular. Denote
?˜L
G
?−1
,
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then˜LH1= Iqand˜LE1= 0. Therefore, from (4) we obtain
ˆA =˜L˜AH1− [ˆD T M]


˜C˜AH1
˜CH1
¯E˜AH1

,
(5)
and
ˆA˜LE1=˜L˜AE1− [ˆD T M]


˜C˜AE1
˜CE1
¯E˜AE1

= 0.
(6)
From (5) and the definition of T, we can see thatˆA andˆB
can be obtained from the knowledge ofˆD, T and M. Now
(6) and (c2) can be rewritten as
?
ˆDTM
?Σ = Θ,
˜C˜AE1
˜CE1
¯E˜AE1
(7)
where
Σ =


˜C
0
˜E

,
and Θ = [˜L˜AE1
only if
˜L ]. There exists a solution to (7) if and
rank
?
Σ
Θ
?
= rank[Σ].
(8)
Take
S1:=
?
[H1 E1]
0
0
I
?
,
then we have
rank





˜C˜A
˜C
˜L˜A
˜L
¯E˜A
˜C˜AH1
˜CH1
˜L˜AH1
Iq
¯E˜AH1
˜C
0
˜L
0
˜E


˜C˜AE1
˜CE1
˜L˜AE1
0
¯E˜AE1

S1
=rank

˜C
0
˜L
0
˜E



=
q + rank


?

˜C˜AE1
˜CE1
˜L˜AE1
¯E˜AE1
?
˜C
0
˜L
˜E



=
q + rank
Σ
Θ
.
Similarly, we have
rank



˜C˜A
˜C
˜L
¯E˜A
˜C
0
0
˜E


= q + rank[Σ].
Hence the first equality in condition (3) holds if and only if
ˆD, T and M are solvable from (7) which in turn givesˆA
andˆB.
Step (2). We prove the second equality of condition (3) is
equivalent to thatˆA is Hurwtiz under the first equality of
condition (3). Notice that
rank






?



0
0
C
0
0
D
−B
˜C
˜L
0
˜E
C
L
0
E
D
0
0
0
λE − A



= rank

0
0
˜C
λ˜E −˜A


0
0
0
I

=rank

0
I
0
0
0
λI
0
I
0
−I
0
0






0
0
˜C
˜C
˜L
0
˜Eλ˜E −˜A



I
0
I
˜A − λI
λ˜L −˜L˜A
˜C˜A
˜C
¯E˜A
?
−˜L
˜C
0
˜E
= rank



.
On the other hand, we obtain the general solution of (7) as
?
ˆDTM
?= ΘΣ†+ Z(I − ΣΣ†),
(9)
where Z is an arbitrary matrix with appropriate dimensions.
Substituting (9) into (5) results in
ˆA
=
˜L˜AH1− ΘΣ†




˜C˜AH1
˜CH1
¯E˜AH1
˜C˜AH1
˜CH1
¯E˜AH1




−Z(I − ΣΣ†)
=Λ − ZΓ,
where
Λ =˜L˜AH1− ΘΣ†


˜C˜AH1
˜CH1
¯E˜AH1

,
Γ = (I − ΣΣ†)


˜C˜AH1
˜CH1
¯E˜AH1

.
There exists a matrix parameter Z such that all eigenvalues
of matrixˆA = Λ − ZΓ be Hurwitz if and only if for ∀λ ∈
C,ℜ(λ) ≥ 0,
rank
?
λI − Λ
Γ
?
= q.
(10)
Consider the full column rank matrix
S2:=


I
0
0
ΘΣ†
I − ΣΣ†
ΣΣ†

,
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and nonsingular matrix
S3:=



I
˜C˜AH1
˜CH1
¯E˜AH1
0
−Σ†




I


.
Therefore, we have
rank







λ˜L −˜L˜A
˜C˜A
˜C
¯E˜A
λIq−˜L˜AH1
˜C˜AH1
˜CH1
¯E˜AH1
λIq−˜L˜AH1




?
0
λIq− Λ
Γ
−˜L
˜C
0
˜E


S1
−˜L˜AE1
˜C˜AE1
˜CE1
¯E˜AE1
=rank

−˜L
˜C
0
˜E



=rank

−Θ
˜C˜AH1
˜CH1
¯E˜AH1
λIq−˜L˜AH1


λIq− Λ
Γ


Σ



=rankS2

−Θ
˜C˜AH1
˜CH1
¯E˜AH1


Σ


S3
=rank


?
?
0
Σ


=rank
?
+ rank[Σ].
Furthermore, from the proof of Step (1), we establish the
result in Step (2).
Now we are in the position to construct a finite-time filter
of the following form:
?







˙ˇζ
ˆ z
=
=
=
=
ˇAˇζ +ˇBy,
ˇζ +ˇDy,
K[ˆ z(t) − eˇ
φ(t), t ∈ [−d,0],
ˇ z(t)
ˆ z(t)
Adˆ z(t − d)],
(11)
where d > 0,ˇζ ∈ R2qis the state vector, ˆ z ∈ R2qis
the output, and ˇ z ∈ Rqis the estimate of z, and φ(·) is a
continuous function.ˇA,ˇB,ˆD and K are unknown matrices
of appropriate dimensions, which are to be determined such
that ˇ z estimates z in finite time d, that is, ˇ z(t) = z(t) for
t ≥ d.
Theorem 2: There exists a finite-time filter for system (1)
if for any λ ∈ C the rank condition (3) holds.
Proof. From the proof of Step (2) in Theorem 1, if for any
λ ∈ C the rank condition (3) holds, then there exists Z such
that all eigenvalues of matrixˆA can be assigned arbitrarily.
Take two different matrices denoted by Z1 and Z2, which
satisfy thatˆAi= Λ−ZiΓ are stable i = 1,2. Then we obtain
the corresponding matricesˆDi, Ti, MiandˆBi. Denote
ˇA =
?
ˆA1
0
0
ˆA2
?
,
ˇB =
?
¯ z1
¯ z2
ˆB1
ˆB2
?
,
ˇD =
?
ˆD1
ˆD2
?
?
,
N =
?
Iq
Iq
?
,
ˆ z =
??
,
ˇζ =
?
ζ1
ζ2
.
(12)
Since
have
d
dt(¯ zi(t)−z(t)) =ˆAi(¯ zi(t)−z(t)) for i = 1,2, we
d
dt(ˆ z(t) − Nz(t)) =ˇA(ˆ z(t) − Nz(t)),
Therefore,
t ≥ 0.
ˆ z(t) − Nz(t) = eˇ
Ad[ˆ z(t − d) − Nz(t − d)],t ≥ d. (13)
Take K =?
such that
proof can be established along an argument similar to that
in [5], hence omitted.
Remark 2 From the proof of Theorem 2, it is known that
the finite-time filter (11) is 2q-th order.
Remark 3
To construct the finite-time observer gain
K,
?
?
?
0
eˆ
Iq
0
??
N
?
eˇ
AdN
?−1
. The existence of d
?
Neˇ
AdN
is nonsingular and the left of the
?
?
Neˇ
AdN
?
?
is required to be nonsingular. Since
Neˇ
AdN
is row equivalent to
Ieˆ
A1d
A1d− eˆ
A2d
?
,
its nonsingularity is satisfied if and only if eˆ
not have unit eigenvalues.
Remark 4 SinceˇA is Hurwitz and the initial function φ(t) is
a given continuous function, eˇ
On the other hand, since all eigenvalues ofˆAi= Λ − ZiΓ
can be assigned arbitrarily under condition of Theorem 2, Zi
can be chosen such that
A2de−ˆ
A1ddoes
?
Adˆ z(t−d) → 0 when d → ∞.
ℜ(λj(ˆA2)) < σ < ℜ(λj(ˆA1)),j = 1,2,...,q,
for some σ < 0. In this case, eˆ
Therefore, we have
A2de−ˆ
A1d→ 0 when d → ∞.
K
=
?
?
?
?
Iq
0
??
?
N,eˇ
AdN
?−1
=
Iq
0
?
I
0
eˆ
A1d
eˆ
A1d− eˆ
A2d
?−1?
Iq− (eˆ
I
I
0
−I
?−1
A1d)−1
=
Iq− (Iq− eˆ
0
Iq
A2de−ˆ
d → ∞.
A1d)−1
A2de−ˆ
?
→
?,
(14)
Therefore, it is obtained that
ˇ z(t) →?
0
Iq
?ˆ z(t) = ¯ z2(t),d → ∞,
where ¯ z2(t) is the output of one of the infinite-time filters,
that is, ¯ z2(t) → z(t), t → ∞.
A schematic diagram illustrating the filter design proce-
dures for a finite-time filter of form (11) or an infinite-time
filter of form (2) for system (1) is shown in Figure 1. Detailed
?
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algorithms for the design of finite-time filters and infinite-
time filters are given below.
Algorithm “Finite-time filter”
Step 1. Given d > 0. Choose Zi such that ˆAi are
stable, i = 1,2, such that eˆ
eigenvalues.
Step 2. ObtainˆDi, Ti and Mi from (9), and subse-
quently, obtainˆBi= Ti+ˆAiˆDifrom Ti=ˆBi−
ˆAiˆDi.
Step 3. Obtain
ˇA,
ˇB,
ˇD from (12) and K
?
coefficientsˇA,ˇB,ˇD and K obtained above.
A2de−ˆ
A1dhas no unit
=
Iq
0
??
N,eˇ
AdN
?−1
.
Step 4. Obtain a finite-time filter of form (11) with the
Algorithm “Infinite-time filter”
Step 1. Choose Z such thatˆA is stable.
Step 2. ObtainˆD, T and M from (9), and subsequently,
obtainˆB = T +ˆAˆD from T =ˆB −ˆAˆD.
Step 3. Obtain an infinite-time filter of form (2) with
the coefficientsˆA,ˆB andˆD obtained above.
First equality in (4)
No
No solution
No solution
Algorithm
“Infinite-time filter”
Algorithm
“Finite-time filter”
),(
??
Controllable
Stabilizable
Neither
controllable nor
stabilizable
Yes
Fig. 1.Schematic Diagram for Filter Design
IV. NUMERICAL EXAMPLE
In this section, we present a numerical example to illus-
trate the developed method. Consider system (1) with the
following data:
E =



1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0


,A =



−1
−1
0
0
1
0
0
0
0
1
0
1
−1
0
−1
0


,
B =



6
−1
0
−1
?


?
,
,
C =
?
1
0
0
0
0
0
0
1
?
,
D =
0
1
L =?
0100
?.
It is easy to check that first equality in (3) is sat-
isfied. Solving (7) and substituting the solution into
(5), we get Λ=0.0952 and Γ=
?
0.04760
0.0476
is obvious that the rank condition in (10) holds for any
λ ∈ C. Taking
−0.1429
−0.047600
−0.14290
?T. It
Z1=?
500000000
?,
and
Z2=?
1000000000
?,
we have
ˆA1= −0.1429,
ˆB1=?
ˆB2=?
0
?,
−0.8776
−0.4286
?,
?,
ˆA2= −0.3810,
−0.7642
−1.1429
ˆD1=?
0.1429
ˆD2=?
0.38100
?.
Take d = 5, it is easy to verify that
det
?
N,eˇ
AdN
?
= det
?
1
1
e−0.1429d
e−0.3810d
?
?= 0.
Therefore, we obtain a finite-time observer (11) with the
following coefficients:
ˇA =
?
−0.1429
0
0
−0.3810
?
,
ˇB =
?
−0.8776
−0.7642
−0.4286
−1.1429
?
,
ˇD =
?
0.1429
0.3810
0
0
?
,
ˇ N =
?
1
1
?
,
and d = 5, whose output ˇ z(t) estimates exactly to the
estimated output z(t) in finite time d = 5. Figure 2 is drawn
under w(t) = 0.3sin(t) and x(0) =
ˆ z(t) =?
?
1
−220
?T,
00
?T, t ∈ [−5,0], ζ(0) = 0.
0123456
−3
−2
−1
0
1
2
3
time [s]
Fig. 2.
z(t) (solid) and its estimate ˇ z(t) (dashed).
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