IET Control Theory & Applications
Brief Paper
On the implementation of delayed output
feedback for linear systems with multiple
input delays
Chuanchuan Xu1
1
ISSN 1751-8644
doi: 0000000000
www.ietdl.org
Bin Zhou1
Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China
E-mail: binzhou@hit.edu.cn, binzhoulee@163.com
Abstract: This paper is concerned with the safe implementation of delayed output feedback (DOF) for linear systems with multiple
input delays. It is observed that the numerical implementation of the distributed delay terms involved in the DOF controller may
lead to instability of the closed-loop system. To solve this problem, a generalized low-pass filter (LPF) is introduced into the control
loop. It is shown that the stability of the closed-loop system can be maintained if the numerical integration is sufficiently precise. An
augmented system is then constructed to design the LPF by applying the traditional pole assignment algorithm for linear systems
without delay. Two illustrative examples are worked out to show the effectiveness of the proposed approach.
1
Introduction
Time-delay phenomenon exists widely in many fields, such as chemical process control, industrial metallurgy process, networked systems, biological community systems and mechanical transmission
systems [2, 7, 9]. Time-delay can be caused by signal transmission,
control calculation, measurement feedback and so on. In many cases,
time-delay can cause performance degradation and even instability of the control systems [14]. Hence, the time-delay problem has
received much attention over the past few decades (see, for example,
[5, 10, 13, 15, 21, 35, 36] and the references therein). The presence
of time-delay makes the stability analysis and stabilization of the
closed-loop system hard in both mathematics and practice.
For systems with a pure input delay, a simple and efficient
approach for compensating the input delay is called Smith predictor, which was initially proposed in [32]. By the Smith predictor, the
closed-loop system can act like a delay-free system. However, the
Smith predictor becomes invalid to stabilize the whole system when
the open-loop system is unstable. The requirement on the open-loop
stability by the Smith predictor can be removed by predictor feedback [17, 18, 20, 22]. The idea of this approach is to estimate the
future state by using the system information and then to stabilize the
system through state feedback which uses the estimated future state.
It is necessary to point out that the predictor feedback needs to take
advantage of the system states, which sometimes are unavailable or
difficult to be measured in practice. To overcome this shortcoming, a
more feasible approach named delayed output feedback (DOF) was
proposed in [25] and [26], which only uses the current and delayed
measurements of the system inputs and outputs to observe the future
system states so that a predictor-like state feedback can be constructed. This approach possesses the advantage that it can estimate
the future states of the system in finite time. Actuator and sensor
fault detection and isolation were studied in [29] under the condition
that the DOF was used as states estimators. To obtain a more general description, the pseudo-differential operator was used to express
the DOF approach in a unified form [27]. Owing to its numerous outstanding advantages, the DOF may have broad application prospects,
such as states estimation and control of systems with both input and
output delays.
Though the DOF approach is theoretically appealing, the implementation problem of the resulting controllers has been neglected
in the previous studies. In the study of predictor feedback (or finite
spectrum assignment) problem, it was pointed out that the implementation of controllers involving distributed delay terms can be
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
a problem: if the integration in the controller is implemented by
numerical integration approaches, such as the backwards rectangular
rule, the trapezoidal rule and so on, the resulting closed-loop system
is unstable no mater how precise the numerical integration is [4],
[6, 19, 33, 34]. This instability phenomenon was well explained in
[30] where it was shown that the closed-loop characteristic quasipolynomial may be of neutral type if the integration in the controller
is implemented by approximated numerical methods.
The aim of this paper is to study the implement issue for the DOF
approach. We first show that, similar to the predictor feedback (or
finite spectrum assignment) approach, numerical implementation of
the integrations in the DOF may cause instability of the closed-loop
system. Then, motivated by the work in [30], we introduce a generalized low-pass filter (LPF) into the control-loop to solve this problem.
By this way, the characteristic quasi-polynomial of the closed-loop
system is of retarded type, and its stability can be guaranteed if
the numerical integration is precise enough. We also provide methods to design the filter gains. To use this method, we need only to
apply a pole assignment algorithm to design stabilizing gains for
an augmented linear system without delay. Two numerical examples
demonstrate the effectiveness of the proposed approaches.
It should be pointed out that, differently from [30] where the predictor feedback control was designed for linear systems with a single
input delay by using state feedback, in this paper we consider predictor feedback for linear systems with multiple input delays by using
delayed output feedback. As a result, our controllers contain more
distributed delay terms than that in [30], which makes the proof
for the stability of the closed-loop system (with controllers approximated by numerical methods) more difficult. Therefore, we provide
an elementary transformation based proof which turns out to be more
easy than the right-coprime factorization based proof in [30]. Finally,
we point out that the restrictive controllability assumption required
by the right-coprime factorization based approach in [30] has been
replaced by a stabilizability assumption in this paper.
The remainder of this paper is organized as follows. The problem
formulation and some preliminary results are shown in Section 2. In
Section 3, we discuss the safe implementation of the controller by
adding a LPF in the control loop and the LPF gains are designed
with the help of an augmented system. Two numerical examples are
worked out in Section 4 to validate the effectiveness of the proposed
approach and Section 5 concludes the paper.
1
2
Delayed output feedback and numerical
implementation
≈
Nij
∑
ηij (k) e
Ak
τi +hj
Nij
k=0
In this paper, we consider the following continuous-time linear
system with multiple input delays

p−1
(
)
∑

ẋ (t) = Ax (t) +
B j u t − hj ,
(1)
j=0

y (t) = Cx (t) ,
where A ∈ Rn×n , Bj ∈ Rn×r , j ∈ I [0, p − 1] , C ∈ Rm×n are
real constant matrices, and hj , j ∈ I [0, p − 1] are some nonnegative
constants denoting the delays in the input. Without loss of generality,
we assume that
0 ≤ h0 < h1 < · · · < hp−2 < hp−1 < ∞.
We impose the following assumption on the system.
Assumption 1. The pair (A, C) is observable and the pair (A, B)
is stabilizable, where
p−1
∑
B=
e
−Ahj
Bj .
(2)
j=0
It is well known that system (1) is stabilizable if and only
if (A, B) satisfies the above assumption [37]. Under the above
assumption, one of the existing approaches for stabilizing system
(1) is the so-called delayed output feedback (DOF) [28], by which
the stabilizing controller is designed as
u (t) = F xd (t) ,
(3)
(
)
τ i + hj
Bj u t − k
,
Nij
(7)
where Nij and ηij (k) , i ∈ I [0, n − 1] , j ∈ I [0, p − 1] are respectively the number of integral steps and the coefficients that depend
on the selected integration rule. Therefore, the DOF controller (3)
can be approximated as
u (t) = F xdN (t) ,
(8)
in which xdN (t) is the numerical implementation of xd (t) with the
distributed delay terms approximated by (7).
It was observed in [34] that, for the traditional predictor feedback
controller, the approximated computation of the integration in the
form of (7) may cause instability of the closed-loop system. Next,
we will illustrate by a simple example that this phenomenon also
exists in the DOF scheme.
For simplicity, we consider the following scalar linear timeinvariant system
{
ẋ (t) = Ax (t) + B0 u (t − h0 ) ,
y (t) = Cx (t) ,
(9)
where A = 1, B0 = 1, C = 1 and h0 = 1. Let F = −2e and τ0 =
1. Then the DOF controller is
)
(
∫
u (t) = −2e2
2
y (t − 1) +
eθ−2 u (t − θ) dθ
.
(10)
0
As a result, the characteristic equation of the closed-loop system
consisting of (9) and (10) can be obtained as
where F ∈ Rr×n is any matrix such that A + BF is Hurwitz, and
xd (t) = W
n−1
∑(
−1
Ce
−Aτi
s + 1 = 0,
)T
Yi (t) ,
(4)
i=0
in which
W =
n−1
∑(
Ce−Aτi
)T
Ce−Aτi ,
(5)
i=0
(
and
+C
p−1
∑
e−A(τi +hj )
∫ τi +hj
eAθ Bj u (t − θ) dθ,
(6)
0
j=0
where
0 ≤ τ0 < τ1 < · · · < τn−1 < ∞,
are the selected delayed output time that can be chosen almost
arbitrarily. Moreover, the observability matrix W is almost always
nonsingular under Assumption 1 [28].
Similar to the other existing predictor feedback methods for timedelay systems (see, for example, [1], [3], [16] and [38]), controller
(3) contains the prediction of the state, which can realize finite spectrum assignment. Moreover, the DOF based controller possesses the
dead-beat property, namely, xd (t) approaches the true state function
in finite time [28], which is important in practice.
However, since the DOF controller (3) involves the integrations
of control signals, namely, the distributed delay terms, they must be
implemented via numerical integration. By applying some fixed-step
numerical integration rules, the distributed delay terms contained in
xd (t) can be approximated as
∫ τi +hj
e
0
Aθ
(
)
1 ( −2
e u (t) + u (t − 2)
2
(
)))
N
−1
∑
2l
2l/N −2
+
e
u t−
,
(11)
N
u (t) = − 2e2 y (t − 1) +
Yi (t) = y (t − τi )
2
namely, the closed-loop system possesses finite spectrum s = −1,
which implies that the closed-loop system is asymptotically stable.
However, as we pointed out before, the controller in (10) must
be implemented via numerical integration. Without loss of generality, we choose the trapezoidal rule for numerical integration to
approximate the distributed delay term, namely,
Bj u (t − θ) dθ
2
N
l=1
in which, N represents the number of integral steps. Taking Laplace
transformation on (9) and (11) under zero initial conditions gives

0 = (s − 1) X (s) −(e−s U (s)(,



(
)


 0 = 2e2−s X (s) + 1 + 4e2 1 e−2 + e−2s
2
N
))

N∑
−1


2l(1−s)/N −2

+
e
U (s) .


(12)
l=1
It follows from (12) that the corresponding characteristic quasipolynomial can be expressed as
(
∆ (s) = (s − 1) 1 +
+
N
−1
∑
4e2
N
(
e2l(1−s)/N −2
)
1 ( −2
e + e−2s
2
))
+ 2e2(1−s) ,
(13)
l=1
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
which is a quasi-polynomial of neutral type since the coefficient of s
with the highest order is not a constant, but a polynomial in e−s . As
pointed out in [30, 38], this quasi-polynomial equation implies that
the control law (11) is sensitive to round-off errors, and may further
lead to instability of the closed-loop system.
For the illustration purpose, we carry out a simulation by setting the initial state as x(0) = 1, u(θ) = 0, y(θ) = 0, θ ≤ 0 and the
numerical integral steps as N = 200 in the calculation of the controller. The state trajectory of the closed-loop system is recorded in
Fig. 1. It follows that the closed-loop system is unstable. Moreover,
the instability situation cannot be changed by increasing N .
where xdN (t) is the numerical implementation of xd (t) with the
distributed delay terms approximated by (7).
Next we will analyze the stability of the closed-loop system consisting of (1) and (16). Substituting (4), (5) and (6) into (15) and
taking Laplace transformation on (1) and (15) give


 0 = (sIn − A) X (s) − B(s)U (s) ,




 0 = Y (s) − CX (s) ,







0 = Bf W −1 V T Q (s) Y (s) − (sIr − Af ) Z (s)
−1
(17)
T
+Bf W V EH (s) U (s) ,
0 = U (s) − Z (s) ,
in which
8
B(s) =
6
p−1
∑
Bj e−shj ,
(18)
j=0

4


V =

x(t)
2
0
Ce−Aτ0
Ce−Aτ1
..
.
Ce−Aτn−1






 , Q (s) = 


e−τ0 s Im
e−τ1 s Im
..
.
e−τn−1 s Im



,

(19)
and
−2
−4
−6
0
2
4
6
8
10
t(s)
Fig. 1: State trajectory of the closed-loop system consisting of (9)
and (11)
3
Implementation by adding low-pass filters
Based on the above analysis, we know that the numerical integration
of integral terms in the DOF controllers may lead to instability of the
closed-loop system. This phenomenon exists also in the implementation of classical predictor feedback control for systems with input
delay (see, for example, [30] and [34]). A simple method overcoming the instability problem is to introduce a low pass filter (LPF) in
the control loop, which can make the approximated control law not
sensitive to round-off errors in the implementation process, namely,
stability can be maintained if the approximation accuracy is high
enough [30]. Motivated by the results in [30], in this section, we will
solve the implementation problem in the DOF approach by adding a
LPF.
By noting that any strictly proper linear system (Af , Bf , Cf , 0)
has a LPF property, we can design the following controller for
system (1):
{
ż (t) = Af z (t) + Bf xd (t) ,
(14)
u (t) = Cf z (t) ,
r×r
r×n
r×r
where Af ∈ R
, Bf ∈ R
and Cf ∈ R
are some constant
matrices to be designed, and z (t) denotes the state of the LPF.
Usually, we can choose Cf = Ir . Then it follows from (14) that
{
i
ż (t) = Af z (t) + Bf xd (t) ,
u (t) = z (t) .
(15)
ż (t) = Af z (t) + Bf xdN (t) ,
u (t) = z (t) ,
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
(16)
i,0
i,1
i,p−1


∫τ +h



Hij (s) = 0i j e(A−sIn )θ Bj dθ,






i ∈ I [0, n − 1] , j ∈ I [0, p − 1] .
(21)
By substituting the second and fourth equation of (17) into the third
equation of (17), we get


 0 = (sIn − A) X (s) − B(s)U (s) ,


(22)
0 = Bf W −1 V T CQ (s) X (s)

(
)


−1 T

− sIr − Af − Bf W V EH (s) U (s) .
From (22), the characteristic matrix of the closed-loop system
consisting of (1) and (15) can be expressed as
D1 (s)
[
=
It follows from (4) and (6) that controller (15) also contains distributed delay terms. Hence it should also be implemented by numerical integration algorithm as shown in (7). Therefore, the controller
(15) should be rewritten as
{



E0 0 . . .
0



 0 E


0

1 ···





,

E
=
..

..
 ..



.


.
.
0





0
0
0 En−1

(20)
]
[


 Ei = Ei,0 Ei,1 · · · Ei,p−1 ,








Eij = Ce−A(τi +hj ) ,






i ∈ I [0, n − 1] , j ∈ I [0, p − 1] ,

[
]T
T


H (s) = H0T (s) H1T (s) · · · Hn−1
,
(s)






[
]T


 H (s) = H T (s) H T (s) · · · H T
,
(s)
sIn − A
Bf W −1 V T CQ (s)
B(s)
sIr − Af − Bf W −1 V T EH (s)
]
.
(23)
Similarly, the characteristic matrix of the closed-loop system consisting of (1) and (16) can be expressed as
DN (s)
[
=
sIn − A
Bf W −1 V T CQ (s)
B(s)
sIr − Af − Bf W −1 V T EHN (s)
]
,
(24)
3
in which HN (s) is the numerical implementation of H (s) and is
defined in a similar way as (21) where Hij (s) is replaced by
+
p−1
∑

−1 −sIn (τi +hj )
(sIn − A)
e
Bj 
j=0
N
Hij
(s) =
N
ij
∑
ηij (k) e
(A−sIn )k
τi +hj
Nij
Bj ,
(25)
k=0
− Bf (sIn − A)−1 B(s)
= Bf W −1
i ∈ I [0, n − 1] , j ∈ I [0, p − 1] .
n−1
∑(
Ce−Aτi
)T
Ce−Aτi (sIn − A)−1 B
i=0
To go further, we need the following two technical lemmas.
− Bf (sIn − A)−1 B(s)
= Bf W −1 W (sIn − A)−1 B
Lemma 1. Consider the linear time-invariant system
ẋ (t) = Ax (t) ,
(26)
where A ∈ Rn×n is a Hurwitz matrix. Let A (s) = sIn − A be the
characteristic matrix of system (26). Then
A−1 (s) = 0.
lim
(27)
Re(s)≥0,|s|→∞
[
]
Proof: Let A∗ (s) = Aij (s) be the adjoint matrix of A (s). It
is clear that deg Aij (s) ≤ n − 1, i, j ∈ I [1, n]. As det A (s) has
no roots in {s : Re (s) ≥ 0} by the assumption that system (26) is
asymptotically stable, the conclusion (27) follows from A−1 (s) =
A∗ (s)
directly [37].
det A(s)
T1 (s) =
[
T2 (s) =
(
In
)
Bf − Bf W −1 V T CQ (s) (sIn − A)−1
In
0
−1
(sIn − A)
(B − B(s))
[
Φ(s) =
sIn − A
Bf
0
Ir
]
,
Ir
and
= Bf (sIn − A)−1 (B − B(s)) ,
in which the explicit expressions of V, E, B(s), Q (s) and H (s)
defined in (18), (19), (20) and (21) have been used. Here we have
also used the equation
e−A(τi +hj ) Hij (s)
(
)
= (sIn − A)−1 e−A(τi +hj ) − e−sIn (τi +hj ) Bj .
Then we have
[
T1 (s) D1 (s) T2 (s) =
Lemma 2. Let D1 (s) be defined in (23) and
[
− Bf (sIn − A)−1 B(s)
B
sIr − Af
[
]
×
,
In
0
B(s)
sIr − Af − D2 (s)
(sIn − A)−1 (B − B(s))
Ir
]
]
= Φ(s).
(28)
The proof is finished.
(29)
We are now ready to present the following main result in this
paper.
]
.
Then there holds
T1 (s) D1 (s) T2 (s) = Φ(s).
sIn − A
Bf
(30)
Theorem 1. Assume that the ideal closed-loop system consisting
of (1) and (15) is asymptotically stable. Let the distributed delay
terms contained in xd (t) be implemented by numerical integration as shown in (7). Then there exists a large enough integer
N ∗ > 0 such that the closed-loop system consisting of (1) and (16) is
also asymptotically stable for all Nij ≥ N ∗ , i ∈ I [0, n − 1] , j ∈
I [0, p − 1].
Proof: Through a simple matrix multiplication, we have
[
T1 (s) D1 (s) =
sIn − A
Bf
B(s)
sIr − Af − D2 (s)
Proof: Let H∆ (s) , H (s) − HN (s). Then we obtain
]
(
)
det (DN (s)) = det (D1 (s)) det In+r + D1−1 (s) D∆ (s) ,
in which
(
D2 (s) = Bf W −1 V T EH (s) + CQ (s) (sIn − A)−1 B(s)
− Bf (sIn − A)
= Bf W −1
n−1
∑(
−1
B(s)
Ce−Aτi
)T

C
i=0
p−1
∑
p−1
∑
e−A(τi +hj ) Hij (s)

Bj e−sIn (τi +hj ) 
j=0
− Bf (sIn − A)
= Bf W −1
n−1
∑(
i=0
−1
B(s)
Ce−Aτi
)T

C
[
0n×n
0r×n
0n×r
Bf W −1 V T EH∆ (s)
p−1
∑
j=0
(sIn − A)−1
]
.
(32)
In the characteristic polynomial (31), if
(
)
σ D1−1 (s) D∆ (s) < 1,
(33)
holds for all s ∈ C with Re (s) ≥ 0, then the equation
(
)
det In+r + D1−1 (s) D∆ (s) = 0
(
)
× e−A(τi +hj ) − e−sIn (τi +hj ) Bj
4
(31)
where we have used (23) and (24) and
D∆ (s) =
j=0
+ (sIn − A)−1
)
(34)
has no roots in the closed right-half plane. On the other hand,
since the ideal closed-loop system consisting of (1) and (15) is
asymptotically stable by assumption, the characteristic equation
det (D1 (s)) = 0 also has no roots in the closed right-half plane.
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
Consequently, we can conclude from (31) that all the roots of
equation det (DN (s)) = 0 locate in the open left-half plane,
namely, the closed-loop system consisting of (1) and (16) is asymptotically stable. Thus it remains to show that (33) is true for large
enough Nij , i ∈ I [0, n − 1] , j ∈ I [0, p − 1].
For any s ∈ C with Re (s) ≥ 0, and i ∈ I [0, n − 1] , j ∈
I [0, p − 1], there exist constants Mij > 0 independent of Nij (see
Proposition 4 in [30]) such that
}
{
H∆(i,j) (s)
sup
Re(s)≥0
,
{ ∫
τi +hj
sup
−
Nij
∑
ηij (k) e
∫ τi +hj
×
lim
Re(s)≥0,|s|→∞
=
(A−sIn )k
τi +hj
Nij
Bj
eAθ Bj dθ +
0
Nij
∑

ηij (k) e
k
τi +hj
Nij
A
Bj
≤ Mij , i ∈ I [0, n − 1] , j ∈ I [0, p − 1] .
(35)
= 0.
(40)
It thus follows from (40) and (36) that there exists a constant R0 > 0
such that
,
T
H∆(0,0)
(s)
sup
Re(s)≥0
T
H∆(n−1,p−1)
(s)
···
n−1
∑ p−1
∑
≤
···
S (f ) − SN (f ) = −
}
H∆(i,j) (s)
≤ npMmax ,
max
i∈I[0,n−1],j∈I[0,p−1]
H∆(i,j) (s)
{∥D∆ (s)∥} ≤ Bf W −1 V T E
Re(s)≥0
{∥H∆ (s)∥}
sup
Re(s)≥0
≤ np Bf W −1 V T E Mmax .
∫ τi +hj
= T2 (s) Φ
(36)
lim
∥T1 (s)∥ = 1,
(s)T1 (s) .
lim
Re(s)≥0,|s|→∞
Bj dθ −
Nij
∑
ηij (k) e
(A−sIn )k
τi +hj
Nij
Bj
k=0
ηξ
q ∑
∑
( (
)
Zξρ S (−θ)ρ−1 eθ(λξ −s)
ξ=1 ρ=1
(37)
By taking into account the forms of T1 (s) and T2 (s) as shown in
(28) and (29), we can see that
Re(s)≥0,|s|→∞
(A−sI)θ
0
Now by applying Lemma 2, we can verify that
−1
e
=
=
D1−1 (s)
(42)
{Mij }. With this and (32),
we have
sup
hβ+1 (γ)
f
(θ0 ) ,
αN β
where k is a constant. Let λξ (ξ = 1, · · · , q) denote the eigenvalues
( of matrix A and[ ηξ the
]) order of multiplicity of λξ , and let Zξρ
ξ ∈ I [1, q] , ρ ∈ I 1, ηξ be the components of matrix A [8]. We
have
i=0 j=0 Re(s)≥0
where Mmax =
(41)
where θ0 ∈ [0, h], and α, β and γ are some positive integers depending on the selected numerical integration approach (for instance, in
the Simpson approach, β = 4, γ = 4 and α = 180) [24]. This result
can be easily extended to a smooth complex function g : R → C as
[30]
{
}
k
|S (g) − SN (g)| ≤ β max
g (γ) (θ) ,
(43)
N θ∈[0,h]
T
H∆(i,j)
(s)
]T }
{
sup
(
)
σ D1−1 (s) D∆ (s) < 1,
for all values of Nij , i ∈ I [0, n − 1] , j ∈ I [0, p − 1].
Given a R0 satisfying (41), it remains to prove that there always
exist large enough Nij , i ∈ I [0, n − 1] , j ∈ I [0, p − 1] such that
the inequality in (33) is also satisfied for s satisfying Re (s) ≥ 0 and
|s| ≤ R0 . For a smooth real function f and its integration S (f ) =
∫h
0 f (θ) dθ, let SN (f ) be the fixed-step numerical integration with
N being the number of integral steps. Then it is well known that [30]
{∥H∆ (s)∥}
{ [
Φ−1 (s)
lim
Re(s)≥0,|s|→∞
According to the properties of the matrix norm, we have
Re(s)≥0
Φ−1 (s)
lim
Re(s)≥0,|s|→∞
∥T1 (s)∥
Re(s)≥0,|s|>R0
k=0
sup
∥T2 (s)∥
sup


k=0
≤
lim
Re(s)≥0,|s|→∞
e(A−sIn )θ Bj dθ
0
Re(s)≥0
≤
∥T2 (s)∥ = 1.
(38)
From Lemma 2, we know that Φ(s) is the characteristic matrix of the
closed-loop system consisting of (1) and (15), which, by assumption,
is asymptotically stable. Hence, it follows from Lemma 1 that
(
))
− SN (−θ)ρ−1 eθ(λξ −s)
Bj ,
(44)
where i ∈ I [0, n − 1] , j ∈ I [0, p − 1]. By replacing g with
(−θ)ρ−1 eθ(λξ −s) in (43), we claim that there exist constants
mij , i ∈ I [0, n − 1] , j ∈ I [0, p − 1] , such that
{
sup
}
H∆(i,j) (s)
≤
Re(s)≥0,|s|≤R0
mij
β
Nij
Bj .
(45)
So we have from (32) and (36) that
−1
lim
Φ
(s) = 0.
Re(s)≥0,|s|→∞
(39)
sup
{∥D∆ (s)∥}
Re(s)≥0,|s|≤R0
Then we get from (37), (38) and (39) that
lim
Re(s)≥0,|s|→∞
=
lim
Re(s)≥0,|s|→∞
D1−1 (s)
T2 (s) Φ−1 (s)T1 (s)
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
≤
Bf W −1 V T E
{∥H∆ (s)∥}
sup
Re(s)≥0,|s|≤R0
≤
Bf W −1 V T E np
mmax
β
Nmin
Bmax ,
(46)
5
where
where
mmax =
Nmin =
max
i∈I[0,n−1],j∈I[0,p−1]
min
i∈I[0,n−1],j∈I[0,p−1]
[
{
}
mij ,
A1 =
{
Hi =
[
}
Nij ,
{
max
j∈I[0,p−1]
Bj
}
0
Af
0
Bf Hi
.
Since the characteristic equation of the closed-loop system,
say, det (D1 (s)) = 0, has no roots in the compact set ΩR0 =
{s : Re (s) ≥ 0, |s| ≤ R0 }, ∥D1−1 (s) ∥ is bounded for all s ∈
ΩR0 . Combining this fact with (46), we conclude that there exists
a large enough constant N ∗ > 0 such that
(
)
sup
σ D1−1 (s) D∆ (s) < 1,
(47)
is satisfied for all Nij ≥ N ∗ , i ∈ I [0, n − 1] , j ∈ I [0, p − 1]. The
proof is finished.
N∗
∗
The existence of such an
has been proven in Theorem 1, and
regarding the calculation of N , we will give the following remark.
Remark 1. The value of N ∗ (and the corresponding R0 ) can be
computed exactly by using (36), (40), (41), (46) and (47). In practice,
especially when the dimension of the system is high, the computation
of N ∗ may be difficult and the computed N ∗ may be more larger
than the true value. Hence we may try a trial-and-error method
instead to determine N ∗ .
Remark 2. The mechanism of the instability phenomenon and the
safe implementation by adding LPF can be also explained as follows.
Noticing that y(t) = Cx(t), the closed-loop system consisting of (1)
and (8) possesses the following form

p−1
(
)
∑



Bj u t − hj ,
 ẋ(t) = Ax(t) +
j=0
(48)
N
n−1

∑
∑1


F
x
(t
−
τ
)
+
u
(t)
=
G
u
(t
−
r
)
,

i
i
k
k
k=0
[
Ẋ (t) = A1 X (t) +
p−1
∑
Bj X (t − hj )
]
(51)
Af
]
. (52)
Then we can write
Φ = A + BF .
This implies that the existence of the filter gain F is equivalent to
the stabilizability of the matrix pair (A, B).
Proposition 1. (A, B) is stabilizable if and only if (A, B) is.
Proof: Without loss of generality, we assume that (A, B) is stabilizable but not completely controllable. Let rank (Qc (A, B)) = k
with Qc (A, B) being the controllability matrix of (A, B). Then
there exists a nonsingular matrix T ∈ Rn×n such that
T AT −1 =
[
A1
0
A12
A2
]
[
, TB =
B1
0
]
,
(53)
where A1 ∈ Rk×k , A2 ∈ R(n−k)×(n−k) , A12 ∈ Rk×(n−k) , B1 ∈
Rk×r , (A1 , B1 ) is controllable, and A2 is Hurwitz. Then it follows
from (52) and (53) that, for all s satisfying Re{s} ≥ 0,
Hi X (t − τi ) +
]
sIn+r − A B
[
[
] [
] ]
A −B
0
rank sIn+r −
0 0r×r
Ir
[
]
sIn − A B
0
rank
0
sIr Ir
[
]
sIn − A B 0
rank
0
0 Ir


sIk − A1
−A12
B1 0
0
sIn−k − A2 0
0 
rank 
0
0
0 Ir


sIk − A1 B1
−A12
0
0
0 sIn−k − A2 0 
rank 
0
0
0
Ir
[
]
rank sIk − A1 B1 + r + n − k
rank
=
=
=
=
=
j=0
6
−B
Af
A
−Bf
is Hurwitz. Let
[
]
[
]
[
A −B
0
A=
, B=
, F = −Bf
0 0r×r
Ir
r×r
where Hi ∈ R
and Kk ∈ R
, k ∈ I[0, N1 ] are some
constant matrices. By defining a new state vector X (t) =
[xT (t), uT (t)]T , (49) can be expressed as
i=0
Φ=
=
r×n
+
]
0 Bj
,
0 0
]
[
]
0
0
0
, Kk =
.
0
0 Bf Kk
k=0
where Fi ∈ Rr×n and Gk ∈ Rr×r are some constant matrices,
N1 is some integer, and rk , k ∈ I[0, N1 ] are some constants. Obviously, (48) is a coupled differential-difference equations [11], which,
as well-known, is equivalent to some neutral time-delay systems
[12, 23]. In a similar way, the closed-loop system with a LPF in
the control loop (namely, (1) and (16)) can be rewritten as

p−1
(
)
∑



ẋ(t) = Ax(t) +
B j u t − hj ,



j=0

(n−1


∑
u̇ (t) = Af u (t) + Bf
Hi x (t − τi )
(49)

i=0 )



N

∑1


Kk u (t − rk ) ,
+


n−1
∑
[
, Bj =
The stability conclusion stated in Theorem 1 is based on the
assumption that the closed-loop system consisting of (1) and (15) is
asymptotically stable. In the following, we will discuss the design of
the filter gains Af and Bf . It follows from Lemma 2 that the closedloop system consisting of (1) and (15) is asymptotically stable if and
only if the following matrix
Re(s)≥0,|s|≤R0
i=0
]
It is clear that (50) is a retarded time-delay system. Since a retarded
linear time-delay system is robust (in terms of stability) with respect
to small perturbations including round-off errors, implementation by
approximated integration is safe, as proved in Theorem 1.
and
Bmax =
A
0
[
= r + n,
N1
∑
k=0
Kk X (t − rk ),
(50)
which implies that (A, B) is stabilizable. The converse can be shown
in a similar way. The proof is finished.
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
Therefore, it follows from Assumption 1 that there indeed exists
F = [−Bf , Af ] such that A + BF is Hurwitz and the design of F
can be accomplished by using any existing stabilizing methods (for
example, pole assignment) for linear time-invariant systems.
20
x(t)
4
Two illustrative examples
4.1
Stabilization of an oscillation system with DOF
x1 (t)
x2 (t)
10
0
−10
−20
In this part, we give a numerical example to demonstrate the
approach developed in this paper. Consider the following oscillation
system [31]
10
15
20
t(s)
25
30
35
40
0
5
10
15
20
t(s)
25
30
35
40
100
(54)
where ω > 0 and h1 > 0 are constants.
Let x1 (t) = y (t) and x2 (t) = ẏ (t), which represent the position and velocity, respectively. Then (54) becomes
{
ẋ (t) = Ax (t) + B0 u (t) + B1 u (t − h1 ) ,
(55)
y (t) = Cx (t) ,
5
200
u(t)
ÿ (t) + ω 2 y (t) = u (t) + u (t − h1 ) ,
0
0
−100
−200
Fig. 2: State trajectories and control signal of the closed-loop system
without an LPF
where
0
−ω 2
1
0
]
[
, B0 = B1 =
0
1
]
,C =
[
1
0
]
2
(56)
By using (2) and (56) we obtain
B = B0 + e−Ah1 B1 =
[
−
sin(ωh1 )
ω
]T
1 + cos (ωh1 )
−3
For the simulation purpose, we choose ω = π2 rad/s and h1 = 3s. In
the simulations for this example, we choose the Simpson rule for the
numerical implementation of the distributed delay terms and use the
method given in Remark 1 to calculate the value of N ∗ , by which,
the integral step is set as ∆h = (τ1 + h1 ) /N11 = 0.1. For the sake
of simplicity, we choose τ0 = 0 and τ1 = 1 as the delayed output
time, and let the initial conditions be x (0) = [1, 2]T and u(θ) =
0, y(θ) = 0, θ ≤ 0.
In the case of traditional DOF based controller without LPF, the
state feedback gain is designed as F = [−2.123, −1.649], which
means that the eigenvalue set of the closed-loop system is {−1, −2}.
The corresponding simulation result based on numerical implementation is recorded in Fig. 2. We can see that the state trajectories
of the closed-loop system consisting of (55) and (8) diverge. This
result indicates that the numerical implementation of the integral
terms indeed lead to instability of the closed-loop system.
In the case that the LPF is introduced into the control loop,
we let Af = −6 and Bf = [−2.299, −7.069], which are such
that the corresponding augmented system possesses the eigenvalue
set {−1, −2, −3}. The state trajectories and control signals are
recorded in Fig. 3. It follows that the closed-loop system is indeed
asymptotically stable.
Comparison with an existing method
Consider a linear system with a single input delay in the form of (1),
with p = 1, h0 = 1 and
[
]
[
]
[
]
0 1
0
A=
, B0 =
,C = 1 0 .
−2 3
1
It is easy to verify that λ(A) = {1, 2}, namely, the original openloop system (1) is not asymptotically stable. We use two different
0
5
10
15
20
t(s)
25
30
35
40
0
5
10
15
20
t(s)
25
30
35
40
2
u(t)
1
ωh1 = (2k + 1) π.
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
0
−1
−2
.
It is easy to verify that the original open-loop system (55) is not
asymptotically stable and (A, B) is controllable for any ω and h1
except for those satisfying
4.2
x1 (t)
x2 (t)
1
x(t)
[
A=
0
−1
Fig. 3: State trajectories and control signal of the closed-loop system
with an LPF
methods, namely, the approach proposed in this paper and the LPF
based predictor feedback proposed in [30], to design stabilizing controllers for this system. Since both controllers involve distributed
integral terms, we choose the Simpson rule to compute them approximately. For both methods, the integration step is set as ∆h = (τ1 +
h0 )/N10 = 0.02, the initial condition is set as x0 = [1, 2]T and
u(θ) = 0, y(θ) = 0, θ ≤ 0, and the poles of closed-loop systems are
placed at {−1, −2, −3}.
The LPF based predictor feedback controller proposed in [30] is
{
(
)
∫h
ż(t) = Af z(t) + Bf eAh0 x(t) + 0 0 eAθ B0 u(t − θ)dθ ,
u(t) = z(t),
(57)
in which Af = −9, Bf = [12, −36] and the distributed delay terms
are approximated by (7). In the LPF based DOF controller (16), the
LPF gains are designed as Af = −9, Bf = [312.9, −378.1] and the
delayed output time are chosen as τ0 = 0 and τ1 = 1 for simplicity.
The state responses of the closed-loop systems by using these two
different controllers are recorded in Fig. 4. It follows that in both
cases the state trajectories converge to zero asymptotically and the
convergence time are comparable. Notice that different from the state
feedback approach in [30], our approach uses only the output and
delayed output signals for feedback.
7
20
x1 (t)
0
controller (16)
Mondié & Michiels (2003)
−20
−40
−60
0
5
10
t(s)
15
20
50
x2 (t)
0
controller (16)
Mondié & Michiels (2003)
−50
−100
0
5
10
t(s)
15
20
Fig. 4: State trajectories of the closed-loop systems with different
controllers
5
Conclusion
This paper has studied the implementation problem of the DOF
based controller for linear systems with multiple input delays. It was
observed that the stability of the closed-loop system is sensitive to
the round-off errors when the integral terms were implemented by
approximated integration methods. To solve this problem, a generalized low-pass filter was introduced into the control loop and it was
proved that the stability of the closed-loop system can be guaranteed if the numerical implementation of the integral terms is precise
enough. The design of the low-pass filter was converted into the
problem of state feedback stabilization of an augmented linear system without time-delay. The effectiveness of the proposed method
was illustrated by two numerical examples.
6
Acknowledgements
This work was supported in part by the National Natural Science
Foundation of China under grant numbers 61273028, 61322305 and
61573120, by the Fok Ying Tung Education Foundation under grant
number 151060, and by the Foundation for the Author of National
Excellent Doctoral Dissertation of China under Grant 201343.
7
References
1 Artstein, Z.: ‘Linear systems with delayed controls: A
reduction’, IEEE Trans. Autom. Control, 1982, 27, (4),
pp. 869-879
2 Bothra, S., Rogers, B., Kellam, M., Osburn, C.M.: ‘Analysis of the effects of scaling on interconnect delay in ulsi
circuits’, IEEE Trans. Electron Devices, 1993, 40, (3),
pp. 591-597
3 Bresch-Pietri, D., Krstic, M.: ‘Delay-adaptive predictor feedback for systems with unknown long actuator
delay’, IEEE Trans. Autom. Control, 2010, 55, (9), pp.
2106-2112
4 Cardelli, M., Boudey, J.: ‘Some Necessary and Sufficient
Conditions of Stability on the Approximation of a Distributed Delay Control Law’, IFAC Proc. Vol., 2008, 41,
(2), pp. 2619-2624
5 Chen, J., Fu, P., Niculescu, S.I., Guan, Z.: ‘An eigenvalue
perturbation approach to stability analysis, part ii: when
will zeros of time-delay systems cross imaginary axis?’,
SIAM J. Control Optim., 2010, 48, (8), pp. 5583-5605
8
6 Engelborghs, K., Dambrine, M., Roose, D.: ‘Limitations
of a class of stabilization methods for delay systems’,
IEEE Trans. Autom. Control, 2001, 46, (2), pp. 336-339
7 Fridman, E.: ‘A refined input delay approach to sampleddata control’, Automatica, 2010, 46, (2), pp. 421-427
8 Gantmacher, F.R.: ‘The theory of matrices’, (Chelsea
Publishing Co., 1959), pp. 76-89
9 García-Gabín, W., Zambrano, D., Camacho, E.: ‘Sliding
mode predictive control for chemical process with time
delay’, IFAC Proc. Vol., 2005, 38, (1), pp. 627-632
10 Gu, K.: ‘A review of some subtleties of practical relevance for time-delay systems of neutral type’, ISRN
Applied Mathematics, 2012, pp. 1-46
11 Gu, K.: ‘Stability problem of systems with multiple delay
channels’, Automatica, 2010, 46, (4), pp. 743-751
12 Gu, K., Liu, Y.: ‘Lyapunov-Krasovskii functional for
uniform stability of coupled differential-functional equations’, Automatica, 2009, 45, (3), pp. 798-804
13 Hao, S.L., Liu, T., Paszke, W., Galkowski, K.: ‘Robust
iterative learning control for batch processes with input
delay subject to time-varying uncertainties’, IET Control
Theory Appl., 2016, 10, (15), pp. 1904-1915
14 Hale, J.K., Lunel, S.M.V.: ‘Introduction to functional differential equations’, (Springer-Verlag, New York, 2013),
pp. 17-25
15 Huang, L., Xie, X., Tan, C.: ‘Improved stability criteria
for TĺCS fuzzy systems with time-varying delay via convex analysis approach’, IET Control Theory Appl., 2016,
10, (15), pp. 1888-1895
16 Karafyllis, I.: ‘Stabilization by means of approximate
predictors for systems with delayed input’, IFAC Proc.
Vol., 2010, 43, (2), pp. 9-14
17 Kojima, A., Uchida, K., Etsujiro S., Shintaro I.: ‘Robust
stabilization of a system with delays in control’, IEEE
Trans. Autom. Control, 1994, 39, (8), pp. 1694-1698
18 Krstic, M.: ‘Delay compensation for nonlinear, adaptive,
and pde systems’, (Springer, Birkhäuser Boston, 2009),
pp. 17-31
19 Li, Z.Y., Zhou, B., Lin, Z.L.: ‘On exponential stability of
integral delay systems’, Automatica, 2013, 49, (11), pp.
3368-3376
20 Liu, Q., Zhou, B.: ‘Delay compensation of discrete-time
linear systems by nested prediction’, IET Control Theory
Appl., 2016, 10, (15), pp. 1824-1834
21 Liu, X.: ‘Stability of perturbed switched homogeneous
systems with delays and uncertainties’, IET Control Theory Appl., 2016, 10, (6), pp. 684-691
22 Manitius, A., Olbrot, A.: ‘Finite spectrum assignment
problem for systems with delays’, IEEE Trans. Autom.
Control, 1979, 24, (4), pp. 541-552
23 Martinez-Amores, P.: ‘Periodic solutions for coupled
systems of differential-difference and difference equations’, Annali di matematica pura ed applicata, 1979,
121, (1), pp. 171-186
24 Mathews, J.H.: ‘Numerical methods for mathematics,
Science and Engineering’, (Prentice-Hall, Englewood,
1992)
25 Medvedev, A.V., Toivonen, H.T.: ‘Feedforward timedelay structures in state estimation: finite memory
smoothing and continuous deadbeat observers’, IEE
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
26
27
28
29
30
31
Proceedings-Control Theory Appl., 1994, 141, (2), pp.
121-129
Medvedev, A.V., Toivonen, H.T.: ‘Continuous-time deadbeat observation problem with application to predictive
control of systems with delay’, Kybernetika, 1994, 30,
(6), pp. 669-688
Medvedev, A.V.: ‘Continuous least-squares observers
with applications’, IEEE Trans. Autom. Control, 1996,
41, (10), pp. 1530-1537
Medvedev, A.V.: ‘Disturbance attenuation in finitespectrum-assignment controllers’, Automatica, 1997, 33,
(6), pp. 1163-1168
Medvedev, A.V.: ‘State estimation and fault detection
by a bank of continuous finite-memory filters’, Int. J.
Control, 1998, 69, (4), pp. 499-517
Mondié, S., Michiels, W.: ‘Finite spectrum assignment of
unstable time-delay systems with a safe implementation’,
IEEE Trans. Autom. Control, 2003, 48, (12), pp. 22072212
Popescu, D.: ‘On the implementation of distributed
delay control laws’, International Symposium on System Theory-SINTES-11, October, Craiova, 2003, pp.
973-8043
IET Control Theory Appl., pp. 1–9
c The Institution of Engineering and Technology 2015
⃝
32 Smith, O.J.: ‘A controller to overcome dead time’, ISA
Journal, 1959, 6, (2), pp. 28-33
33 Santos, O., Mondié S.: ‘Control laws involving distributed time delays: robustness of the implementation’,
Proc. Am. Control Conf., Chicago, illinois USA, June
2000, pp. 2479-2481
34 Van Assche, V., Dambrine, M., Lafay, J.F., Richard,
J.P.: ‘Some problems arising in the implementation of
distributed-delay control laws’, Proc. 38th IEEE Conf.
Decis. Control, Phoenix, Arizona, USA, December 1999,
pp. 4668-4672
35 Wu, Z. G., Park, J. H., Su, H., Chu, J.: ‘Discontinuous Lyapunov functional approach to synchronization of
time-delay neural networks using sampled-data’, Nonlinear Dynamics, 2012, 69, (4), pp. 2021-2030
36 Zhou, B., Egorov, A. V. : ‘Razumikhin and Krasovskii
stability theorems for time-varying time-delay systems’,
Automatica, 2016, 71, pp. 281-291
37 Zhou, B.: ‘Truncated predictor feedback for time-delay
systems’, (Springer-Verlag, Berlin, 2014), pp. 11-15
38 Zhou, B.: ‘Input delay compensation of linear systems
with both state and input delays by nested prediction’,
Automatica, 2014, 50, (5), pp. 1434-1443
9