1. No category

Systems & Control Letters 62 (2013) 837–844
Contents lists available at ScienceDirect
Systems & Control Letters
journal homepage: www.elsevier.com/locate/sysconle
Truncated predictor feedback control for exponentially unstable
linear systems with time-varying input delay✩
Se Young Yoon ∗ , Zongli Lin
Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743, USA
article
info
Article history:
Received 16 April 2012
Received in revised form
22 March 2013
Accepted 31 May 2013
Available online 21 July 2013
Keywords:
Truncated predictor feedback
Exponentially unstable system
Time-varying delay
abstract
The stabilization of exponentially unstable linear systems with time-varying input delay is considered
in this paper. We extend the truncated predictor feedback (TPF) design method, which was recently
developed for systems with all poles on the closed left-half plane, to be applicable to exponentially
unstable linear systems. Assuming that the time-varying delay is known and bounded, the design
approach of a time-varying state feedback controller is developed based on the solution of a parametric
Lyapunov equation. An explicit condition is derived for which the stability of the closed-loop system with
the proposed controller is guaranteed. It is shown that, for the stability of the closed-loop system, the
maximum allowable time-delay in the input is inversely proportional to the sum of the unstable poles in
the plant. The effectiveness of the proposed method is demonstrated through numerical examples.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
The control of most dynamic systems in the real world is affected by time-delays, which degrade the closed-loop performance
and stability characteristics. The introduction of economical and
robust digital controller implementations broadens the reach of
control theory to many modern industrial applications and allows
the development of remotely controlled and complex networked
systems. However, with the added complexity and the time required to complete the digital computations and communications
in the control loop, many applications need to deal with substantial time-delays. A straightforward approach to dealing with delays
in control systems is to treat it as a stability robustness problem.
In such an approach, the information of the delay is typically not
used in the design of the controller. However, the difficult problem
remains of establishing the conditions for stability and the corresponding bound on the allowable delays. This difficulty is more
evident for multiple input multiple output systems, for which
the concept of gain/phase margin becomes indefinite. As a result,
methods such as the predictor feedback, which explicitly use the
delay information to design the stabilizing controller, have been
widely explored in the literature.
The control of linear and nonlinear systems with time-delays
has been a topic of extensive research, where [1–3] and all other
✩ This work is supported in part by the National Science Foundation under grant
CMMI-1129752.
∗ Corresponding author. Tel.: +1 434 242 7480.
E-mail addresses: [email protected], [email protected] (S.Y. Yoon),
[email protected] (Z. Lin).
0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.sysconle.2013.05.013
references cited in this section are only a small sample of the
available literature on this topic. The stabilization of a linear
oscillator system was explored in [4,5]. The stabilization of a
delayed chain of integrators was discussed in [6,7]. One extensively
explored method that has proven to be efficient in dealing with
time-delays is the predictor feedback. Although much of the
work in this field has focused on systems with constant timedelays as found in [8,9], research on the stabilization of systems
with time-varying delays has been very active since the work of
Artstein in [10]. A small sample of the work found in the literature
on the stabilization of linear and nonlinear systems with timevarying delays can be found in [11–14]. When the time-delays are
unknown, adaptive predictor feedback methods can be developed
as in [15,16], for linear and nonlinear systems, respectively.
Lin and Fang in [13] developed a low gain feedback approach
to the stabilization of a class of linear systems with constant input delays. By using an eigenstructure assignment based low gain
feedback design [17], the authors of [13] show that a stabilizable
and detectable linear system with an arbitrarily large time-delay
in the input can be asymptotically stabilized either by linear state
feedback or by linear output feedback as long as the open-loop system is not exponentially unstable. A salient feature of this low gain
design is that it takes the structure of a predictor feedback control law but with the distributed portion of the predictor feedback
control dropped, and hence the resulting feedback law is of a finite dimension. A simple example was also constructed in [13] to
show that such a result would not be true if the open-loop system is exponentially unstable. Another by-product of this low gain
feedback design is that, with no additional conditions, the resulting
linear feedback laws would also semi-globally asymptotically stabilize such systems when they are also subject to input saturation.
838
S.Y. Yoon, Z. Lin / Systems & Control Letters 62 (2013) 837–844
The low gain feedback design approach proposed in [13] was
further developed in [14], where a parametric Lyapunov equation
based low gain feedback design was developed and the design
method is termed ‘‘truncated predictor feedback (TPF)’’. In addition, time-varying delays are allowed.
In this paper we revisit the problem of asymptotically stabilizing a linear system with time-varying bounded input delay. The
truncated predictor feedback (TPF) controller developed in [14]
is extended to the general linear systems with poles in the open
right-half plane. An explicit condition is established that guarantees the global asymptotic stability of the closed-loop system. We
will see that for the special case where the system poles are contained in the closed left-half plane, the developed feedback law and
the stability condition reduce to the same results presented in [14],
and the upper bound on the delay function can be arbitrarily large.
On the other hand, if the considered system is exponentially unstable, the maximum of the delay function bound is inversely proportional to the sum of the unstable poles.
By extending the results in [14] to exponentially unstable systems, this paper offers the following contributions.
1. It establishes a systematic method to design a stabilizing control law for exponentially unstable systems with time-varying
input delay. Many real-world control applications such as magnetic levitation, flow stabilization and flight control, involve exponentially unstable systems.
2. The results in this paper provide an insight into how the original TPF control is affected when the limitation in the open-loop
poles is removed. In particular, we investigate how the presence
of exponentially unstable poles constrains the delay tolerable
by the closed-loop system.
The extension of the TPF controller to exponentially unstable systems is not straightforward, and many of the simplifying assumptions employed in the above mentioned papers do not apply for
systems with eigenvalues in the open right-half plane. However,
by manipulating the structure and utilizing the intricate properties
of the state space matrices, we were able to demonstrate that exponentially unstable systems with time-varying input delays can
be stabilized using a similar controller design procedure as in [14].
The remainder of this paper is organized as follows. The control
problem to be studied in this paper is defined in Section 2. Preliminary results necessary for presenting our main results are included
in Section 3. Section 4 contains the main results of this paper for the
state feedback case, and Section 5 demonstrates the case of output
feedback. Numerical examples are included in Section 6 to verify
the theoretical derivation. Finally, Section 7 draws the conclusion
to this paper.
2. Problem definition
where x(t ) ∈ R is the state vector and u(t ) ∈ R is the
input vector, and the pair (A, B) is assumed to be controllable.
The time-varying delay function φ(t ) : R+ → R is assumed to
be exactly known, continuously differentiable and invertible, with
d
φ(t ) > 0 for all t > 0 [11,14]. Here, we define the delay function
dt
to have the standard form
m
φ(t ) = t − D(t ),
(2)
¯
for a bounded function D(t ) : R+ → R+ , where 0 ≤ D(t ) ≤ D.
Without loss of generality, we also assume that the state matrix
in (1) is structured as,

A−
A = blkdiag {A1 , A2 , . . . , Al } =
0

0
,
A+
−1
−1
u(t ) = −BT PeA(φ (t )−t ) x(t ),
AT P + PA − PBBT P = −γ P ,
(4)
(5)
where γ > 0. Differently from the derivation in [14], the solution
to (5) may not be strictly positive definite because A is allowed to
have eigenvalues in the open right-half plane. The parameter γ is
related to the minimum rate of decay of the closed-loop system.
The role of γ and the condition for P > 0 are discussed in detail
in [20].
In order to simplify the notation, we define the following
matrices. Let the matrix

A˜ = A +
γ 
2
I .
(6)
Because of the assumed structure of matrix A in (3), A˜ is also a block
diagonal matrix,

0
.
A˜ +
(7)
As before, the first block A˜ − ∈ Rp˜ טp contains all the stable eigenvalues in the open left-half plane max{Re(λ(A˜ − ))} < 0, and
A˜ + ∈ Rq˜ טq has all eigenvalues in the closed right-half plane
min{Re(λ(A˜ + ))} ≥ 0. The diagonal blocks A′− ∈ Rp˜ טp and A′+ ∈
Rq˜ טq of the state matrix A are defined, such that
γ
A˜ − = A′− + I ,
2
γ
and A˜ + = A′+ + I .
2
(8)
A′−
It is important to notice here that
may not be equal to A− for
γ > 0, but the eigenvalues of A′− are always in the open left-half
plane and p˜ ≤ p. The input matrix B in (1) is partitioned accordingly as

(3)
∀t ≥ 0 .
The semi-positive definite matrix P is the solution to the parametric algebraic Riccati equation (ARE)
0
(1)
n
K eA(φ (t )−t ) x(t ), for all t ≥ 0.
The low gain feedback approach was explored in [13], based on
the eigenstructure assignment developed in [17], to determine the
feedback gain K for the stabilization of a delayed linear system
with all poles in the closed left-half plane. A simpler approach
using a parametric Lyapunov equation based design method was
developed in [14,19] for the same type of linear systems. In this
paper, we extend the results presented in [14] for systems with
poles in the open right-half plane.
The structure of the TPF type controllers proposed in this paper
is given as

˜
˜A = A−
Consider a linear time-invariant system with input delay,
x˙ (t ) = Ax(t ) + Bu (φ (t )) ,
where each block Ai for i = 1 to l contains the eigenvalues of A with
an equal real part. The existence of a coordinate transformation
to obtain the system realization in (3) was demonstrated in [17,
18], which include discussions on how to obtain the corresponding
transformation matrices. We further assume that the blocks are
ordered such that Re(λ(A1 )) ≤ Re(λ(A2 )) ≤ · · · ≤ Re(λ(Al )).
Therefore, matrix A can be divided into the A− ∈ Rp×p block with all
the eigenvalues in the open left-half plane, i.e., max{Re(λ(A− ))} <
0, and the A+ ∈ Rq×q block with all the eigenvalues in the closed
right-half plane, i.e., min{Re(λ(A+ ))} ≥ 0.
The predictor feedback is an approach used in the stabilization of a delayed system, where the delay in the system is compensated by predicting the future trajectory of the states from the
system equations and initial conditions. In what is known as the
‘‘truncated predictor feedback’’ (TPF) approach, the state prediction is simplified by eliminating the input dependent term from
the computation of the stabilizing control law. As a result, the
controller equation for the feedback gain K simplifies to u(t ) =

B−
B=
,
B+
(9)
S.Y. Yoon, Z. Lin / Systems & Control Letters 62 (2013) 837–844
where B− ∈ Rp˜ ×m and B+ ∈ Rq˜ ×m . With the matrices as defined in
(7) and (9), the ARE in (5) can be rewritten as
A˜ T P + P A˜ − PBBT P = 0.
(10)
839
for an arbitrary t ≥ 0, and a positive scalar ω such that

ω≥2
tr A˜ +

− 1.
γ
(20)
Furthermore, for the special case where P is positive definite, the
Riccati equation in (10) can be transformed into the Lyapunov
equation [20]. The advantage of the Lyapunov equation over (10)
is that the matrix equation becomes linear with respect to the unknown positive definite matrix.
Proof. Let ω be a positive scalar, P be the solution to the parametric
ARE in (10), and the matrix Q (ω) be defined as
3. Preliminary results
Then, it was shown in [19] that
In this section we present some properties of the solution to
the ARE (10), as well as some basic theories for time-delay systems
that will be valuable in establishing our main results. The first two
lemmas we will introduce are extensions of the results presented
in [14,21,22] on a system with all poles on the imaginary axis to a
general time-invariant linear system (1).
eA t PeA
Lemma 1. Given matrices A˜ and B as defined in (7) and (9), the
parametric ARE in (10) has a positive semidefinite solution P ≥ 0
in the form of


0
0
P =
0
,
P+
(11)
(12)
Additionally, it follows that
tr BT PB = 2 tr (A˜ + ),


(13)
(14)
Proof. The first part of the proof is straightforward and involves
substituting the P in (11) into (10), which gives the same
expression as in (12). The existence and uniqueness of P+ has been
established in [20]. Additionally, it was demonstrated in [20] that,
with A˜ + as defined in (7), the solution to the Riccati equation (12)
is positive definite.
The second part of the proof is obtained from the ARE in (12)
after multiplying both sides of the equality to the right by the
inverse of P+ . Then, by the properties of the trace operation, it
follows that

T
tr B+ P+ B+



= 2 tr A˜ + ,
(15)
and it follows from [14] that


P+ B+ BT+ P+ ≤ 2 tr A˜ + P+ .
(16)
Also, because of the structure of P, it holds that


tr BT PB = 2 tr A˜ + ,


(17)
and


PBBT P ≤ 2 tr A˜ + P ,
(18)
which completes the proof.
Lemma 2. Assume that we are given the state space system as defined
in (1), where the pair (A, B) is controllable, and P is the solution (11) to
the parametric ARE in (10). Then, it holds that
T
eA t PeA
Tt
≤ eωγ t P ,
Tt
− eωγ t P = −eωγ t
t

T
e−ωγ s eA s Q (ω)eAs ds
(22)
0
is true for all t ≥ 0. Additionally, the right-hand side of the equality
is greater than or equal to zero if


Q (ω) ≥ (ω + 1)γ − 2 tr A˜ +

P ≥ 0.
(23)
Since P ≥ 0, the above inequality is satisfied if

ω≥2
tr A˜ +

− 1.
γ
(24)
Lemma 3 ([23]). For any positive definite matrix P > 0, any scalars
γ1 and γ2 such that γ2 ≥ γ1 , and a vector valued function x :
[γ1 , γ2 ] → Rn such that the integrals in the following are well defined,

γ2
γ1
where ‘‘ tr’’ represents the trace of a matrix, and
PBBT P ≤ 2 tr (A˜ + )P .
T
(21)
Thus, the inequality in (19) holds if (24) is satisfied.
where P+ > 0 is the unique positive definite solution to
A˜ T+ P+ + P+ A˜ + − P+ B+ BT+ P+ = 0.
Q (ω) = γ P + ωγ P − PBBT P .
(19)
 
xT (s)ds P
≤ (γ2 − γ1 )
γ1
γ2

γ2
γ1
x(s)ds

xT (s)Px(s)ds.
(25)
4. State feedback control
The linear system with time-delay (1) can be written in
subsystems
x− (t ) = A′− x− (t ) + B− u(φ(t )),
(26)
x+ ( t ) =
(27)
A′+ x+ (t )
+ B+ u(φ(t )),
A′−
where all the eigenvalues of
are in the open left-half plane,
and some eigenvalues of A′+ can be in the open right-half plane.
Additionally, due to the assumed structures of the matrices P , A
and B, the TPF control input only depends on the trajectory of x+ (t ),
′
−1
u(t ) = B+ P+ eA+ (φ (t )−t ) x+ (t ).
(28)
Based on the TPF control input given above and how the linear
system is structured in (26) and (27), we observe that the subsystem in (26) is asymptotically stable. The size of the matrix A′− may
vary with the value of γ as defined in (8), but its eigenvalues are
always contained in the open left-half plane. The input to the subsystem (26) is a feedback law of the states of (27). Therefore, the
input u(t ) is an external signal to (26), which is bounded and converging when (27) is stabilized. This implies that the full system
will be asymptotically stable if we establish the asymptotic stability of (27).
Theorem 1. Consider the linear system (1) with the corresponding
subsystems in (26) and (27). If there exist δ > 0 and δ > 0 such that



δ 1 − nδ eδ eδ − 1 > 2 tr (A+ ) D¯
(29)
840
S.Y. Yoon, Z. Lin / Systems & Control Letters 62 (2013) 837–844
holds for δ ∈ (δ, δ), then there exist γ > 0 and γ > 0 such that

 
tr A˜ + γ
=
δ
,
¯
2D


tr A˜ + (γ ) =
δ
¯
2D
.
¯ ) ≥ t − 2D.
¯ Thus, under the condition
As noted in [14], φ(t − D
that
V (x+ (t + θ )) < ηV (x+ (t )) ,


∀θ ∈ −2D¯ , 0 ,
Furthermore, for any γ ∈ (γ , γ ) the TPF control in (4) asymptotically
stabilizes the delayed system (1).
for t ≥ φ (φ
sion in (34) as
Proof. The proof of this theorem is similar to the results presented
in [14], except for the modifications introduced in Lemmas 1 and
2. Because of the similarities, we will only present the steps in
the proof that are critical to arriving at our results. To simplify the
notation, we will denote K = −BT P.
Given the linear time-invariant system (1) with an arbitrary
initial condition x(t ) = ϕ(t ), t ∈ [φ(0), 0], and the TPF control in
(4), it was demonstrated in [14] that the closed-loop system states
are bounded for all time t ∈ [0, φ −1 (φ −1 (0))). Thus, the stability
can be established by considering t ≥ φ −1 (φ −1 (0)).
The state trajectory at time t of the time-delay system in (1),
with the TPF control (4) and initial condition x(φ(t )), can be found
explicitly from the system equation. By substituting this explicit
solution back into (4), we obtain
 2
¯ ωγ D¯ η
λT (t )P λ(t ) ≤ 4 tr A˜ + De
 t
eωγ (t −s) dsV (x+ (φ(s))) .
×
u (φ(t )) = K

x(t ) −
t


φ(t )
eA(t −s) BK eA(s−φ(s)) x (φ(s)) ds .
x˙ (t ) = (A + BK )x(t )
t
− BK
φ(t )
= (A + BK ) x(t ) − BK λ(t ).
(31)
Although the Lyapunov function only depends on the trajectory
of x+ (t ), we will later find that the expression in (31) is more
convenient for including the case when the dimensions of A˜ + and
x+ (t ) change with the value of γ . Therefore, (31) will be used
throughout the remainder of this proof. The time derivative of
the Lyapunov function along the trajectory of the system (1) is
expressed as
V˙ (x+ (t )) ≤ −γ V (x+ (t )) + λT (t )PBBT P λ(t ),


≤ −γ V (x+ (t )) + 2 tr A˜ + λT (t )P λ(t ),
(32)
where the properties derived in Lemma 1 were used.
Next, we define the scalar parameter ω = 2 tr (A˜ + )/γ . Referring to Lemma 3, we can simplify the term λT (t )P λ(t ) in (32) as,
T (t −s)
× eA
t
φ(t )
The substitution of the inequality in (36) into (32) results in

 3
˙V (x+ (t )) ≤ − γ − 8 tr A˜ + De
¯ ωγ D¯ η

t

×
¯
t −D
eωγ (t −s) ds V (x+ (φ(s))) .
(37)
Here we notice that there are sufficiently small values of η > 1 and
ϵ > 0, such that
(38)
if it holds that


 3
¯ ωγ D¯ 1 eωγ D¯ − 1 > 0.
γ − 8 tr A˜ + De
ωγ
(39)
(40)
PeA(t −s) BBT PeA(s−φ(s)) x (φ(s)) ds.
 2
¯ ωγ D¯
λT (t )P λ(t ) ≤ 4 tr A˜ + De
 t
×
eωγ (t −s) V (x+ (φ(s))) ds.
and the inequality in (39) simplifies to
 


γ − 2 tr A˜ + δ eδ eδ − 1 > 0.

γ ≥2
tr A˜ +
n

−2
tr (A+ )
n
.
(42)
Then, the inequality in (41) will be satisfied if



δ 1 − nδ eδ eδ − 1 > 2 tr (A+ ) D¯ .
(34)
(43)
From observing the above expression, we can deduce that the lefthand side of the inequality in (43) is a concave function of δ . Therefore (38) holds if there exist δ and δ such that (43) is true for all
δ ∈ (δ, δ), and the delayed input system is asymptotically stable by
the Razumikhin Stability Theorem [24] and the assumption in (35).
Next we demonstrate that if (43) is satisfied for some δ , then it
is always possible to find a γ > 0, such that
2
(33)
(41)
In order to eliminate γ on the left-hand side of the above inequality, we observe that
 

γ 
δ = 2 tr A˜ + D¯ = 2 tr A′+ + I D¯ .
T
xT (φ(s)) eA (s−φ(s)) PBBT
The above inequality can be further simplified by employing the
results of Lemmas 1 and 2, and by making use of the upper bound
information of D(t ),
¯
t −D
(36)
¯
t −D
(30)
V (x+ (t )) = xT+ (t )P+ x+ (t ) = xT (t )Px(t ).

(0)) and some η > 1, we can simplify the expres-
 
δ = ωγ D¯ = 2 tr A˜ + D¯ ,
As mentioned earlier, the subsystem (26) is asymptotically stable. Consider the following Lyapunov function for the subsystem (27)
λT (t )P λ(t ) ≤ (t − φ(t ))
(35)
We define a new variable δ as
eA(t −s) BK eA(t −φ(s)) x (φ(s)) ds,
def
−1
V˙ (x+ (t )) ≤ −ϵ V (x+ (φ(s))) ,
With the above expression as the input to the system equation in
(1), the closed-loop state equation becomes,

−1
(44)
Consider the case where (43) is satisfied. In this case, there exists
a δ such that



δ 1 − nδ eδ eδ − 1 = 2 tr (A+ ) D¯ .
(45)
We assume that this δ is outside the range of γ > 0 given by (44).
Since tr (A˜ + ) is a continuous and nondecreasing function of γ with
limγ →∞ tr (A˜ + ) = ∞, the only possibility is that
 
δ < 2 tr A′+ D¯ = 2 tr (A+ ) D¯ .
(46)
S.Y. Yoon, Z. Lin / Systems & Control Letters 62 (2013) 837–844
Then, it follows that
δ
The truncated predictor output feedback law is constructed as
δ
δ < δ 1 − nδ e e − 1 .



(47)
Simplifying the above inequality, we conclude that the following
must hold,
nδ e
δ

δ
e − 1 < 0,

(48)
which is a contradiction since the left-hand side of the above inequality is positive for all δ > 0. Therefore, there is a γ > 0 such
¯ = δ.
that 2 tr (A˜ + )D
Remark 1. Consider the stability condition in (29). We observe
from the right-hand side of (29) that the existence of a stabilizing
¯
TPF controller for (1) depends on the product of the delay bound D
and the sum of the unstable poles of the plant, that is, the bound
on the allowable input delay is inversely proportional to the sum
of the unstable poles of (1). Thus, the stability of the closed-loop
system results from the trade-off between the magnitude of the
unstable poles in the plant and the maximum delay in the input
signal. This relationship may also be deduced from the fact that
the distributed term in the classical prediction feedback may not
be negligible when one or more open-loop poles are in the open
right-half plane, limiting the acceptable prediction horizon of the
TPF control. Indeed, if the eigenvalues of matrix A in (1) are all in
the closed left-half plane, then the trace of A+ equals zero, and
(43) becomes the same as the condition developed in [14], where
ω = n − 1.
¯ in the stability
Remark 2. The relationship between γ and D
conditions (43) and (44) can be inverted numerically to obtain the
maximum delay for a given gain γ . Given a gain γ , the maximum
¯ can be found from the stability conditions by solving,
D


eγ ω D eγ ω D − 1 >
¯
¯
841
1
nγ ω

1−
2 tr (A+ )
γω

.
(49)
This can be helpful in combining the TPF control with other
design methods for satisfying additional stability and performance
¯ that the closed-loop
objectives. On the other hand, the maximum D
system under the TPF control can tolerate for some γ > 0 can be
found directly from (29). The left-hand side of (29) is a concave
function for δ > 0, thus a global maximum exists for



δ 1 − nδ e δ e δ − 1
f (δ) =
,
2 tr (A+ )
¯ resulting
at some δ = δ ∗ > 0, and f (δ ∗ ) equals the maximum D
from the TPF control law.

x˙ˆ (t ) = Axˆ (t ) + Bu(φ(t )) + L(y(t ) − C xˆ (t )),
−1
u(t ) = −BT PeA(φ (t )−t ) xˆ (t ),
(51)
∀t ≥ 0,
where xˆ is the estimate of the state vector in (1), the positive semidefinite matrix P is the solution of the ARE in (5), and the observer
gain L ∈ Rn×r is selected such that all the eigenvalues of (A − LC )
are in the open left-half plane. The condition for stability of the
delayed system with the above output feedback TPF control can be
readily derived using the results in Theorem 1.
Theorem 2. Consider the time-delay system in (1). Assume that the
pair (A, B) is controllable and the pair (A, C ) is detectable. If (A − LC )
¯
is Hurwitz,
 and there exist δ and δ such that (29) is satisfied for
¯ δ , then there exist γ¯ > 0 and γ > 0 such that the
δ ∈ δ,
closed-loop system under the output
TPF control in (51) is
 feedback

asymptotically stable for all γ ∈ γ¯ , γ .
Proof. Define the observer error vector as
e(t ) = x(t ) − xˆ (t ).
(52)
We can rewrite the time-delay system in (1) with the output
feedback TPF control law as,
x˙ (t ) =

Ax(t ) + BK eA(t −φ(t )) x(φ(t ))

− BK eA(t −φ(t )) e(φ(t )),
(53)
e˙ (t ) = (A − LC ) e(t ),
(54)
T
where K = B P.
We observe that the subsystem (53) is the closed-loop system of
(1) under the state feedback TPF control (4) and in the presence of
an external input signal as a function of the error e(t ). Furthermore,
this external input is bounded and converges to zero exponentially
since A − LC is Hurwitz. Therefore, the delayed system under
the output feedback TPF control is asymptotically stable if (53) is
asymptotically stable in the absence of e(t ).
 
¯ δ ,
Since δ¯ and δ are such that (29) is satisfied for δ ∈ δ,
then, by Theorem 1, there exist γ¯ > 0 and γ > 0 such that the


subsystem (53) is asymptotically stable for all γ ∈ γ¯ , γ .
Remark 3. The proof of Theorem 2 determines the stability of the
delayed system by a straightforward coordinate transformation. A
more direct proof of stability, which results in a Lyapunov function,
can be carried out as in [14].
A stability condition for the delayed system (1) with TPF control
was derived in this section. The existence of a stabilizing TPF
controller when the above mentioned condition is satisfied was
also demonstrated here. The condition for stability in (43) is
relatively easy to test numerically because it only has one variable
γ and the left-hand side of the inequality is a concave function.
In this section we have presented the design of an output
feedback controller for systems with time-varying input delay.
The design of similar stabilizing controllers for systems with
time-varying output delays does not come straightforwardly from
the presented results, and it requires an extensive discussion on
observer design for systems with delayed inputs/outputs. Further
discussion on this topic can be found in [25–29] and the references
therein.
5. Output feedback control
6. Numerical examples
In this section, we extend the state feedback results to the case
where the TPF controller is based on the output signal of the timedelay system. Let the output of the time-delay system in (1) be
defined as
We consider the case of a delayed double oscillator system, as
considered in the numerical examples in [14], with a positive real
pole added to the plant equation. The resulting state space matrices
A and B as in (1) are given by
y(t ) = Cx(t ),
C ∈ R r ×n .
(50)
In addition to the controllability of the pair (A, B) and the matrix
structures described in (3), (7) and (9), we further assume that the
pair (A, C ) is detectable.

p
0

A = 0
0
0
1
0
−ω
0
0
0
ω
0
0
0
0
0
1
0
−ω

0
0

0 ,
ω
0
 
0
0
 
B = 0 .
0
1
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S.Y. Yoon, Z. Lin / Systems & Control Letters 62 (2013) 837–844
(a) Case 1.
(b) Case 2.
Fig. 1. Time-varying delay function D(t ) = t − φ(t ) and the corresponding signal φ −1 − t used in the control signal.
(a) x1 (t ) and x2 (t ).
(b) x3 (t ), x4 (t ) and x5 (t ).
Fig. 2. Time response of the system states corresponding to the first case of the delay function φ(t ).
The scalar p = 0.1 represents the location of the unstable real
pole, and ω = 1 locates the resonance frequency of the double
oscillator. We assume that the initial condition of the states is given
¯ , 0 ].
by x(θ) = [−1, 2, 2, −1, 2]T , for all θ ∈ [−D
Two time-delay functions φ(t ) are considered here, which are
similar to the examples presented in [12,14]. The upper bounds of
the delay functions were reduced, in order to accommodate the
additional constraints introduced by the open right-half plane pole
in the plant.
Fig. 3. Original and delayed control signals of the closed-loop system corresponding to the first case of the delay function φ(t ).
6.1. Delay function case 1
In this first case, the time-varying delay function is given by
φ(t ) = t − 0.3
t +1
2t + 1
,
∀t ≥ 0.
(55)
Both the delay D(t ) and the inverse delay signal φ −1 (t ) − t used
in the TPF control (4) are plotted in Fig. 1. Here we clearly observe
¯ = 0.3 s.
that the upper bound of the delay signal D(t ) is D
¯ is found, it is possible to determine
Once the value of D
from (44) and (43) the range of γ for which the stability of the
delayed closed-loop system is guaranteed. In this case, the stability
condition for the delayed system with the TFP control is satisfied if
0 < γ < 0.17. For this example we select a value of γ = 0.15.
The time response of the states for the closed-loop system is
shown in Fig. 2. The figure shows that initially the amplitude of
the system oscillatory response increases due to the positive real
pole in the plant. Eventually, we observe that the closed-loop states
asymptotically approach zero. The closed-loop control signal u(t )
and the delayed input to the plant u(φ(t )) are shown in Fig. 3.
In the second case, the inverse of the delay function is specified
as,

φ (t ) = ρ(t ) = t + d 1 +
6.3. Improvement in delay compensation with TPF
In this subsection, the TPF controller in (4) is compared to the
state feedback control
u(t ) = Kx(t ).
6.2. Delay function case 2
−1
where the scalar d = 0.3. Fig. 1 illustrates the time evolution of
the delay function D(t ), and the inverse delay signal φ −1 (t ) − t
employed in the TPF control signal. We can observe that the delay
¯ = 0.45 s.
signal in this case is oscillatory, with an upper bound of D
Once again, we find the range of γ such that the stability
¯ In this case, we observe
condition in (43) is satisfied for the given D.
that the stability condition is satisfied if 0.002 < γ < 0.092. For
this example, we select γ = 0.09.
The time response of the closed-loop states with the TPF
controller is shown in Fig. 4. The same as in the previous case,
the amplitude of the state oscillation increases initially, but the
states are later brought back to zero asymptotically. The closedloop control signal u(t ) and the delayed input to the plant u(φ(t ))
are shown in Fig. 5.
1
2

cos(t ) ,
(56)
(57)
The feedback gain K = −BT P is taken to be the same as in the
example in Case 2 of this section for both the TPF control and (57).
The purpose of this comparison is to examine the contribution of
the prediction factor in the TPF control for the stabilization of the
delayed system. The phase margin of the system under the control
S.Y. Yoon, Z. Lin / Systems & Control Letters 62 (2013) 837–844
(a) x1 (t ) and x2 (t ).
843
(b) x3 (t ), x4 (t ) and x5 (t ).
Fig. 4. Time response of the system states corresponding to the second case of the delay function φ(t ).
Fig. 5. Original and delayed control signals of the closed-loop system corresponding to the second case of the delay function φ(t ).
input u(t ) = Kx(t ) is found to be 68.1°, which corresponds to a
delay margin of 0.809 s.
Fig. 6 demonstrates the state responses of the closed-loop
systems under the TPF controller (4) and under the static feedback
(57), as both systems are subjected to the delay function (56) with
¯ =
d = 1. This corresponds to a bound on the delay function of D
1.5 s, which is larger than the delay margin of the system under the
control law (57). It is observed in the figure that the system under
the feedback (57) is unstable for the given time-delay. On the other
hand, the simulation results under the TPF controller show a stable
response.
Here we note that it may be possible to find a different feedback
gain K such that the closed-loop system under the control law (57)
is stable when subjected to the same delay as above. However,
the fact remains that no systematic approach exists to find such
a K . In view of this, the comparison presented in this subsection is
considered to be a fair benchmark test to identify the benefits of
the TPF controller for exponentially unstable systems with timevarying delay.
6.4. Comparison between the predicted and the actual maximum
delay bounds
¯
As discussed in Remark 2, the maximum of the delay bound D,
satisfying the stability condition (29) for the closed-loop system
under the TPF controller, can be determined from (49). The
(a) Under the TPF control (4).
maximum delay bound corresponding to the TPF controller in
¯ max = 0.458 s. In this subsection
Section 6.2 with γ = 0.09 is D
we investigate how this sufficient bound on the delay function
compares to the actual bound observed in numerical simulation.
Consider the sinusoidal delay function in (56). The maximum
delay bound for the closed-loop system with (56) was found
through simulation, where the value of d was gradually increased
until the closed-loop system became unstable. Fig. 7 shows the
state response of the closed-loop system with the maximum
tolerable delay at d = 1.7. The corresponding upper bound of the
¯ = 2.55 s. The control input u(t ) and the delayed
delay function is D
input u(φ(t )) are plotted in Fig. 8.
The simulation results in Figs. 7 and 8 demonstrate that
the closed-loop system under the TPF control law is stable for
¯ greater than theoretical maximum
delay functions with bounds D
¯ max . However, the stability condition (29) and the corresponding
D
maximum delay bound in (49) were derived for any delay
function satisfying the assumptions listed in Section 2, while the
bound obtained by simulation is specific to the particular delay
function (56).
7. Conclusions
The stabilization of general linear systems with time-varying
input delays was examined in this paper. The truncated predictor
feedback (TPF) method, which had recently been developed for
systems with poles in the closed left-half plane and is based on
low gain feedback, was extended to be applicable to exponentially
unstable systems. An explicit design procedure of the control law
was presented along with a stability condition for the closed-loop
system.
In the special case where the system poles are all in the closed
left-half plane, it was observed that the stability condition obtained
here reduces to the result presented in [14], and a stabilizing
controller can be found for an arbitrarily large delay. On the other
hand, if the system has exponentially unstable poles, then the
closed-loop stability condition reveals a clear trade-off between
the sum of the unstable poles and the maximum allowable delay in
(b) Under the static feedback (57).
Fig. 6. Comparison between the TPF control in (4) and the static gain feedback u(t ) = Kx(t ).
844
S.Y. Yoon, Z. Lin / Systems & Control Letters 62 (2013) 837–844
(a) x1 (t ) and x2 (t ).
(b) x3 (t ), x4 (t ) and x5 (t ).
Fig. 7. Time response of the system states corresponding to the delay function (56) with d = 1.7.
Fig. 8. Original and delayed control signals of the closed-loop system corresponding to the delay function (56) with d = 1.7.
the system input. Numerical examples validated the mathematical
derivation in this paper.
The results presented in this paper are built upon the existence
of φ −1 (t ) and the assumption that φ(t ) is continuously differentiable. Such assumptions are satisfied in many applications, such
as the case of systems with constant delays. However, the assumptions may not be appropriate in some applications such as control
over networks, where the delay function is generally not continuous. As discussed in the introduction, some methods exist for the
stabilization of delayed system with minimum information on the
delay, but those methods mostly focus on constant delays. The limitations that come from the assumptions on φ(t ) will need to be
addressed in future research.
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