Research Journal of Applied Sciences, Engineering and Technology 4(18): 3201-3208, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 23, 2011
Accepted: January 21, 2012
Published: September 15, 2012
Stabilization of State-derivative Feedback Control with Time Delay
1
1
Witchupong Wiboonjaroen and 2Sarawut Sujitjorn
Department of Electronic Engineering, Rajamangala University of Technology Isan, Thailand
2
Control and Automation Research Unit; Power Electronics, Machines and Control Research
Group, School of Electrical Engineering, Suranaree University of Technology, Thailand
Abstract: This study considers the problem of state-derivative feedback controller design via eigenvalue
assignment for LTI systems of linear Delay Differential Equations (DDEs) with a single delay. Unlike simple
LTI systems, the systems described by DDEs have an infinite eigenspectrum and it is not feasible to assign all
closed-loop eigenvalues. The paper proposes a method to assign a critical subset of them using an approach of
the matrix Lambert W function. The solution has an analytical form expressed in terms of the parameters of
the DDE. With the proposed method, one can extend a conventional eigenvalue assignment method for a
feedback controller to a delayed LTI system. A scheme including filtered state-derivative feedback is proposed
to overcome the destabilizing effect of feedback delays. Proofs of the proposed method and numerical examples
are presented.
Keywords: Delay differential equation, eigenvalue assignment, feedback controller, lambert W function,
neutral system, state-derivative feed back
INTRODUCTION
Eigenvalue assignment has been an important design
method of a Linear Time-Invariant system (LTI) system.
One approach is to use state feedback in which the gain
matrix is calculated via Ackermann’s formula. The
concept has been extended to state-derivative feedback
that is useful for various practical systems including
control of vibration (Abdelaziz and Valasek, 2003a;
Moreira et al., 2010a; Kwak et al., 2002a; Reithmeier and
Leitmann, 2003b). One advantage over the conventional
state feedback is that it results in smaller gains. A linear
quadratic regulator to achieve the state-derivative
feedback was also developed Abdelaziz and Valasek
(2005). Despite the benefit, the question of stability of the
system employing state-derivative feedback has been
raised, especially computer-controlled systems. This is
due to inevitable latencies caused by sampling, data
conversion and instruction-execution processes. Delayed
systems are rather sensitive to instability (Chen et al.,
1995; Park, 1999; Niculescu, 2001a; Richard, 2003c).
Stabilization of such systems via a classic method (Chen,
1984) is not straightforward because the transcendental
term causes the number of eigenvalues to be infinite.
Furthermore, a control system that uses control laws
involving state-derivative feedback could lead to an
extreme sensitivity of closed-loop stability w.r.t. small
delays. A state-derivative feedback control system with
time-delay is also referred to as a neutral system, which
has many characteristic roots located to the right of the
stability boundary. Moreover, the positions of these roots
are very sensitive to changes in delay (Hale and Verduyn
Lunel, 2001b, 2002b; Michiels et al., 2004a).
During recent decades, the stabilization of systems of
linear DDEs using feedback control has been studied
extensively. The problem of robust stabilization of timedelayed systems, or the stabilization problem via delayed
feedback control, is solved by Richard (2003c). Recently,
the Lambert W function (Corless et al., 1996) method has
gained the popularity in various fields of science and
engineering (Caillol, 2003d; Boutat et al., 2007a; Corless
et al., 2000) including the investigation of time-delay
systems (Chen and Moore, 2002c; Cheng and Hwang,
2006a; Jarlebring and Damm, 2007b; Shinozaki and Mori,
2006b). An approach for finding the solution to LTI
systems of DDEs has been developed using the Lambert
W function (Asl and Ulsoy, 2003e; Yi and Ulsoy, 2006c;
Yi et al., 2006d). In papers such as Yi et al. (2010a) the
results for state-feedback controller design for a class of
delayed systems are presented.
In this study, we apply the matrix Lambert W
function-based approach for the solution to DDEs to
stabilize linear time-delayed systems. To be more
specific, we present a new approach for state-derivative
feedback controller design via eigenvalue assignment for
delayed systems and illustrate the method with two
numerical examples. Upon the authors’ knowledge, the
concept of eigenvalue assignment via state-derivative
Corresponding Author: Sarawut Sujitjorn, Control and Automation Research Unit; Power Electronics, Machines and Control
Research Group, School of Electrical Engineering, Suranaree University of Technology, Thailand
3201
Res. J. Appl. Sci. Eng. Technol., 4(18): 3201-3208, 2012
feedback has not been introduced to a delayed system
before. Using the proposed approach, we can move a
subset of eigenvalues to desired locations in a manner
similar to pole placement for systems of ODEs. For a
given completely controllable system, which is
represented by a DDE, the solution to the system is
obtained based on the Lambert W function and stability is
determined. If the system is unstable, a stabilizing statederivative feedback is designed by assigning eigenvalues.
In this way, a time-delayed system can be stabilized.
Tf u&(t ) + u(t ) = Kd x&(t − τ )
where, Tf is the time constant of the filter and Kd0Rn is a
row gain vector for the derivative feedback element.
Proposition 2.1: The control u (t) with a first-order lowpass filter in the form of (5) such that Tf >> 1, a LTI
systems of DDEs, with a single constant delay can be
approximated by:
MATERIALS AND METHODS
x&(t ) = Ax (t ) +
Consider the generalization to free system of DDEs
in matrix-vector form, with a single constant delay, J:
x (t ) = g (t ),
t=0
(1)
U ( s) =
t ∈ [ − τ , 0]
x (t ) =
U ( s) ≈
∑eτ
k = −∞
Kd e −sτ sX ( s )
Tf s
C
, therefore:
Kd − sτ
e X ( s)
Tf
(3)
where u(t),R, A and B are n×n and n×1 real coefficient
matrices, respectively. For the delay state-derivative
feedback, the control u (t) is of the form:
Taking the inverse Laplace transform of Eq. (8), we
obtain:
u( t ) =
Kd
x (t − τ )
Tf
(9)
Substitution of (9) into (3), a LTI system of DDEs with
state-derivative feedback can be approximated as (6).
This completes the proof.
Proposition 2.2: The system (3) is subject to the control
input u (t) in the form of (5). There exists the solution to
Eq. (6) of the following form:
∞
x (t ) = ∑ e sk t CkI
k =−∞
u(t ) = Kd x&(t − τ )
(8)
(2)
where CIk is constant n×1 vector (determined numerically
from the preshape function and initial state). However,
this solution is only valid when the matrices A and Ad
commute (Yi and Ulsoy, 2006c). Therefore, the solution
(2) in terms of a matrix Lambert W function Wk cannot be
used for general delayed systems.
Consider a LTI system having single input of the
form:
x&(t ) = Ax (t ) + Bu(t ), x (t 0 ) = x 0
(7)
,
U ( s) =
I
k
Kd e − sτ sX ( s )
Tf s+ 1
For Tf>>1, the control U(s) can be approximated by
1
( Wk ( Ad τ e − Aτ ) + A ) t
(6)
(Tf s + 1)U ( s) = Kd e − sτ sX ( s)
where A and Ad are n×n real coefficient matrices, x(t) is
an n×1 state vector, g(t) and x0 are an n×1 preshape
(initial) vector function and an initial state, respectively.
Under a special circumstance that A and Ad commute, the
solution is given as Asl and Ulsoy (2003e):
∞
BKd
x (t − τ )
Tf
Proof: For the control u (t) in (5), taking the Laplace
transform of both sides, we obtain:
x&(t ) = Ax (t ) + Ad x (t − τ ), t > 0
x ( t ) = x0 ,
(5)
(10)
(4)
where,
when applying a first order low-pass filter to Eq. (4), the
control u (t) becomes:
3202
Sk =
1
τ
wk (
BKd
τQk ) + A
Tf
(11)
Res. J. Appl. Sci. Eng. Technol., 4(18): 3201-3208, 2012
The coefficient C lk is a function of A, B, Kd, J, g(t)
and x0. The numerical method for computing CIk was
developed in papers such as Asl and Ulsoy (2003e). The
matrix Qk is obtained from the following condition, which
can be used to solve for unknown matrix Qk:
wk (
BKd
τQk ) + Aτ
Tf
wk (
BKd
τQk )e
Tf
=
BKd τ
Tf
w( H )e w( H ) = H
( S − A)τe ( S − A)τ =
where S is an n×n matrix and CI is a constant n×1 vector
(Hale and Verduyn Lunel, 1977). Substitution of (13) into
(6) yields:
w(
BKd
w(
τQ)e
Tf
(14)
So,
w(
BKd
τQ) is
Tf
(15)
we find:
Multiplying Eq. (15) by eSJ we represent the
transcendental characteristic Eq. (6) in the following
form:
S=
( S − A)e Sτ =
BK d
Tf
( S − A)τe e
BKd − Aτ
=
τe
Tf
1
τ
w(
BKd
τQ )
Tf
=
BKd
τQ
Tf
(22)
the value of the matrix Lambert W
H= (
BKd
τQ) . Solving Eq. (21) for S,
Tf
BKd
τQ) + A
Tf
(23)
(16)
Performing further transformation, we multiply both side
of Eq. (16) by Je-AJ . This yield:
− Aτ
(21)
Substituting (23) into (15) we obtain the condition, which
can be used to solve for the unknown matrix Q:
1
Sτ
(20)
BKd
τQ)
Tf
function w(H), where
BKd − sτ
e = 0
Tf
BKd
τQ
Tf
and
Since e St C I ≠ 0 , we get:
S − A−
(19)
Comparing Eq. (19) and (20), we see that:
( S − A)τ = w(
BKd S ( t − τ ) I
Se St C I − Ae St C I −
e
C =0
Tf
(18)
where H is an n×n matrix variable, we introduce,
following (Asl and Ulsoy, 2003e), an unknown matrix Q
that satisfies the condition:
(13)
BKd − Sτ Sτ I
(S − A −
e )e C = 0
Tf
− A )τ
Consequently, the solution to (17) can be written in
terms of the matrix Lambert W function W(H), which
satisfies the equality:
(12)
Proof: Finding the solution to Eq. (6), to begin with, we
assume that it has the form:
x (t ) = e st C I
( S − A)τe Sτ e − Aτ ≠ ( S − A)τe ( S
τ
BK
w(
BKd − ( w Tf d τQ ) + Aτ }
BKd
e
τQ) + A − A −
=0
Tf
Tf
BKd
BKd τ − w(
w(
τQ) =
e
Tf
Tf
(17)
When S and A commute, we can write the solution in
terms of a Lambert W function. In general, the matrices A
and BKd do not commute and neither do the S and A.
Hence:
3203
w(
w(
BKd
τQ)e
Tf
BKd
τ Q ) + Aτ
Tf
=
BKd
τ Q ) + Aτ }
Tf
BKd τ
Tf
(24)
Res. J. Appl. Sci. Eng. Technol., 4(18): 3201-3208, 2012
The matrix Lambert W function, w(H), is complex
valued, with a complex argument H and has an infinite
number of branches wk(Hk), where k = -4, ..., -1, 0, 1, ...,4
(Corless et al., 1996). Corresponding to each branch, k, of
the Lambert W function, wk, there is a solution Qk from
BKd
Eq. (24) and for H k =
τQk , the Jordan canonical form
Tf
Jk is computed from Hk = zkJk z-1k. Jk = diag
( J k 1 ( λ$1 ),..., J k 2 ( λ$2 ),..., J kp ( λ$p )) where, J ki (λ$i ) is an
m×m Jordan block and m is multiplicity of the eigenvalue
λ$i . Then, the matrix Lambert W function can be
computed as (Pease, 1965):
{ (
)}
wk ( Hk ) = zk diag wk ( J K1 (λ$1 )),..., wk ( J kp (λ$p )) Zk− 1 (25)
where,
⎡
' $
$
⎢ wk ( λi ) wk (λi )
⎢
$
wk ( λ$i )
wk ( J ki ( λi )) = ⎢ 0
⎢
⎢ M
M
⎢
0
⎣ 0
1
⎤
w( m−1) (λ$i ) ⎥
(m − 1)! k
⎥
1
wk( m−2 ) (λ$i ) ⎥
K
(m − 2)!
⎥
⎥
O
M
⎥
$
wk ( λi )
L
⎦
(26)
the unknown matrix Qk is numerically obtained from:
BKd
τQk ) + Aτ
Tf
=
BKd τ
Tf
(27)
The obtained Qk can be substituted into Eq. (23) to obtain
Sk of the form:
BKd
τQk ) + A
S k = wk (
τ
Tf
1
∑e
Sk t
CkI
(29)
This completes the proof.
Note that the Eq. (27) has a unique solution Qk for
each branch k of the matrix Lambert W function. To
obtain the solution, we use the fsolve function in
MATLAB. The solution forms in (27)-(29) reveal that the
stability condition for the system of Eq. (6) depends on
the eigenvalues of the matrix Sk. In other words, a timedelayed system is asymptotically stable if all the
eigenvalues of , have negative real parts. However, Sk k
= -4, ...,-1, 0, 1, ..., 4 computing the matrix Sk for an
infinite number of branches, k = -4, ...,-1, 0, 1, ..., 4 is not
feasible. It is shown by Shinozaki and Mori (2006b) and
Yi et al. (2010a) that only the coefficients of the principal
branch (k = 0) of the Sk need to be known because they
represent the rightmost eigenvalues in the complex plane
and determine the stability of the system (6). This gives:
max[Re{λ ( S o )}] ≥ Re{λ ( S k )}
k = − ∞ ,...,− 1, 0, 1, ... ∞
,
w k ( H k ) e wk ( H k ) = H k
∞
k = −∞
K
With the matrix Lambert W function, wk, given in (25), Sk
is computed from (23). The principal (k = 0) and other
(k…0) branches of the Lambert W function can be
calculated from a series definition or using commands
already embedded in various commercial software
packages, such as MATLAB, Maple and Mathematica.
With wk satisfiying:
wk (
BK
wk ( d τQk )e
Tf
x(t ) =
(28)
(30)
where, 8(S0) and 8 (Sk) are eigenvalues of S0 and all other
eigenvalues of Sk, respectively. One of the major results
for a LTI system is that, with full state feedback, one can
specify all the closed-loop eigenvalues by selecting the
gains. A direct application of the idea is not feasible for a
time-delayed system because a DDE always has an
infinite number of eigenvalues. Fortunately, for a
completely controllable delayed system, the Lambert W
function approach can be used to specify the first matrix,
S0, corresponding to the principal branch, k = 0 and is
observed to be critical to the solution (29), by choosing
the state-derivative feedback gain and designing a
feedback controller. Since J-nonstabilizable neutral
systems cannot return to stability for any J >0 (Olgac and
Sipahi, 2004b), we focus only on the J-nonstabilizable
class of neutral systems in this paper.
The gain, Kd, to assign the rightmost eigenvalues, is
determined as follows:
First: select desired eigenvalue 8, i,desired, for i = 1, …, n
and set an equation so that the selected eigenvalues
become those of the matrix S0 as:
λi ( S o ) = λi ,desired
i = 1,..., n
(31)
where, 8 i(S0) is the ith eigenvalue of the matrix S0.
Substitution of Sk into Eq. (13) results in the free solution
to Eq. (6) of the form:
Second: apply the two coefficient matrices A and BKd in
(6)-(12) and solve numerically to obtain the matrix Q0 for
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Res. J. Appl. Sci. Eng. Technol., 4(18): 3201-3208, 2012
Start
1
Initialization A, B, τ, Tf and λ i, desired, i = 1,.., n
Compute S = τ +A
Randomly assign value to Kd
2
Compute eigenvalue λ (S0)
Randomly assign value to Q 0
Find λi, max = max λi (S0)
Compute Ad = BKd
H = Ad Qo /Tf
Evaluate JKd
JKd = ∑⎜λ i, max - λ i, desired ⏐
Compute W0 using “lambertw (0, H)”
i
JKd ≤ c 2
Evaluate J
JQ = ⏐W0 e(W0+Ax ) -Adτ ⏐
No
No
2
Yes
Stop
JQ ≤ c1
Yes
1
Fig. 1: Flow diagram represents the computing process to obtain the state-derivative feedback gain. (Remark: c1,c2 : Termination
tolerance = 1e-8, 8i(S0): ith eigenvalue of the matrix S0)
the principal branch (k = 0). Note that Kd is an unknown
matrix with all unknown elements in it and the matrix Q0
is a function of the unknown Kd.
Third: substitute the matrix Q0 from (12) into (11) to
obtain S0 and its eigenvalues as the function of the
unknown matrix Kd .
Example 1: Consider the van der Pol equation, which has
become a prototype for systems with self-excited limit
cycle oscillation and has the form:
RESULTS AND DISCUSSION
Two numerical examples are presented with focusing
on stabilization of state-derivative feedback control
system with time-delay.
(32)
f ( x , t ) = ε ( x 2 (t ) − 1)
(33)
with
Finally: solve (13) with the matrix S0 for the unknown,
Kd using a numerical method such as fsolve in MATLAB.
As mentioned by Corless et al. (1996), depending on
the structure or parameters of a given system, there exists
a limitation in that some values are not proper for the
rightmost eigenvalues. In such a case, the above approach
does not yield any solution for Kd. To resolve the
problem, one may try with fewer desired eigenvalues, or
different values of the desired rightmost eigenvalues.
Then the solution, Kd, is obtained numerically for a
variety of initial conditions by an iterative trial and error
procedure. Figure 1 depicts the flow diagram representing
the computing procedures to obtain the gain Kd.
x&&(t ) + f ( x , t ) x&(t ) + x (t ) = g ( x , t ; τ )
For the dynamics of the van der Pol equation under
the effect of state-derivative feedback with time-delayed,
the left-hand side of (32) can be written as:
g ( x , t ; τ ) = k d 1 x&(t − τ ) + k d 2 &&(
x t − τ)
(34)
Then, with the damping coefficient function in (33) and
feedback in (34), the system description (32) becomes:
x&&(t ) + x (t ) = ε ( x 2 (t ) − 1) x&(t ) + k d 1
x&(t − τ ) + k d 2 &&(
x t − τ)
(35)
Linearizing the system (35) about the zero equilibrium
yields the equation for infinitesimal perturbation:
3205
x&&(t ) + x (t ) = εx&(t ) + kd 1x&(t − τ ) + kd 2 &&(
x t − τ)
(36)
Res. J. Appl. Sci. Eng. Technol., 4(18): 3201-3208, 2012
Response without feedback
Stabilized response
0.15
States
0.10
X1
0.05
X1
0 X
2
-0.05
-0.10
X2
-0.15
0
2
4
6
8
10 12
Time (s)
14
16
18
20
Fig. 3: Pendulum system
Fig. 2: Comparison of system states before (dashed) and after
(solid) applying feedback with Kd = [0.6886 -8.8357].
The chosen feedback gain stabilizes the system
or, equivalently, by defining x1 = x and x 2 = x& , one
0.3
0.2
⎡ 0 1⎤
⎡ 0
x&(t ) = ⎢
x (t ) + ⎢
⎥
⎣− 1 ε⎦
⎣ kd1
States
obtains the state Eq. (37).
0 ⎤
x&(t − τ )
k d 2 ⎥⎦
⎡ 0⎤
⎢ 1 ⎥ u( t )
⎣ ⎦
-0.4
0
⎡ 0 ⎤
⎢ 1 ⎥ T (t − τ )
⎢ 2⎥
⎣ ml ⎦
(39)
(40)
1
2
3
4
5
6
Time (s)
7
8
9
10
Fig. 4: Comparison of system states before (dashed) and after
(solid) applying feedback with Kd = [3.28-36.26]. The
chosen feedback gain stabilizes the system
(38)
where, 2 is the pendulum angle, l is the length of the
pendulum, g is gravitational acceleration, m is the
pendulum mass and T is the input torque. A constant
delay between the sensor and the controller is considered
in the model. By defining x1 = 2 and x2 = θ& , it is desired
to keep the pendulum at x1 = xe. One can show that, the
nonlinear model can be linearized around xe as:
1⎤
⎥ x (t ) +
0⎥
⎦
X2
-0.3
The controller is designed to be:
T (t − τ ) = Kd x&(t − τ )
(41)
Thus, the closed-loop system can be described by:
0
⎡
x&(t ) = ⎢ g
⎢⎣ − l cos x1
1⎤
⎥ x (t ) +
0⎥
⎦
⎡ 0
⎢ kd1
⎢⎣ ml 2
0 ⎤
k d 2 ⎥ x&(t − τ )
ml 2 ⎥⎦
(42)
which can also be expressed in the form of (3) as:
0
⎡
x&(t ) = ⎢ g
−
cos
x1
⎢⎣ l
Example 2: Consider the equation of motion of a simple
pendulum represented in Fig 3 as follows:
0
⎡
x&(t ) = ⎢ g
⎢⎣ − l cos x1
X2
-0.2
(37)
where, u (t) = Kd x&(t − τ ) , Without the state-derivative
feedback term (i.e., kd1 = kd2 = 0) the system (37) is
unstable when g = 0.1 and its eigenvalues are
0.0500±0.9987j. For example, when J = 0.085 s and
Tf = 5 s, if the real-part of the desired rightmost
eigenvalue is -1, which is arbitrarily selected, then, the
required gain is found to be Kd = [0.6886 -8.8357].
Figure 2, the responses without feedback control is
unstable. Applying the proposed state-derivative feedback
controller stabilizes the system. The rightmost
eigenvalues have exactly their real-parts at -1 and all the
other eigenvalues are to the left.
T (t − τ ) − mgl sin θ (t ) = ml 2θ&&(t )
0
-0.1
which can also be expressed in the form of (3) as:
⎡ 0 1⎤
x&(t ) = ⎢
⎥ x (t ) +
⎣− 1 ε⎦
X1
X1
0.1
1⎤
⎥ x (t ) +
0⎥
⎦
⎡ 0 ⎤
⎢ 1 ⎥ u(t )
⎢ 2⎥
⎣ ml ⎦
(43)
where, u(t ) = K d x&(t − τ ) . Consider the linearized
pendulum system in (43) with parameters l = 2 m, m = 1
kg, g = 10 ms-2 and xe = 20º = 0.3491 rad. Thus, the
closed-loop system becomes:
0
1⎤
⎡
x&(t ) = ⎢
x (t ) +
4
6985
0⎥⎦
−
.
⎣
⎡ 0 ⎤
⎢ 0.25⎥ u(t )
⎣
⎦
(44)
Without the state-derivative feedback, the system is
unstable and its eigenvalues are ±2.1676j. As an example,
when J = 0.05 s and Tf = 5 s, if the real-part of the
rightmost eigenvalues is -1, then the required gains are
found to be Kd = [3.28 -36.26]. Figure 4, the states
3206
Res. J. Appl. Sci. Eng. Technol., 4(18): 3201-3208, 2012
without feedback control are unstable. Applying the
designed feedback controller stabilizes the system. The
rightmost eigenvalues have exactly the desired real-parts
and all the other eigenvalues are to the left. If the desired
eigenvalues are -1.0000 ± 2 j or -1.0000 ± j , then the
corresponding gains are Kd = [8.76 -33.94] or Kd = [-21 60.90], respectively.
The above examples serve to demonstrate the
effectiveness of the proposed method to compute the gain
matrix of the state-derivative feedback control system
with time-delayed. This is an important issue because a
LTI system with state-derivative feedback is prone to
instability due to delay.
CONCLUSION
In this paper, new results for state-derivative
feedback controller design for a class of time-delayed
systems are presented. For a given system, which can be
represented by a DDE, based on the Lambert W Function,
the solution to the system is obtained and stability is
determined. If the system is unstable, after the
controllability of the system is checked, a stabilizing
feedback is designed by assigning eigenvalues and finally
the closed-loop system can be stabilized. All of these
results are based upon the Lambert W function-based
approach. Numerical examples are presented to illustrate
the approach. Although DDEs have an infinite
eigenspectrum and it is not possible to assign all closedloop eigenvalues, we assign a subset of them that are
critical in determining the stability for the system of
DDEs.
ACKNOWLEDGMENT
The authors gratefully acknowledge the financial
supports by the Office of Higher Education Commission,
Thailand, under the NRU Project and Suranaree
University of Technology (SUT).
REFERENCES
Abdelaziz, T.H.S. and M. Valasek, 2003a. A direct
algorithm for pole placement by state-derivative
feedback for single-input linear systems. Acta
Polytech., 43(6): 52-60.
Abdelaziz, T.H.S. and M. Valasek, 2005. State derivative
feedback by lqr for linear time-invariant systems.
Proceeding of 16th IFAC World Congress, 16(1),
Prague, Czech Republic, July.
Asl, F.M. and A.G. Ulsoy, 2003e. Analysis of a system of
linear delay differential equations. J. Dynamic Syst.
Measurement Control, 125(2): 215-223.
Boutat, M., S. Hilout and J. Grilhe, 2007a. Plastic
deformation instabilities: Lambert solutions of
mecking-lucke equation with delay. Math. Probl.
Eng., 13: 329-359.
Caillol, J.M., 2003d. Some applications of the lambert w
function to classical statistical mechanics. J. Phys. A:
Math. General, 36(42): 10431-10442.
Chen, C.T., 1984. Linear System Theory and Design.
Holt, Rinehart and Winston, New York.
Chen, J., G. Guand and C.N. Nett, 1995. A new method
for computing delay margins for stability of linear
delay systems. Syst. Contr. Let., 26: 107-117.
Chen, Y.Q. and K.L. Moore, 2002c. Analytical stability
bound for delayed second-order systems with
repeating poles using Lambert W function.
Automatica, 38(5): 891-895.
Cheng, Y.C. and C. Hwang, 2006a. Use of the Lambert
W function for time-domain analysis of feedback
fractional delay systems. IEEE Proc. Contr. Th.
Appl., 153(2): 167-174.
Corless, R.M., G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey
and D.E. Knuth, 1996. On the Lambert W function.
Adv. Comput. Math., 5(1): 329-359.
Corless, R.M., S.R. Valluri and D.J. Jeffrey, 2000. Some
applications of the Lambert W function to physics.
Can. J. Phys., 78(9): 823 -831.
Hale, J.K. and S.M. Verduyn Lunel, 1977. Introduction to
Functional Differential Equations. Springer-Verlag,
New York.
Hale, J.K. and S.M. Verduyn Lunel, 2001b. Effect of
small delays on stability and control. Oper. Theory,
122: 275-301.
Hale, J.K. and S.M. Verduyn Lunel, 2002b. Strong
stabilization of neutral function differential equation.
IMA J. Math. Control I., 19: 5-23.
Jarlebring, E. and T. Damm, 2007b. The Lambert W
function and the spectrum of some multidimensional
time-delay systems. Automatica, 43(12): 2124-2128.
Kwak, S.K., G. Washington and R.K. Yedavalli, 2002a.
Acceleration feedback-based active and passive
vibration control of landing gear components. J.
Aerospace Eng., 15(1): 1-9.
Michiels, W., K. Engelborghs, D. Roose and D. Dochain,
2004a. Sensitivity to infinitesimal delays in neutral
equations. SIAM J. Control Optim., 40: 1134-1158.
Moreira, M.R., E.I.M. Junior, T.T. Esteves,
M.C.M. Teixeira, R. Cardim, E. Assuncao and F.A.
Faria, 2010a. Stabilizability and disturbance rejection
with state-derivative feedback. Math. Probl. Eng.,
2010(ID 123751): 12.
Niculescu, S.I., 2001a. Delay Effect on Stability.
Springer-Verlag, New York.
Olgac, N. and R. Sipahi, 2004b. A practical method for
analyzing the stability of neutral type LTI-time
delayed systems. Automatica, 40: 847-853.
Park, P., 1999. A delay-dependent stability criterion for
systems with uncertain time-invariant delays. IEEE
T. Automat. Contr., 44: 876-877.
3207
Res. J. Appl. Sci. Eng. Technol., 4(18): 3201-3208, 2012
Pease, M.C., 1965. Methods of Matrix Algebra.
Academic Press, New York.
Reithmeier, E. and G. Leitmann, 2003b. Robust vibration
control of dynamical systems based on the derivative
of the state. Arch. Appl. Mech., 72(11-12): 856-864.
Richard, J.P., 2003c. Time-delay systems: An overview
of some recent advances and open problems.
Automatica, 39: 1667-1964.
Shinozaki, H. and T. Mori, 2006b. Robust stability
analysis of linear time-delay systems by Lambert W
function: Some extreme point results. Automatica,
42: 1791-1799.
Yi, S. and A.G. Ulsoy, 2006c. Solution of a System of
Linear Delay Differential Equations Using the Matrix
Lambert Function. Proceedings of 25th American
Control Conference, Minneapolis, MN, June.
Yi, S., A.G. Ulsoy and P.W. Nelson, 2006d. Solution of
Systems of linear delay differential equations via
laplace transformation. Proceedings of 45th IEEE
Conference on Decision and Control, San Diego, CA,
Dec.
Yi, S., A.G. Ulsoy and P.W. Nelson, 2010a. Eigenvalue
assignment via the Lambert W function for control of
time-delay system. J. Vib. Control, 16(7-8): 961-982.
3208