Research Journal of Applied Sciences, Engineering and Technology 4(14): 2265-2272, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 23, 2012
Accepted: April 08, 2012
Published: July 15, 2012
Stability of Retarded/Neutral System with State-PID Feedback Using
Lienard-Chipart Stability Criterion
1
W. Wiboonjaroen and 2S. Sujitjorn
Department of Electronic Engineering, Ratchamangkala University of Technology, Isarn
2
School of Electrical Engineering, Suranaree University of Technology,
Muang District, Nakhon Ratchasima, Thailand, 30000, Chain
1
Abstract: A retarded/neutral system is sensitive to instability due to delay. A knowledge of maximum
allowable delay ( τ ) or tolerable delay is thus useful for controller design and stabilization. This article
proposes a method to compute a tolerable delay time for a Linear-Time-Invariant-Time-Delayed-System (LTITDS) with state-PID feedback control. The method proposed is based-on Lienard-Chipart criterion, which
reduces the number of determinants to be evaluated. The proposed approach is compared with the matrix pencil
method. Three case studies serve to demonstrate the effectiveness of the method.
Keywords: Retarded/neutral system, state-PID feedback, stability and stabilization, lienard-chipart stability
criterion
INTRODUCTION
Eigenvalue assignment has been an important design
method of a Linear Time-Invariant System (LTI). 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, 2003; Moreira et al.,
2010; Reithmeier and Leitmann, 2003). 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 by
Abdelaziz and Valasek (2005).
Recently, state-PID feedback has been proposed for
regulation problem of an LTI system (Guo et al., 2006;
Sujitjorn and Wiboonjaroen, 2011). The proposed control
scheme introducing an integral element to study with the
gain can effectively eliminate the steady-state errors.
Despite the benefit, the question of stability of the system
employing state-PID feedback has been raised, especially
in 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 (Park, 1999; Niculescu,
2001; Michiels et al., 2002; Richard, 2003; Gu and
Niculescu, 2003). 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-PID feedback could
lead to an extreme sensitivity of closed-loop stability
w.r.t. small delays. A state-PID 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. Additionally, the positions of these
roots are very sensitive to changes in delay (Hale and
Verduyn, 2001, 2002; Michiels et al., 2004). Since a
system can withstand a certain delay time before
becoming unstable, an accurate prediction of a tolerable
delay is therefore practically useful. An early method for
stability test proposed by Rekasius (1980) uses exact
bilinear transformation to represent the transcendental
term. The work was extended to retarded systems (Olgac
and Sipahi, 2002), in which they presented comparative
studies among five stability analysis methods. They
concluded that the Rekasius’ method and the features of
Routh-Hurwitz criterion is the most attractive one due to
simplicity, accuracy and exactness (Sipahi and Olgac,
2006). Moreover, this approach lends itself nicely to the
analysis of neutral systems (Sipahi and Olgac, 2003;
Olgac and Sipahi, 2004, 2005). Computing for
generalized eigenvalues and tolerable delay are also
possible using the matrix pencil with the Rekasius’
substitution (Chen et al., 1995; Fu et al., 2006). Recently,
a new approach based on Lambert W function to compute
eigenvalue spectrum and predict stability of a delayed
system has been proposed (Asl and Ulsoy, 2003; Yi et al.,
2010). Unfortunately, computing based-on the LambertW-function is quite complex and sometimes numerical
solutions cannot be found. This is due to a fundamental
reason that the existence of the function as a solution to
retarded/neutral equations has not yet been rigorously
confirmed.
Corresponding Author: S. Sujitjorn, School of Electrical Engineering, Suranaree University of Technology Muang District,
Nakhon Ratchasima, Thailand, 30000, China
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Res. J. Appl. Sci. Eng. Technol., 4(14): 2265-2272, 2012
This study presents a stability analysis method based
on Lienard-Chipart stability criterion in comparison with
the matrix pencil approach. The Lienard-Chipart criterion
(Dorato, 2000; Gantmacher, 1959) has a decisive
advantage over the Routh-Hurwitz criterion because it
computes about half the number of determinants. Even
though the Hurwitz determinants to be computed can have
a larger size than those required by the Routh-Hurwitz
criterion, computation is no longer a problem due to
available software packages, e.g., MATLAB, Maple,
Mathematica, SciLab etc. Three numerical examples
including an inverted pendulum and a magnetic ball
levitation are demonstrated.
C
x& (t ) = Ax (t ) + Ad x (t − t )
f (s,J) = det (sI - A - Ade-Js) = 0, J>0
CE (s,J) = an(s)e-nJs + an-1(s)e-(n-1)Js + ... + a0(s)
=
Lienard-chipart stability criterion: This criterion
gives condition on the coefficients ci of a polynomial:
(2)
The following calculation procedure is referred to as
matrix pencil method and applies the Rekasius’
substitution. Computing procedures to obtain τ :
(3)
Step 1: Define the system characteristic equation in the
form of (7) and rewrite it in the form of (8) via
the Rekasius’ substitution:
while either of these sets of inequalities may be used,
(2) is more efficient (fewer inequalities to test) for n
even and (3) is more efficient for n odd. Also note
that the inequalities )i>0 need to be tested only up to
i = n-1, where )i is the so-called Hurwitz determinant
and given by:
c1 c3 c5 K
1 c2 c4 K
0 c1 c3 K
∆ i = 0 1 c2 K
CE ( s, T ) =
n + nd
∑ bi (T ) sn +n
d
−i
=0
(8)
i =0
in which n is the system order and nd is the
commensurate degree. Rearrange (8) to:
, cm = 0
(4)
0 0 c1 K
for m>n.
n
∑ ak ( s)e− kτs = 0
where, ak(S) is (n-k)th order of s-polynomial having
real coefficients. For a stable system, all
characteristic roots must lie on the left-half of the splane. The transcendental term results in an infinite
number of characteristic roots commonly referred to
in literatures as characteristic spectrum or Eigenspectrum. For stability test, only the dominant branch
of such spectrum is necessary for justification of
stability because the rest of them lie on the left side
of this dominant branch. Calculation of the
characteristic roots of the dominant branch can be
done easily via applying the Rekasius’ substitution.
(1)
or
cn>0, cn-2>0, ...: )2>0, )4>0
(7)
k =0
in order that all its roots have negative real parts. A
polynomial with this property is commonly referred
to as Hurwitz polynomial. A necessary condition for
the polynomial (1) to be Hurwitz is that all the
coefficients ci be positive. When p(s) is the closedloop characteristic polynomial, p(s)-Hurwitz is
precisely the condition for closed-loop stability. In
particular, the Lienard-Chipart criterion states that
the polynomial (1) has all its roots with negative real
parts if and only if any one of the following sets of
inequalities hold:
cn>0, cn-2>0, ...: )1>0, )3>0
(6)
or in a general form:
This section gives reviews of the Lienard-Chipart
stability criterion and the matrix pencil method:
P(s) = sn + c1sn-1 + c2sn-2 + … +cn
(5)
where x(n×1), A(n×n), Ad(n×n),R and J,R+. A and
Ad are constant matrices. The system characteristic
equation is expressed by:
MATERIALS AND METHODS
C
Matrix pencil method: Consider a Linear-TimeInvariant with single Time-Delay System (LTI-TDS)
represented by:
CE ( s, T ) =
M M M M O
0 0 K K ci
nd
∑ qk ( s) T k
k=0
2266
(9)
Res. J. Appl. Sci. Eng. Technol., 4(14): 2265-2272, 2012
where,
q k ( s) =
n + nd
∑ qk 1sn + n −1 = qk 0sn + n
d
d
1= 0
where x(n×1) is the state vector, u 0 R is the control input,
A(n×n) and B(n×1) are the system matrix and the control
vector, respectively. Suppose that the system (15) is
completely controllable and J-stabilizable (Olgac and
Sipahi, 2004). For the state-PID feedback, the delayed
control input is of the form:
+ qk 1sn + nd − 1
+ ... + qk ( n + nd ) , qk 1
are constants.
Step 2: Construct the Hurwitz matrix:
H (qk ) = ∆ n + nd
∫
u(t − τ ) = K p x (t − τ ) + Ki x (t − τ )dt + Kd x& (t − τ )
(10)
Step 3: Compute the real eigenvalues of the matrix
pencil ' for T = 8k, k = 1,..., m, m#nnd as:
'(8) = U8+V
(16)
where Kp, K1, Kd 0 Rn are row gain vectors for the P, I and
D feedback elements, respectively. The closed-loop
system can be expressed by:
(11)
∫
x& (t ) = Ax (t ) + B[ K p x (t − t ) + Ki x (t − t )dt + Kd x& (t − t )]
(17)
where
⎡I
⎢ O
U=⎢
⎢
I
⎢
⎢⎣
⎤
⎥
⎥ ∈ R nd ( n + nd ) × nd ( n + nd )
⎥
⎥
H (qnd ) ⎥⎦
The system (15) possesses the following characteristic
equation:
(12)
1
⎛
⎞
CE ( s, τ ) = det ⎜ sI − A − B ( K p + Ki + Kd s)e −τs ⎟ = 0,τ ∈ R +
⎝
⎠
s
This class of systems exhibits only a finite number of
possible imaginary characteristic roots for all J ,R+ at
given frequencies. The method must be able to detect all
of them. Let us call this set:
and
⎡0
⎢M
V= ⎢
⎢0
⎢
⎢⎣ H (q0 )
−I
K 0
M
O M
M
0 −I
H (q1 ) K H (qnd
⎤
⎥
⎥ ∈ R nd ( n + nd ) × nd ( n + nd )
⎥
⎥
− 1) ⎥⎦
(13)
{Tc} = {Tc1, Tc2, ... , Tcm}
where, U and V consist of square block matrices
of order n+nd.
Step 4: Compute Tck for (11) that results in ±jTck
eigenvalues for (9).
Step 5: Substitute T and T in (14) by Tck and Tck,
respectively.
τ=
2
ω
[ tan
−1
]
(ωT ) + 1π , 1= 0, 1,... ∞
(18)
(19)
where subscript c indicates imaginary axis crossing. This
finite number m is influenced not only by n, but also the
numerical formation of A, B, Kp, KI and Kd matrices.
Furthermore, each Tck, k = 1,..., m corresponds to
infinitely many periodically spaced J values denoted as:
{Jk} = {Jk1, Jk2,...,Jk¥}, k = 1,..., m
(20)
For the characteristic Eq. (18), we first evaluate the
complete root crossing structure [Tck,{Jk}], k = 1... m for
(14)
Jk0 R+. By substitution of e− τs =
Obtain τ = min(J).
1 − Ts
, τ ∈ R + , T ∈ R,
1 + Ts
we
obtain the system characteristic equation expressed as:
Note that, either manual calculation or symbolic
programming is necessary for Steps 1-3. Steps 4-5 need
conventional numerical computing.
1
1 − Ts ⎞
⎛
CE ( s,τ ) = CE ( s, T ) = det ⎜ sI − A − B( K p + Ki + Kd s)
⎟
⎝
s
1 + Ts ⎠
= 0, τ ∈ R + , T ∈ R
(21)
RESULTS AND DISCUSSION
Equation (22) expresses the relationship between T and J.
Consider a LTI-TDS of the form:
x&(t ) = Ax (t ) + Bu (t − τ )
τ=
(15)
2267
2
ω
[ tan
−1
]
(ωT ) + 1π , 1= 0, 1,... ∞
(22)
Res. J. Appl. Sci. Eng. Technol., 4(14): 2265-2272, 2012
Some values of T cause the eigenvalues s = jT with
infinite numbers of J, i.e.,
Tck ↔ s = jω ck ↔ τ k 1 , k = 1, 2.. m, 1 = 0, 1... ∞
Step1: Define the system characteristic equation in the
form of (7) and rewrite it in the form of:
(23)
n
=
[
]
⎫
⎧2
tan −1 (ωT ) + 1π ⎬ = min(τ )
⎭
⎩ω
k
=0
k =0
The maximum delay time ( τ ) can be figured out from:
τ = min ⎨
⎛ 1 − Ts ⎞
∑ ak ( s)⎜⎝ 1 + Ts ⎟⎠
CE ( s, T ) =
n
∑ ak ( s)(1 + Ts)n − k (1 − Ts)k = 0
k =0
(24)
=
2n
∑ bk (T ) sk = 0
(28)
k =0
τ results in critical or marginal stability. This means
that a J-stabilizable system remains stable if and only if
0#J< τ . Applying the Rekasius’ substitution to Eq. (18)
results in a rational polynomial:
n
⎛ 1 − Ts ⎞
CE ( s, T ) = ∑ a k ( s) ⎜
⎟ =0
⎝ 1 + Ts ⎠
k=0
k
(25)
or recasting it into a simpler form:
The following simulation examples illustrate the
effectiveness of the proposed method. Example 1 is
explained in details and the rest are presented in brief.
CE ( s, T ) =
n
∑ ak ( s) (1 + Ts)n− k (1 − Ts)k = 0
(26)
k =0
Example 1: Let us consider a J-stabilizable LTI system
having:
Sorting the terms in power of s, this equation becomes:
CE ( s, T ) =
Step2: Construct the Hurwitz determinant,)i(T),
according to (4)
Step3: Iteratively compute for the values of Tck subject
to either inequality (2) or (3) that results in ±jTck
being the eigenvalues of (28)
Step 4: Substitute T and T in (24) by Tck and Tck
obtained from Step 3, respectively. Obtain τ =
min(J)
⎡0
A= ⎢
⎣ 20.6
2n
∑ bk (T ) sk = 0
(27)
1⎤
0 ⎥⎦
⎡ 0⎤
and B = ⎢ ⎥
1
⎣ ⎦
(29)
k =0
where, bk being the elements of A, BKp, BKI and BKd
matrices. Assuming A, BKp, BKI and BKd are given
constant matrices, bk are parameterized in T 0 R only.
Therefore, the coefficients can be either positive or
negative. It should be noted that the nth degree
transcendental Eq. (18) is now converted into 2n degree
polynomial in Eq. (27) without the transcendental term.
Its purely imaginary characteristic roots coincide with
those of Eq. (18) exactly (Olgac and Sipahi, 2002). These
imaginary roots are determined next.
Due to the Linard-Chipart criterion, the inequality (2)
or (3) holds for a stable system. Since bk in the Hurwitz
determinants, )i, are functions of T, bk(T), the values of
T cause a finite number of sign changes to the
determinants. Therefore, we can compute for Ts that
correspond to imaginary roots ±jTck. Such T values are
used to find the tolerable delay τ = min(J) due to (24).
Computing procedures to obtain τ :
The system is unstable and has its eigenvalues at
±4.539. The system is stabilized by using the state-PID
feedback method (Sujitjorn and Wiboonjaroen, 2011).
The closed-loop poles are -1.8±j2.4 and -8 and can
be achieved via the gain matrices Kp = [-20.6 0], Ki =
[-72 -37.8], Kd = [-11.6 0]. The closed-loop system
without delay is stable. Next, we compute the value τ .
The characteristic polynomial CE(s,J) can be formulated
as:
CE(s,J) = a2(s)e-2Js+a1(s)e-Js+a0(s)
in which a2(s) = 0, a1(s) = 58s2+292s+360 and a0(s) = 5s3130s. Next, CE(s,T) is obtained as:
CE ( s, T ) =
2n
∑ bk (T ) sk = b4 s4 + b3s3 + b2 s2 + b1s + b0
k =0
in which b4(T) = 5T, b3(T) = 5-58T, b2(T) = 58-395T,
b1(T) = 189-360T and b0(T) = 360. For the polynomial
2268
Res. J. Appl. Sci. Eng. Technol., 4(14): 2265-2272, 2012
Using an iterative computing, the set of Ts can be
obtained as T = [0.590, 0.0862, 0.5250, 0.1468]. The
value of T = 0.0590 results in the eigenvalues
0.0042±j10.3034 and -2.6788±j2.0782. For T = 0.0862, T
= 0.5250 and T = 0.1468, the obtained eigenvalues are
[2.8094±j8.263, -2.8099±j1.7518], [-3.4532, -2.0394,
1.4138, 13.7741] and [-2.8588±j1.2528±j4.7698],
respectively. We could say that 0.0042±j10.3034
–±j10.3034(±jTck). Therefore, Tck = 0.0590 s is the critical
time interval and the imaginary-axis crossover frequencies
Tck = ±10.3034 rad/s. Finally, we obtain τ = 106 ms.
Figure 1 illustrates responses of the system states. As
shown in Fig. 1a, the response converges to zero, as the
delay time is less than the maximum allowable delay.
Oscillatory and unstable responses can be observed in
Fig. 1b,c, as the delay times are greater than the maximum
delay.
Now we present the calculation procedures based on
the matrix pencil approach as follows:
One can write the characteristic polynomial of the
system as:
x1
-0.15
x2
-0.10
States
-0.05
-0.00
-0.05
-0.10
-0.15
-0.20
0
0.5
1.0
1.5
2.0 2.5 3.0
Time (s)
3.5
4.0
4.5
5.0
(a) Delay time J = 50 ms< τ
x1
0.6
x2
0.4
States
0.2
0.0
-0.2
-0.4
CE ( s,τ ) = a2 ( s)e − 2τs + a1 ( s)e − τs + a0 ( s)
-0.6
0
1
2
3
4
Time (s)
5
6
8
7
where a2(s) = 0, a1(s) = 58s2+292s+360 and a0(s) = 5s3130s. Using the Rekasius’ substitution, the characteristic
polynomial can be rewritten as:
(b) Delay time J = 106 ms = τ
5
x1
CE ( s, T ) = q0 ( s) + q1 ( s)T + q2 ( s)T 2
States
x2
where q0(s) = 5s3+58s2+189s+360, q1(s) = 5s4-58s3-395s2
-360s and q2(s) = 0. The next step is to form the Hurwitz
matrices and obtained as:
0.0
0
⎡ − 58 − 360
0 ⎤
⎡ 5 189 0
⎢ 5
⎢ 0 58 360 0 ⎥
−
395
0
⎥ , H (q ) = ⎢
H (q0 ) = ⎢
1
⎢ 0 5 189 0 ⎥
⎢ 0
− 58 − 360
⎢
⎥
⎢
58 360⎦
− 395
5
⎣0 0
⎣ 0
-5
0
1
2
3
4
Time (s)
5
6
7
8
and
(c) Delay time J = 110 ms> τ
⎡0
⎢0
H (q2 ) = ⎢
⎢0
⎢
⎣0
Fig. 1: State responses of example 1
CE(s,T), n = 4 is even so we use the inequality (2). The
constructed Hurwitz determinants are as follows:
0 0 0⎤
0 0 0⎥⎥
0 0 0⎥
⎥
0 0 0⎦
Now we can form the matrices U and V as follows:
c4>0, c2>0
⎡1
⎢0
⎢
⎢0
⎢
⎢0
U=⎢
0
⎢
⎢0
⎢0
⎢
⎢⎣ 0
and
c1 c3 0
∆ 1 = c1 = c1 > 0, ∆ 3 = 1 c2 c4 = c1c2c3 − c12c4 − c32 > 0
0 c1 c3
where, c4=b0/b4, c3=b1/b4, c2=b2/ b4 and c1=b3/ b4.
2269
0 0 0 0 0 0 0⎤
1 0 0 0 0 0 0⎥⎥
0 1 0 0 0 0 0⎥
⎥
0 0 1 0 0 0 0⎥
0 0 0 0 0 0 0⎥
⎥
0 0 0 0 0 0 0⎥
0 0 0 0 0 0 0⎥
⎥
0 0 0 0 0 0 0⎥⎦
0⎤
0⎥⎥
0⎥
⎥
0⎦
Res. J. Appl. Sci. Eng. Technol., 4(14): 2265-2272, 2012
0
0
0
0
0⎤
−1
⎡0 0
⎢0 0
0
0
0
0
0 ⎥⎥
−1
⎢
⎢0 0
0
0
0
0
0⎥
−1
⎥
⎢
0
0
0
0
0
0
0
1⎥
−
⎢
V=⎢
5 189 0
0 − 58 − 360
0
0⎥
⎥
⎢
5
0
0⎥
− 395
⎢ 0 58 360 0
⎢ 0 5 189 0
0
− 58 − 360 0 ⎥
⎥
⎢
⎢⎣ 0 0
58 360
0
5
− 395 0 ⎥⎦
x1
8
States
6
Example 2: The magnetic ball suspension system (Kuo,
1987) represented by the diagram in Fig. 2 is adopted as
the second example. This 3rd order system is described by:
⎡ 0 ⎤
⎢
⎥
⎢ 0 ⎥u
⎢⎣100⎥⎦
4
2
0
0.1
0
0.2
0.3
0.4 0.5 0.6
Time (s)
0.7
0.8 0.9
1.0
(a) Delay time J = 1ms< τ
x1
x2
X3
15
10
5
States
As a result, '(8) = 0.0590,0.2921±j0.0447.
Therefore, T = 0.0590 is used. The value of T results in
eigenvalues = 0.0043±j10.3034 and -2.6788±j2.0782,
respectively. We could say that 0.0043 ± j10.3034 – ±
j10.3034(±jTck) and hence Tck = 0.0590 and Tck = ±
10.3034 rad/s. Finally, the maximum allowable delay is
obtained as τ = 106 ms, which is equal to that obtained
previously.
Example 1 illustrates that the proposed direct method
is exact and gives the solutions of the same accuracies as
those of the existing matrix pencil method. To apply the
matrix pencil method, one needs to know matrix algebra
and numerical computation. To apply the proposed direct
method needs only basic knowledge of Lienard-Chipart
criterion and loop iterative computing commonly taught
in undergraduate level.
⎡ 0 1
0 ⎤
⎢
⎥
x& = ⎢ 980 0 − 2.8 ⎥ x +
⎢⎣ 0 0 − 100⎥⎦
x2
X3
10
0
-5
-10
-15
0
0.5
1.0
1.5 2.0 2.5 3.0
Time (s)
3.5
4.0
4.5
5.0
3.5
4.0
4.5
5.0
(b) Delay time J = 4 ms = τ
(30)
150
x1
100
x2
X3
States
50
0
-50
-100
-150
0
0.5
1.0
1.5
(c) Delay time J = 4.2 ms >
2.0 2.5 3.0
Time (s)
τ
Fig. 3: State responses of example 2
Fig. 2: Magnetic ball suspension
where x1 = y, x2 = y& and x3 = i. The system is originally
unstable with its poles at ±31.3050 and -100. The closedloop poles at -10±j3.464, -11 and -11can be achieved via
the state-PID feedback. The proposed direct method is
applied to obtain the tolerable delay. As a result, Kp = [0
3.5 4], KI = [4759.78 17.58 -13.46], Kd = [-2.42 0.15 0],
Tck = 0.0020, Tck±11.50 rad/s and τ = 4 ms. The similar
figures are obtained from using the matrix pencil method.
Fig. 3a illustrates a stable response, while Fig. 3b, c
illustrate the unstable cases.
2270
Res. J. Appl. Sci. Eng. Technol., 4(14): 2265-2272, 2012
60
x1
x3
x2
x4
40
States
20
0
-20
-40
0
1
2
3
Time (s)
4
5
6
(c) Delay time J = 43 ms > τ
Fig. 5: State responses of example 3
Fig. 4: Inverted pendulum on cart
Example 3: An inverted pendulum system (Ogata, 2002)
is adopted as an example and represented by the diagram
in Fig. 4. Its state model is expressed by:
0
⎡
⎢ 20.601
x& = ⎢
⎢
0
⎢
⎣ − 0.4905
1.0
1 0 0⎤
0 0 0⎥⎥
x+
0 0 1⎥
⎥
0 0 0⎦
x1
x3
x2
x4
⎡ 0⎤
⎢ − 1⎥
⎢ ⎥u
⎢ 0⎥
⎢ ⎥
⎣ 0.5⎦
(31)
0.5
States
0.0
-0.5
where, x1 = 2, x2 = θ& , x3 = x and x4 = x& . The system is
inherently unstable with its open-loop poles at 0,0 and
±4.5388. It is stabilized via the state-PID feedback to
achieve the desired closed-loop poles at -2±3.464j,-4,-10
and -10. The proposed direct method is applied to obtain
the tolerable delay. As a result, Kp = [-20.601 0 0 0], KI =
[7123.1490 1490.2386 1956.3781 1043.4560], Kd =
[-120.6720 -24.6841 -313.0368 -49.3195], Tck = 0.0236,
Tck = ±27.62 rad/s and τ = 41.8 ms.
In this case, the matrix pencil method fails to provide
a result since the matrix ' is singular. Figure 5 illustrates
responses of the system states. As shown in Fig. 5a, the
response converges to zero, as the delay time is less than
the tolerable delay. In contrast, oscillatory and unstable
responses can be observed in Fig. 5b,c, as the delays
exceed the tolerable delay.
CONCLUSION
-1.0
-1.5
0
1
2
3
Time (s)
4
5
6
(a) Delay time J = 20 ms < τ
3
x1
x3
x2
x4
2
States
1
0
The maximum allowable delay or tolerable delay can
be predicted for retarded/neutral systems by using our
proposed method. The method employs simple iterative
computing, the Lienard-Chipart stability criterion and the
Rekasius’ substitution to represent the transcendental
term. Computational results are compared with those
obtained from the matrix pencil method. Both methods
have similar accuracies. However, the matrix pencil
method fails to provide a solution for one example in
which a matrix is singular. The direct method is
successful with this case.
-1
ACKNOWLEDGEMENT
-2
-3
0
1
2
3
Time (s)
(b) Delay time J = 41.8 ms = τ
4
5
6
The study was supported by Suranaree University of
Technology (SUT), the Office of the Higher Education
Commission under NRU project of Thailand and
Ratchamangkala University of Technology Isarn,
Thailand.
2271
Res. J. Appl. Sci. Eng. Technol., 4(14): 2265-2272, 2012
REFERENCES
Abdelaziz, T.H.S. and M. Valasek, 2003. 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, Prague,
Czech Republic, 16(1).
Asl, F.M. and A.G. Ulsoy, 2003. Analysis of a system of
linear delay differential equations. J. Dyn. Syst.
Meas. Cont., 125(2): 215-223.
Chen, C.T., 1984. Linear System Theory and Design.
Holt, Rinehart and Winston, New York.
Chen, J., G. Gu and C.N. Nett, 1995. A new method for
computing delay margins for stability of linear delay
systems. Syst. Cont. Lett., 26(2): 107-117.
Dorato, P., 2000. Analytic Feedback System Design-An
Interpolation Approach. Brooks/Cole, pp: 98-100.
Fu, P., S.I. Niculescu and J. Chen, 2006. Stability of
linear neutral time-delay systems: Exact conditions
via matrix pencil solutions. IEEE T. Autom. Cont.,
51(6):1063-1069.
Gantmacher, F.R., 1959. Matrix Theory (2). Chelsea
Publishing Co., pp: 220-225.
Gu, K. and S.I Niculescu, 2003. Survey on recent results
in the stability and control of time-delay system. J.
Dyn. Syst. Meas. Cont., 125: 158-165.
Guo, G., Z. Ma and J. Qiao, 2006. State-PID feedback
control with application to a robot vibration absorber.
Int. J. Modell. Identif. Cont., 1(1): 38-43.
Hale, J.K. and S.M. Verduyn Lunel, 2001. Effect of small
delays on stability and control. Oper. Th. Adv. Appl.,
122: 275-301.
Hale, J.K. and S.M. Verduyn Lunel, 2002. Strong
stabilization of neutral function differential equation.
IMA J. Mathemat. Cont. Inform., 19: 5-23.
Kuo, B.C., 1987. Automatic Control Systems, Prentice
Hall, New York, USA.
Michiels,W., K. Engelborghs, P. Vansevenant, D. Roose,
2002. Continuous pole placement for delay
equations. Automatica, 38(5): 747-761.
Michiels, W., K. Engelborghs, D. Roose and D. Dochain,
2004. Sensitivity to infinitesimal delays in neutral
equations. SIAM J. Cont. 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, 2010. Stabilizability and disturbance rejection
with state-derivative feedback. Math. Probl. Eng., ID
123751: 12.
Niculescu, S.I., 2001. Delay Effect on Stability. SpringerVerlag, New York.
Ogata, K., 2002. Modern Control Engineering. Prentice
Hall, New York, USA.
Olgac, N. and R. Sipahi, 2002. An exact method for the
stability analysis of time-delayed: Inear TimeInvariant (LTI) systems. IEEE T. Autom. Cont.,
47(5): 793-797.
Olgac, N. and R. Sipahi, 2004. A practical method for
analyzing the stability of neutral type LTI-Time
Delayed systems. Automatica, 40(5): 847-853.
Olgac, N. and R. Sipahi, 2005. The cluster treatment of
characteristic roots and the neutral type time-delayed
systems. Trans. ASME, 127(1): 88-97.
Park, P., 1999. A delay-dependent stability criterion for
systems with uncertain time-invariant delays. IEEE
T. Autom. Cont., 44: 876–877.
Rekasius, Z.V., 1980. A stability test for systems with
delays (TP9-A). In Proceeding Joint Automatic
Control Conference, San Franscisco, USA.
Reithmeier, E. and G. Leitmann, 2003. Robust vibration
control of dynamical systems based on the derivative
of the state. Arch. Appl. Mech., 72(11-12): 856-864.
Richard, J.P., 2003. Time-delay systems: An overview of
some recent advances and open problems.
Automatica, 39: 1667-1964.
Sipahi, R. and N. Olgac, 2006. Stability robustness of
retarded LTI system with single delay and exhaustive
determination of their imaginary spectra. SIAM J.
Control Optim., 45(5): 1680-1696.
Sipahi, R. and N. Olgac, 2003. Degenerate cases in using
direct method. J. Dyn. Syst. Meas. Cont., 125(2):
194-201.
Sujitjorn, S. and W. Wiboonjaroen, 2011. State-PID
feedback for pole placement of LTI system. Math.
Probl. Eng., DOI 10.1155/2011/929430.
Yi, S., A.G. Ulsoy and P.W. Nelson, 2010. Eigenvalue
assignment via the Lambert W function for control of
time-delay system. J. Vib. Control, 16(7-8): 961-982.
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