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Short communication
Comments on ?An LMI approach to
non-fragile robust optimal guaranteed
cost control of uncertain 2-D discrete
systems with both state and input
delays?
Transactions of the Institute of
Measurement and Control
1?5
Σ The Author(s) 2017
Reprints and permissions:
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DOI: 10.1177/0142331217732829
journals.sagepub.com/home/tim
Neha Agarwal and Haranath Kar
Abstract
This paper points out some technical errors in a recent paper that appeared in Transactions of the Institute of Measurement and Control entitled ?An LMI
approach to non-fragile robust optimal guaranteed cost control of uncertain 2-D discrete systems with both state and input delays? by Akshata Tandon
and Amit Dhawan (http://dx.doi.org/10.1177/0142331216667476). We reveal that the upper bound of the closed-loop cost function provided by their
Lemma 4 is erroneous. Some critical issues associated with the system initial conditions assumed in their paper are highlighted. The closed-loop cost
bound claimed by their Theorem 1 is found to be incorrect. The optimization problem formulated in their Theorem 2 for the selection of an optimal
guaranteed cost controller is erroneous. Finally, the corrections over their results are made available.
Keywords
2-D discrete systems, guaranteed cost control, linear matrix inequality, Lyapunov methods, robust stability, state-delayed systems, uncertain systems.
Introduction
Notations
During the past two decades, the guaranteed cost control
problem for a two-dimensional (2-D) discrete system has
drawn the attention of many researchers (see Dhawan and
Kar, 2007, 2011; Hien and Trinh, 2017; Tandon and Dhawan,
2016; Xu and Yu, 2009; Ye et al., 2011, and the references
cited therein for a sample of the literature) due to its extensive
application in controlling a real plant. The aim of guaranteed
cost control is to design a controller such that the resulting
closed-loop system is asymptotically stable and an upper
bound on the closed-loop cost function is guaranteed.
This paper focuses on the results proposed in Tandon and
Dhawan (2016) for non-fragile optimal guaranteed cost control of 2-D discrete uncertain systems described by the general
model with both state and input delays. The purpose of this
paper is to point out several technical errors that occurred in
that article. We argue that the upper bound of the closed-loop
cost function obtained in their Lemma 4 is absurd and its
proof is mathematically incorrect. Some critical issues associated with the assumptions on the system initial conditions
made in their paper are highlighted. It is illustrated that the
closed-loop cost bound as claimed by their Theorem 1 is
incorrect. For the selection of a non-fragile robust optimal
guaranteed cost controller, the convex optimization problem
proposed in Theorem 2 of their paper is erroneous. Some
remarks on their illustrative example are also made. In addition, this paper describes the corrected forms of their results.
Unless explicitly mentioned otherwise, the notations and symbols adopted throughout this paper have same meaning as in
Tandon and Dhawan (2016). Moreover, equation numbers
followed by an asterisk ?*?, e.g. (5*), stand for the equations
in Tandon and Dhawan (2016), whereas equation numbers
without an asterisk refer to equations in this paper.
The fallacy in Lemma 4 of Tandon and
Dhawan (2016)
Note that the closed-loop cost function J given by (5a*) is
always a non-negative scalar. Therefore, the closed-loop cost
bound should remain non-negative for all possible solutions
obtainable via (7*) for a given 2-D system. Now, the fallacy
in Lemma 4 of Tandon and Dhawan (2016) can best be visualized with the help of a simple example. Consider a secondorder 2-D delayed system described by (4*), (1e*) and (1f*)
Department of Electronics and Communication Engineering, Motilal
Nehru National Institute of Technology Allahabad, Allahabad, India
Corresponding author:
Neha Agarwal, Department of Electronics and Communication
Engineering, Motilal Nehru National Institute of Technology Allahabad,
Allahabad 211004, India.
Email: nehaagarwal.mnnit@gmail.com
2
Transactions of the Institute of Measurement and Control 00(0)
with g = h = k = p = m = n = l = d = 2, L1 =
L2 = 101, M = I2 , for which (7*) yields the following feasible
solution:
? X
? X
xT πi l, j mήQ4 xπi l, j mή xT πi + 1, j + 1ή
i=0 j=0
Q4 xπi + 1, j + 1ή =
Q = Q1 = Q2 = Q3 = Q4 = Q5 = Q6 = Q7 = I2 ,
xT πi, jή Q4 xπi, jή
i = l j = m
~ = 8I2 , P = 9I2
Q
π1ή
where I2 is the 2 3 2 identity matrix. Pertaining to this example, (10*) leads to
J J = 18381
π2ή
which is absurd. Thus, Lemma 4 in Tandon and Dhawan
(2016) is found to be incorrect.
Critical issues associated with initial
conditions and corrected form of Lemma
4 in Tandon and Dhawan (2016)
A close examination reveals that the last three steps in (58*),
which are crucial to obtain an upper bound on the closed-loop
cost function, are mathematically incorrect. Moreover, it will
be illustrated that the initial conditions given by {(1e*), (1f*)}
for the system (1a*) are inadequate and incapable of eliminating the dependence of guaranteed cost value on initial conditions. Before proceeding further, we assume in this paper that
system (1a*) has a finite set of initial conditions, i.e. there exist
two positive integers L1 and L2 such that
0
0
X
X
? X
?
X
xT πi, jή Q4 xπi, jή,
and
? X
? X
xT πi n, j pήQ7 xπi n, j pή xT πi + 1, j + 1ή
i=0 j=0
Q7 xπi + 1, j + 1ή
=
0
0
X
X
xT πi, jή Q7 xπi, jή i = n j = p
? X
?
X
π3aή
xT πi, jή Q7 xπi, jή, π5ή
i=1 j=1
which are, unfortunately, incorrect. The relation (4) can be
corrected as follows:
? X
?
X
xT πi l, j mήQ4 xπi l, j mή xT πi + 1, j + 1ή
i=0 j=0
Q4 xπi + 1, j + 1ή
?
?
? X
?
X
X
X
=
xT πi, jήQ4 xπi, jή xT πi, jήQ4 xπi, jή,
i = l j = m
=
?
0
X
X
i=1 j=1
xT πi, jήQ4 xπi, jή +
i = l j = m
xπi, jή = 0, 8i L1 , j = u, u + 1, . . . , 0,
xπi, jή = 0, 8j L2 , i = k, k + 1, . . . , 0,
π4ή
i=1 j=1
? X
?
X
? X
?
X
xT πi, jήQ4 xπi, jή
i = l j = 1
xT πi, jήQ4 xπi, jή,
i=1 j=1
and
=
xπi, jή = nij ,
xπi, jή = vij ,
xπi, jή = uij ,
π3bή
and the initial conditions are arbitrary but belong to the set
[ fxπt, jή 2 R : xπt, jή = MN2 ,
0\j\L2 , t = k, k + 1, . . . , 0g
n
[ fxπt, qή 2 R :
xT πi, jήQ4 xπi, jή =
?
0
X
X
? X
?
X
xT πi, jήQ4 xπi, jή
xT πi, jήQ4 xπi, jή,
i=1 j=1
xT πi, jήQ4 xπi, jή +
0 X
?
X
xT πi, jήQ4 xπi, jή:
i = l j = 1
π6ή
NT1 N1 \1,
Using similar steps to those shown in (6), relation (5) can be
corrected as
NT2 N2 \1,
π3cή
? X
?
X
xT πi n, j pήQ7 xπi n, j pή xT πi + 1, j + 1ή
i=0 j=0
t = l, l + 1, . . . , 0,
q = m, m + 1, . . . , 0g,
? X
?
X
i = l j = m
NT3 N3 \1,
xπt, qή = MN3 ,
+
0 X
?
X
i = l j = 1
i=1 j=1
1
n
xT πi, jήQ4 xπi, jή +
i = l j = m
j 2 ½u, 0, 0\i\L1 ,
i 2 ½k, 0, 0\j\L2 ,
i 2 l, 0 , j 2 ½m, 0,
S = fxπi, qή 2 Rn : xπi, qή = MN1 ,
0\i\L1 , q = u, u + 1, . . . , 0g
?
0
X
X
Q7 xπi + 1, j + 1ή
where l = maxfl , n g, m = maxfm , p g, u = maxπk , m ,
h , p ή, k = maxπd , l , g , n ή and M is a given matrix. We
will provide the reasoning behind assumption (3) after making necessary corrections in (58*).
It may be noted that (58*) makes use of the relations
=
?
0
X
X
i = n j = p
xT πi, jήQ7 xπi, jή +
0 X
?
X
xT πi, jήQ7 xπi, jή:
i = n j = 1
π7ή
In view of (3), (6) and (7), the flaw in the last three steps in
(58*) is now corrected as follows:
Agarwal and Kar
J
3
?
X
?
?
X
X
~ xπ0, j + 1ή +
xT π0, j + 1ή P Q
xT πi + 1, 0ή Q xπi + 1, 0ή +
xT πi, 0ή Q1 xπi, 0ή
j=0
+
?
X
i=0
xT π0, jή Q1 xπ0, jή +
?
0
X
X
T
x πi, jή Q4 xπi, jή +
i = l j = m
+
? X
0
X
LX
2 1
+
xT πi + 1, jή Q6 xπi + 1, jή +
xT π0, jή Q1 xπ0, jή +
0 LX
2 1
X
0
X
T
x πi, jή Q4 xπi, jή +
0
X
LX
1 1
xT πi, jή Q6 xπi, jή +
xT πi, jή Q7 xπi, jή
i = n j = 1
i=0
LX
1 1
xT πi, jή Q2 xπi, jή +
0 LX
2 1
X
LX
1 1
0
X
xT πi, jή Q3 xπi, jή
i = 1 j = k
T
x πi, jή Q4 xπi, jή +
0
X
0 LX
2 1
X
π9ή
T
x πi, jή Q5 xπi, jή
i = g j = 1
xT πi, jή Q7 xπi, jή +
i = n j = p
0 LX
2 1
X
xT πi, jή Q7 xπi, jή
i = n j = 1
LX
LX
1 1
1 1
~ xπ0, jή +
xT π0, jή P Q
xT πi, 0ή Q xπi, 0ή +
xT πi, 0ή Q1 xπi, 0ή
LX
2 1
i=1
xT π0, jή Q1 xπ0, jή +
LX
0
2 1
X
i=0
xT πi, jή Q2 xπi, jή +
i = d j = 1
LX
1 1
0
X
xT πi, jή Q4 xπi, jή +
i = l j = m
+
0 X
?
X
xT πi, jή Q7 xπi, jή +
i = l j = 1
j=1
+
?
0
X
X
i = d j = 1
j=1
+
x πi, j + 1ή Q5 xπi, j + 1ή
i = g j = 0
i=1
LX
1 1
LX
2 1
π8ή
T
LX
LX
1 1
1 1
~ xπ0, jή +
xT π0, jή P Q
xT πi, 0ή Q xπi, 0ή +
xT πi, 0ή Q1 xπi, 0ή
i = 1 j = h
x πi, jή Q4 xπi, jή +
i = n j = p
i = l j = m
+
0 X
?
X
T
i = l j = 1
j=1
+
xT πi + 1, jή Q3 xπi + 1, jή
i = 0 j = k
0 X
?
X
j=1
LX
2 1
? X
0
X
xT πi, j + 1ή Q2 xπi, j + 1ή +
i = 0 j = h
=
i=0
i = d j = 0
j=1
+
0 X
?
X
LX
1 1
0
X
i=1
j = h
LX
1 1
0
X
LX
0
2 1
X
xT πi, jή Q4 xπi, jή +
J
LX
1 1
0
X
i = n
j = p
LX
0
2 1
X
π10ή
xT πi, jή Q5 xπi, jή
i = g j = 1
i = l j = 1
xT πi, jή Q6 xπi, jή +
xT πi, jή Q3 xπi, jή
i = 1 j = k xT πi, jή Q7 xπi, jή +
LX
0
2 1
X
i = n
xT πi, jή Q7 xπi, jή
j=1
π11ή
where
n
o
~ M + πL1 1ήlmax MT QM
J = ½πL2 1ήlmax MT P Q
+ πL1 + L2 1ήlmax MT Q1 M
+ πL2 1ήπd + 1ήlmax MT Q2 M
+ πL1 1ήπk + 1ήlmax MT Q3 M
+ fπl + 1ήπL2 1ή + πm + 1ήπl + L1 ήglmax MT Q4 M
+ πL2 1ήπg + 1ήlmax MT Q5 M
+ πL1 1ήπh + 1ήlmax MT Q6 M
+ fπn + 1ήπL2 1ή + πp + 1ήπn + L1 ήglmax MT Q7 M :
π12ή
In the following, we present the reasoning behind assumption
(3). Note that assumption (3a), which is not explicitly
mentioned in Tandon and Dhawan (2016), is essential to
arrive at (9) from (8). In fact, assumption (3a) plays an
important role in the formulation of the guaranteed cost control problem for 2-D systems.
Furthermore, we observe that the upper bound of the cost
function given by (10) depends on a set of initial conditions
which is incompletely specified in {(1e*), (1f*)} but completely
defined in {(3b), (3c)}. As an illustration of this, consider a
specific example where
l = n = m = p = 4, d = g = k = h = 2, L1 = L2 = 3:
π13ή
For this example, the bound of the cost function obtained in
the right hand side of (10) depends on the initial state vectors
x(i, j) 2 X0 , where
4
Transactions of the Institute of Measurement and Control 00(0)
X0 = fxπ4, 0ή, xπ3, 0ή, xπ2, 0ή, xπ1, 0ή, xπ0, 0ή,
xπ1, 0ή, xπ2, 0ή, xπ4, 1ή, xπ3, 1ή, xπ2, 1ή,
xπ1, 1ή, xπ0, 1ή, xπ1, 1ή, xπ2, 1ή,
xπ4, 2ή, xπ3, 2ή, xπ2, 2ή, xπ1, 2ή,
xπ0, 2ή, xπ1, 2ή, xπ2, 2ή,
xπ4, 3ή, xπ3, 3ή, xπ2, 3ή, xπ1, 3ή,
xπ0, 3ή, xπ1, 3ή, xπ2, 3ή,
xπ4, 4ή, xπ3, 4ή, xπ2, 4ή, xπ1, 4ή,
xπ0, 4ή, xπ1, 4ή, xπ2, 4ή,
xπ4, 1ή, xπ3, 1ή, xπ2, 1ή, xπ1, 1ή, xπ0, 1ή,
π14ή
J J =
1
1
1
1
½πL2 1ήlmax fMT P1
1 P1 Y1 P1 P1 Y2 P1
1
1
1
P1
1 Y3 P1 : P1 Y4 P1
xπ4, 2ή, xπ3, 2ή, xπ2, 2ή, xπ1, 2ή, xπ0, 2ήg:
However, for this example, one can easily verify that the set
of initial state vectors given by
X1 = fxπ1, 3ή, xπ1, 4ή, xπ2, 3ή, xπ2, 4ή, xπ4, 1ή,
xπ3, 1ή, xπ4, 2ή, xπ3, 2ήg X0
guaranteed cost control problem if there exist positive scalars
e1 , e2 , a m 3 n matrix U, n 3 n positive definite symmetric
matrices P1 , Y1 , Y2 , Y3 , Y4 , Y5 , Y6 , Y7 and Y8 such that the
LMI given by (11*) is feasible. In this situation, the feedback
gain of the stabilizing non-fragile guaranteed cost control law is
given by (12*). Moreover, the closed-loop cost function satisfies the bound
π15ή
remains undefined in {(1e*), (1f*)}. In this context, it may be
mentioned that the structure of initial conditions in (1f*) or
(3c) has been widely adopted in the literature (Dhawan and
Kar, 2007, 2011; Hien and Trinh, 2017) for removing the
direct dependence of the closed-loop cost bound on the initial
conditions. As the set of initial state vectors given by X1 (see
Equation (15)) are not defined in (1f*), it is not possible to
eliminate the direct dependence of the closed-loop cost bound
on all the initial conditions x(i, j) 2 X1 under assumption
(1f*). Thus, the approach in Tandon and Dhawan (2016) fails
to define adequately the required initial conditions in the present example. On the other hand, one can verify that {(3b),
(3c)} succeeds to define all the initial state vectors x(i, j) 2 X0
for the present case. This illustrates why the initial conditions
for the 2-D system should be assumed as (3) instead of {(1e*),
(1f*)}.
Based on the above discussions, we now correct the
Lemma 4 of Tandon and Dhawan (2016) as follows.
Lemma C1. Suppose there exist n 3 n positive definite symmetric matrices P, Q, Q1 , Q2 , Q3 , Q4 , Q5 , Q6 and Q7 for the
system (4*) with initial conditions (3) and cost function (5*)
such that (7*) holds. Then: i) system (4*) is asymptotically stable and ii) for all admissible uncertainties, the closed-loop cost
function (5*) satisfies the bound J J where J is given by
(12).
Corrections to Theorems 1 and 2 in
Tandon and Dhawan (2016)
We observe that the proof of Theorem 1 in Tandon and
Dhawan (2016) utilizes their Lemma 4 (which, as discussed
previously, is incorrect) to obtain the closed-loop cost bound
(13*) and, consequently, Theorem 1 in their paper is erroneous. Using Lemma C1 of this paper, Theorem 1 in Tandon
and Dhawan (2016) can be corrected as follows.
Theorem C1. Consider system (4*) with initial conditions (3)
and cost function (5*), then there exists a non-fragile state
feedback control law (3*) that solves the addressed robust
1
1
1
1
1
1
1
P1
1 Y5 P1 P1 Y6 P1 P1 Y7 P1 P1 Y8 P1 M
1
+ πL1 1ήlmax MT P1
1 Y1 P1 M + πL1 + L2 1ή
T 1
lmax M P1 Y2 P1
1 M
1
+ πL2 1ήπd + 1ήlmax MT P1
1 Y3 P 1 M
1
+ πL1 1ήπk + 1ήlmax MT P1
1 Y4 P 1 M
+ fπl + 1ήπL2 1ή + πm + 1ήπl + L1 ήg
1
lmax MT P1
1 Y5 P 1 M
1
+ πL2 1ήπg + 1ήlmax MT P1
1 Y6 P 1 M
1
+ πL1 1ήπh + 1ήlmax MT P1
1 Y7 P 1 M
+ fπn + 1ήπL2 1ή + πp + 1ήπn + L1 ήg
1
lmax MT P1
1 Y8 P1 M :
π16ή
The proof of Theorem 2 in Tandon and Dhawan (2016) relies
on their Theorem 1 and, therefore, Theorem 2 of their paper
turns out to be incorrect. Employing Theorem C1 of this
paper and following similar steps as given in the proof of
their Theorem 2, one can easily arrive at the following result.
Theorem C2. Consider system (4*) with initial conditions (3)
and cost function (5*). If the following optimization problem:
minimizefπL2 1ήa + πL1 1ήb + πL1 + L2 1ήg
+ πL2 1ήπd + 1ήd + πL1 1ήπk + 1ήs
+ fπl + 1ήπL2 1ή + πm + 1ήπl + L1 ήgl
+ πL2 1ήπg + 1ήm + πL1 1ήπh + 1ήn
+ fπn + 1ήπL2 1ή + πp + 1ήπn + L1 ήghg
π17ή
subject to (14*) has a feasible solution a . 0, b . 0, g . 0,
d . 0, s . 0, l . 0, m . 0, n . 0, h . 0, e1 . 0, e2 . 0, a m 3 n
matrix U and n 3 n positive definite symmetric matrices P1 ,
Y1 , Y2 , Y3 , Y4 , Y5 , Y6 , Y7 and Y8 , then the control law (3*)
with K = UP1
is the non-fragile robust optimal guaranteed
1
cost control law that ensures the minimization of guaranteed
cost in (16).
Theorem C2 may be treated as the corrected form of
Theorem 2 in Tandon and Dhawan (2016).
Comments on the illustrative example in
Tandon and Dhawan (2016)
Some comments on the illustrative example presented in
Tandon and Dhawan (2016) are in order.
Agarwal and Kar
i)
ii)
5
Let us concentrate on the selection of the matrix M
in the illustrative example of Tandon and Dhawan
0:01
0
(2016). For the chosen M =
by their
0:02 0:05
paper, it can be verified that none of the initial conditions given by (51*) can be expressed in the form
MNk , k = 1, 2, 3, such that NTk Nk \1, k = 1, 2, 3,
holds (see assumptions (1f*) and (3c)). For instance, as
0:1
demanded by (3c) or even (1f*), xπ2, 2ή =
0:1
is required to be of the form xπ2, 2ή = MN3 satisfying NT3 N3 \1. On the other hand, a routine calculation shows that NT3 N3 = 104 . 1 in the present case.
In other words, the choice of M is inappropriate in
their example.
Although the illustrative example of Tandon and
Dhawan (2016) considers a 2-D state-space system
having two state variables, the state response for a single state of the closed-loop system has been shown in
their Figure 2. Now, we would like to make a comment on the following statement appearing at the last
paragraph of their ?Illustrative example? section: ?. it
can be observed from Figure 2 that the closed-loop
system is asymptotically stable.? This statement is illogical because no conclusion on the stability of a system
characterized by two state variables can generally be
made by observing the state response for single state
only. Moreover, asymptotic stability of the closed-loop
system cannot be ensured merely on the basis of the
convergence of the system state trajectories starting
from some specific values of initial conditions.
Additional comments
We would like to remark that (2b*) should be corrected as
t ij = uT πi, j + 1ή
uT πi + 1, jή
uT πi, jή
uT πi g, j + 1ή
uT πi + 1, j hή uT πi n, j pή T
π18ή
and (2e*) should be replaced by
jij = xT πi, j + 1ή xT πi + 1, jή
xT πi, jή
xT πi d, j + 1ή
T
π19ή
xT πi + 1, j k ή xT πi l, j mή
to make (2a*) dimensionally compatible.
Conclusion
In this paper, errors in Tandon and Dhawan (2016) have been
pointed out and corrections have been proposed.
Acknowledgements
The authors wish to thank the Editor-in-Chief and the anonymous reviewers for their constructive comments.
Declaration of conflicting interest
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
Note
1. It may also be noted that NTk Nk \I, k = 1, 2, 3 was
assumed in (1f*). However, a routine verification shows
that NTk Nk , k = 1, 2, 3, is always a scalar quantity.
Therefore, we have assumed NTk Nk \1 (instead of
NTk Nk \I), k = 1, 2, 3 in (3c).
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Hien LV and Trinh H (2017) Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser
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Tandon A and Dhawan A (2016) An LMI approach to non-fragile
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