Multidim Syst Sign Process (2012) 23:415–419DOI 10.1007/s11045-011-0151-6

COMMUNICATION BRIEF

Comment on “Robust guaranteed cost control for a classof two-dimensional discrete systems with shift-delays”

Manish Tiwari · Amit Dhawan

Received: 26 January 2011 / Accepted: 26 March 2011 / Published online: 12 April 2011© Springer Science+Business Media, LLC 2011

Abstract This paper points out some technical errors in the derivation of main results inthe above paper and present the results in corrected form.

Keywords 2-D discrete systems · Shift-delay · Guaranteed cost control · Linear matrixinequality

1 Introduction

In the work of Ye et al. (2009), the problem of robust guaranteed cost control via static-statefeedback for a class of two-dimensional (2-D) discrete shift-delayed system in Fornasini-Marchesini (FM) second model with norm-bounded parameter uncertainty, is considered andlinear matrix inequality (LMI) based sufficient condition for the existence of a guaranteed costcontroller is derived. However, there are several technical errors in the proposed main resultsthat need to be pointed out. Unless explicitly mentioned otherwise, the notations/symbolsand equation numbers used throughout this paper have same meaning as in Ye et al. (2009).

Consider the uncertain 2-D discrete shift-delayed system and its associated cost func-tion given by Eqs.(1) and (2). The main objective of Ye et al. (2009) is to derive sufficientconditions for the existence of a static-state feedback controller for system (1) with costfunction (2) such that the closed-loop system is asymptotically stable and the cost functionof closed-loop system is lower than a specified upper bound.

M. Tiwari (B) · A. DhawanDepartment of Electronics and Communication Engineering, Motilal Nehru National Instituteof Technology, Allahabad 211004, Indiae-mail: manishtiwari_me@yahoo.com

A. Dhawane-mail: amit_dhawan2@rediffmail.com

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416 Multidim Syst Sign Process (2012) 23:415–419

2 Revisions

In this section, some technical errors that occurred in main results of Ye et al. (2009) arepointed out and corrected.

Let us concentrate on the proof of Lemma 3 in Ye et al. (2009). In the proof of Lemma 3,the following cost upper bound is defined

J0 < −∞∑

i=0

∞∑

j=0

ξ Ti jΓ ξ i j

= −∞∑

i=0

∞∑

j=0

[xT (i + 1, j + 1) Px (i + 1, j + 1) − xT (i + 1, j)

×(P − Q − Q1 − Q2)x (i + 1, j)

−xT (i, j + 1) Qx (i, j + 1) − xT (i − d, j + 1) Q1x (i − d, j + 1)

−xT (i + 1, j − k) Q2x (i + 1, j − k)].

It can be easily seen from the above expression that state vector associated with (P − Q −Q1 − Q2) is x (i + 1, j) and that with Q is x (i, j + 1). However, ξ i j and Γ are defined inYe et al. (2009) as

ξ i j = [xT (i, j + 1) xT (i + 1, j) xT (i − d, j + 1) xT (i + 1, j − k)

]T,

and

Γ =

⎡

⎢⎢⎣

AT�1

AT�2

AT�1d

AT�2k

⎤

⎥⎥⎦ × P[

A�1 A�2 A�1d A�2k] −

⎡

⎢⎢⎣

P − Q − Q1 − Q2 0 0 00 Q 0 00 0 Q1 00 0 0 Q2

⎤

⎥⎥⎦ < 0.

In the light of above definitions (ξ i j and Γ ), it is obvious that the authors have made afundamental mistake in associating the state vectors with (P − Q − Q1 − Q2) and Q in thederivation of Lemma 3, which should be corrected.

In what follows, the correction for the proof of Lemma 3 is presented.

J0 < −∞∑

i=0

∞∑

j=0

ξ Ti jΓ ξ i j

= −∞∑

i=0

∞∑

j=0

[xT (i + 1, j + 1) Px (i + 1, j + 1) − xT (i, j + 1)

×(P − Q − Q1 − Q2)x (i, j + 1)

−xT (i + 1, j) Qx (i + 1, j) − xT (i − d, j + 1) Q1x (i − d, j + 1)

−xT (i + 1, j − k) Q2x (i + 1, j − k)]

=∞∑

i=0

∞∑

j=0

[xT (i, j + 1) (P − Q − Q1 − Q2)x (i, j + 1)

−xT (i + 1, j + 1) (P − Q − Q1 − Q2)x (i + 1, j + 1)

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Multidim Syst Sign Process (2012) 23:415–419 417

+ xT (i + 1, j) Qx (i + 1, j) − xT (i + 1, j + 1) Qx (i + 1, j + 1)

+ xT (i − d, j + 1) Q1x (i − d, j + 1) − xT (i + 1, j + 1) Q1x (i + 1, j + 1)

+ xT (i + 1, j − k) Q2x (i + 1, j − k) − xT (i + 1, j + 1) Q2x (i + 1, j + 1)]

=⎡

⎣∞∑

j=0

xT (0, j + 1) (P − Q − Q1 − Q2)x (0, j + 1) +∞∑

i=0

xT (i + 1, 0) Qx (i + 1, 0)

+∞∑

j=0

0∑

i=−d

xT (i, j + 1) Q1x (i, j + 1) +∞∑

i=0

0∑

j=−k

xT (i + 1, j) Q2x (i + 1, j)

⎤

⎦ ,

when i, j → ∞, we obtain

J0 ≤⎡

⎣r1−1∑

j=1

xT (0, j) (P − Q − Q1 − Q2)x (0, j) +r2−1∑

i=1

xT (i, 0) Qx (i, 0)

+r1−1∑

j=1

0∑

i=−d

xT (i, j) Q1x (i, j) +r2−1∑

i=1

0∑

j=−k

xT (i, j) Q2x (i, j)

⎤

⎦ .

On the ground of such derivation, the corrected version of Lemma 3 in Ye et al. (2009) isstated as follows:

Lemma 3 If there exist matrices P > 0, Q > 0, Q1 > 0, Q2 > 0, satisfying (6) for system(3) with the initial conditions (1b) and the cost function (4), then system (3) is asymptoticallystable with a guaranteed cost for all admissible uncertainties. The cost function satisfies thebound

J0 ≤⎡

⎣r1−1∑

j=1

xT (0, j) (P − Q − Q1 − Q2)x (0, j) +r2−1∑

i=1

xT (i, 0) Qx (i, 0)

+r1−1∑

j=1

0∑

i=−d

xT (i, j) Q1x (i, j) +r2−1∑

i=1

0∑

j=−k

xT (i, j) Q2x (i, j)

⎤

⎦ .

In the view of above lemma, the corrected versions of Theorem 1 and Theorem 2 in Ye et al.(2009) are stated as the follows:

Theorem 1 If there exist a scalar ε > 0, positive definite symmetric matrices P1, Y1, Y2,and Y3 such that the following LMI is feasible:

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418 Multidim Syst Sign Process (2012) 23:415–419

Z =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−P1 A1P1 A2P1 A1dP1 A2kP1 L 0 0 0 0 0∗ Z22 0 0 0 0 P1MT

11 P1S1/21 0 0 0

∗ ∗ −Y1 0 0 0 P1MT12 0 P1S1/2

2 0 0∗ ∗ ∗ −Y2 0 0 P1MT

13 0 0 P1S1/23 0

∗ ∗ ∗ ∗ −Y3 0 P1MT14 0 0 0 P1S1/2

4∗ ∗ ∗ ∗ ∗ −I 0 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ −I 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

then system (3) with initial conditions (1b) and cost function (4) is robust asymptoticallystable with a guaranteed cost. Here, Z22 = −P1 +Y1 +Y2 +Y3. Moreover the cost functionsatisfies the bound:

J0 < ε

⎡

⎣r1−1∑

j=1

xT (0, j) (P2 − P2Y1P2 − P2Y2P2 − P2Y3P2)x(0, j)

+r2−1∑

i=1

xT (i, 0) P2Y1P2x (i, 0) +r1−1∑

j=1

0∑

i=−d

xT (i, j) P2Y2P2x (i, j)

+r2−1∑

i=1

0∑

j=−k

xT (i, j) P2Y3P2x (i, j)

⎤

⎦ ,

where P2 = P−11 , P1 = εP−1, Y1 = ε−1P1QP1, Y2 = ε−1P1Q1P1 and Y3 = ε−1P1Q2P1.

Theorem 2 If there exist a scalar ε > 0, positive definite symmetric matrices P1, Y1,Y2, Y3,and U such that the following LMI is feasible:

T =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−P1 A1 A2 A1d P1 A2kP1 L 0 0 0 0 0 0 0

∗ T22 0 0 0 0 M11 P1S1/21 0 0 0 UT R1/2

1 0

∗ ∗ −Y1 0 0 0 M12 0 P1S1/22 0 0 0 UT R1/2

2∗ ∗ ∗ −Y2 0 0 P1MT

13 0 0 P1S1/23 0 0 0

∗ ∗ ∗ ∗ −Y3 0 P1MT14 0 0 0 P1S1/2

4 0 0∗ ∗ ∗ ∗ ∗ −I 0 0 0 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ −I 0 0 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −εI

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

then there exists a static-state controller u(i, j) = Kx(i, j) such that system (1) is asymp-totically stable with a guaranteed cost and for admissible uncertainties the cost functionsatisfies the bound

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Multidim Syst Sign Process (2012) 23:415–419 419

J0 < ε

⎡

⎣r1−1∑

j=1

xT (0, j) (P2−P2Y1P2−P2Y2P2−P2Y3P2)x (0, j)+r2−1∑

i=1

xT (i, 0) P2Y1P2x(i, 0)

+r1−1∑

j=1

0∑

i=−d

xT (i, j) P2Y2P2x (i, j) +r2−1∑

i=1

0∑

j=−k

xT (i, j) P2Y3P2x (i, j)

⎤

⎦ ,

where A1 = A1P1 + B1U, A2 = A2P1 + B2U, M11 = (P1MT11 + UT MT

21), M12 =(P1MT

12 + UT MT22), K = UP−1

1 , T22 = −P1 + Y1 + Y2 + Y3, P2 = P−11 .

In Sect. 4, to illustrate Theorem 2 in Ye et al. (2009), the authors have taken a numericalexample. By applying Theorem 2, the cost bound should be corrected as

J0 < 8.3691.

3 Conclusion

In this paper, some technical errors in the derivation of main results in Ye et al. (2009) arepointed and corrected.

Reference

Ye, S., Wang, W., & Zou, Y. (2009). Robust guaranteed cost control for a class of two-dimensionaldiscrete systems with shift-delays. Multidimensional Systems and Signal Processing, 20, 297–307. doi:10.1007/s11045-008-0063-2.

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