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: sagepub.co.uk/journalsPermissions.nav 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). References Dhawan A and Kar H (2007) LMI-based criterion for the robust guaranteed cost control of 2-D systems described by the Fornasini?Marchesini second model. Signal Processing 87(3): 479?488. Dhawan A and Kar H (2011) An improved LMI-based criterion for the design of optimal guaranteed cost controller for 2-D discrete uncertain systems. Signal Processing 91(4): 1032?1035. Hien LV and Trinh H (2017) Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser model. Nonlinear Analysis: Hybrid Systems 24: 45?57. Tandon A and Dhawan A (2016) 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 Controlhttp://dx.doi.org/10.1177/ 0142331216667476. Xu J and Yu L (2009) Delay-dependent guaranteed cost control for uncertain 2-D discrete systems with state delay in the FM second model. Journal of the Franklin Institute 346(2): 159?174. Ye S, Wang W, Zou Y, et al. (2011) Non-fragile robust guaranteed cost control of 2-D discrete uncertain systems described by the general models. Circuits, Systems and Signal Processing 30(5): 899?914.

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