276 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005 Comments on “Quadratic Stability and Stabilization of Dynamic Interval Systems” Guang-Hong Yang and Kai-Yew Lum Abstract—In the above paper, it was claimed that the necessary and suf- ficient conditions for the quadratic stability and stabilization of dynamic interval systems are given in terms of linear matrix inequalities (LMIs). In this note, numerical examples are presented to show that the necessity of the conditions given in [1] does not hold. Index Terms—Interval systems, linear matrix inequality (LMI), quadratic stability, quadratic stabilization. I. INTRODUCTION Mao abd Chu [1] considered the following dynamic interval system: (1) where is the state and is the control input, and (2) (3) with , , , and . Denote (4) (5) Moreover, let and denote the column vectors in which the th element equals 1 and the others equal 0, being an ap- propriate number in each case. Then, Mao and Chu [1] claimed the following results for the quadratic stability and stabilization [2] of dy- namic interval system (1). Proposition 1 [1]: The dynamic interval system (1) is quadratically stable if and only if there exist a symmetric positive–definite matrix and real scalars satisfying (6) where (7) Manuscript received August 21, 2003; revised January 6, 2004, July 6, 2004, and October 19, 2004. Recommended by Associate Editor A. Datta. This work was supported by Grant DSTA POD1820, Singapore. The authors are with the Temasek Laboratories, National University of Singapore, Singapore 117508, Singapore (e-mail: tslygh@nus.edu.sg, tsllumky@nus.edu.sg). Digital Object Identifier 10.1109/TAC.2004.841921 and (8) Proposition 2 [1]: The dynamic interval system (1) is quadratically stabilizable if and only if there exist a matrix , a symmetric positive–definite matrix , and real scalars satisfying (9) where and are given by (7) and (8), and and . If the aforementioned condi- tion holds, then a quadratically stabilizing state feedback control is given by with . Note that conditions (6) and (9) in Propositions 1 and 2 are linear matrix inequalities (LMIs), and the problems of checking for the exis- tence of solutions to these LMIs are globally solvable using the LMI Control Toolbox [4]. However, in Section II, we will present numerical examples to show that the necessity of the conditions given in Propo- sitions 1 and 2 does not hold. II. COUNTEREXAMPLES Consider the uncertain linear system described by (10) where (11) with , and (12) , , , and are known real matrices. Remark 1: It is easy to see that the dynamic interval system (1) can be described by (10). In fact, from (2)–(5), the matrices and in (1) can be written as with and . Denote (13) (14) 0018-9286/$20.00 © 2005 IEEE