IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 5, MAY 2003 797 The matrices and are obtained as It is easy to check that is not positive definite. Also, is not diagonally row dominant. Thus, the conditions in [16] and [18] do not hold. The solution trajectory is shown in Fig. 2. III. CONCLUSION In this note, GAS of a class of delayed neural networks is studied. We established a new criteria which is milder than known results. It can apply to neural networks with a broad range of the activation functions which do not assume differentiable or strictly monotonously increasing and even do not satisfy the Lipschitz condition. Therefore, our result can be used to the design and applications of globally stable neural networks with delays. REFERENCES [1] Q. Zhang, R. Ma, and J. Xu, “Stability of cellular neural networks with delay,” Electron. Lett., vol. 37, pp. 575–576, 2001. [2] J. Cao, “A set of stability criteria for delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 48, pp. 494–498, Apr. 2001. [3] , “Global stability analysis in delayed cellular neural networks,” Phys. Rev. E, vol. 59, pp. 5940–5944, 1999. [4] J. Cao and D. Zhou, “Stability analysis of delayed cellular neural net- works,” Neural Networks, vol. 11, pp. 1601–1605, 1998. [5] J. Cao, “On stability of delayed neural networks,” Phys. Lett. A, vol. 261, pp. 303–308, 1999. [6] H. Lu, “On stability of nonlinear continuous-time neural networks with delays,” Neural Networks, vol. 13, pp. 1135–1143, 2000. [7] T. Roska and L. O. Chua, “Cellular neural networks with nonlinear and delay type template,” Int. J. Circuit Theory Applicat., vol. 20, pp. 469–481, 1992. [8] T. Roska, C. W. Wu, and L. O. Chua, “Stability of cellular neural net- works with dominant nonlinear and delay-type template,” IEEE Trans. Circuits Syst. I, vol. 40, pp. 270–272, Apr. 1993. [9] T. Roska, C. W. Wu, M. Balsi, and L. O. Chua, “Stability and dynamics of delay-type general and cellular neural networks,” IEEE Trans. Cir- cuits Syst. I, vol. 39, pp. 487–490, June 1992. [10] M. Gilli, “Stability of cellular neural networks and delayed cellular neural networks with nonpositive templates and nonmonotonic output functions,” IEEE Trans. Circuit Syst. I, vol. 41, pp. 518–528, Aug. 1994. [11] C. Guzelis and L. O. Chua, “Stability analysis of generalized cellular neural networks,” Int. J. Circuit Theory Applicat., vol. 21, pp. 1–33, 1993. [12] P. P. Civalleri and M. Gilli, “On the dynamic behavior of two-cell cel- lular neural networks,” Int. J. Circuit Theory Appl., vol. 21, pp. 451–471, 1993. [13] P. V. D. Driessche and X. Zhou, “Global attractivity in delayed hopfield neural network nodels,” SIAM J. Appl. Math., vol. 58, pp. 1878–1890, 1998. [14] M. Joy, “Results concerning the absolute stability of delayed neural net- works,” Neural Networks, vol. 13, pp. 613–616, 2000. [15] S. Arik, “Stability analysis of delayed neural networks,” IEEE Trans. Circuits Syst., vol. 47, pp. 1089–1092, 2000. [16] S. Arik and V. Tavsanoglu, “On the global asymptotic stability of de- layed cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 47, pp. 571–574, Apr. 2000. [17] X. B. Liang and J. Si, “Global exponential stability of neural networks with globally Lipschitz continuous activations and its application to linear variational inequality problem,” IEEE Trans. Neural Networks, vol. 12, pp. 349–359, Mar. 2001. [18] Q. Hong, J. Peng, and Z. B. Xu, “Nonlinear measures: a new approach to exponential stability analysis for Hopfield-type neural networks,” IEEE Trans. Neural Networks, vol. 12, pp. 360–370, 2001. [19] B. Kosko, Neural Networks and Fuzzy System—A Dynamical Systems Approach to Machine Intelligence. New Delhi, India: Prentice-Hall of India, 1994. [20] C. Feng and R. Plamondon, “On the stability of delayed neural networks systems,” Neural Networks, vol. 14, pp. 1181–1188, Mar. 2001. [21] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, 2nd ed. London, U.K.: Cambridge Univ. Press, 1952. On the Synthesis of Controllers for a Nonovershooting Step Response Swaroop Darbha and S. P. Bhattacharyya Abstract—In this note, we show how a two-parameter compensator can always be designed for anylinear time-invariant plant, that does not have a zero at the origin, to render its step response nonovershooting. Index Terms—Nonminimum phase systems, overshoot, step response, time response, two-parameter compensator synthesis. I. INTRODUCTION The problem of achieving nonovershooting step response is im- portant in control systems and several results have been reported [1], [5]–[9]. Although these references provide useful design techniques, the question of whether overshoot can be eliminated or not has not been clearly answered for continuous-time linear time-invariant (LTI) systems. The discrete-time counterpart of this question was answered by Deodhare and Vidyasagar [3]. They established that there is a deadbeat closed loop system that can be synthesized which has a nonovershooting step response. However, they conclude that the continuous-time counterpart of their results for discrete-time LTI systems can lead to controllers with irrational transfer functions. In this note, we show, by construction, that overshoot can always be eliminated by proper, rational two parameter controllers. II. CONTINUOUS TIME LTI SYSTEMS Given any plant ; consider a two-parameter compensator as shown in Fig. 1. To avoid trivially unsolvable problems, we only consider plants that satisfy the following conditions: 1) are stabilizable by feedback controllers; 2) do not have a zero at the origin. Plants satisfying these conditions will be called admissible plants. Theorem: For every admissible plant, there is a two-parameter com- pensator, as shown in Fig. 1, that renders the closed-loop step response nonovershooting. Manuscript received June 19, 2002; revised September 11, 2002. Recom- mended by Associate Editor Z. Lin. The work of S. Darbha was supported by the National Science Foundation under Grant CMS 0127941. The work of S. P. Bhattacharyya was supported by the National Science Foundation under Grant ECS 9903488. S. Darbha is with the Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123 USA (e-mail: dswaroop@ mengr.tamu.edu). S. P. Bhattacharyya is with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3123 USA (e-mail: bhatt@ee.tamu.edu). Digital Object Identifier 10.1109/TAC.2003.811256 0018-9286/03$17.00 © 2003 IEEE