Research Article State Observer Design for Delayed Genetic Regulatory Networks Li-Ping Tian, 1 Zhi-Jun Wang, 2 Amin Mohammadbagheri, 3 and Fang-Xiang Wu 3,4 1 School of Information, Beijing Wuzi University, Beijing 101149, China 2 College of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, Hebei 050061, China 3 Division of Biomedical Engineering, University of Saskatchewan, Saskatoon, SK, Canada S7N 5A9 4 Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK, Canada S7N 5A9 Correspondence should be addressed to Fang-Xiang Wu; faw341@mail.usask.ca Received 14 March 2014; Accepted 6 May 2014; Published 22 May 2014 Academic Editor: Zhongming Zhao Copyright © 2014 Li-Ping Tian et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Genetic regulatory networks are dynamic systems which describe the interactions among gene products (mRNAs and proteins). he internal states of a genetic regulatory network consist of the concentrations of mRNA and proteins involved in it, which are very helpful in understanding its dynamic behaviors. However, because of some limitations such as experiment techniques, not all internal states of genetic regulatory network can be efectively measured. herefore it becomes an important issue to estimate the unmeasured states via the available measurements. In this study, we design a state observer to estimate the states of genetic regulatory networks with time delays from available measurements. Furthermore, based on linear matrix inequality (LMI) approach, a criterion is established to guarantee that the dynamic of estimation error is globally asymptotically stable. A gene repressillatory network is employed to illustrate the efectiveness of our design approach. 1. Introduction Recently nonlinear diferential equations have been proposed to model genetic regulatory networks. Based on this model, stability of genetic regulatory networks has been intensively studied, which is believed useful in designing and controlling genetic regulatory networks. In [1], suicient and necessary local delay-independent stability conditions are given for several types of simpliied genetic regulatory networks with a single time delay. In [2, 3], we present some suicient and necessary conditions of local delay-independent stability conditions for general genetic regulatory networks with a single time delay and multiple time delays. Some suicient conditions for global stability of genetic regulatory networks have been derived based on LMI approaches [4–6] and M- matrix theorem [7, 8]. On the other hand, to understand the dynamic behavior of genetic regulatory networks, measurements of all internal states are very useful. he internal states of a genetic regu- latory network consist of the concentrations of mRNA and proteins involved in it. However, because of some limitations such as experiment techniques, not all internal states of genetic regulatory network can be efectively measured. As a result, the internal states of genetic regulatory networks cannot be completely available. herefore, the state estima- tion problem can play an important role in understanding the dynamic behaviors of genetic regulatory networks. he state estimation problem addressed is to estimate the states based on available output measurements such that the dynamic of estimation error is globally asymptotically stable. Actually, the state estimation methods have been very important in understanding, designing, and controlling dynamic systems such as engineering control system [9], neural networks [10, 11], and complex systems [12]. In this study, we will study the state estimation of genetic regulatory networks with time delays modeled by nonlinear diferential equations. Section 2 briely describes delayed genetic regulatory networks with SUM regulatory logic. In Section 3 we design a full-order state observer to estimate the states of delayed genetic regulatory networks. Some properties of this observer are discussed. In Section 4, based on LMI approach we establish a suicient condition under which the dynamic of estimation error for designed state observer is asymptotically and delay-independently stable. Hindawi Publishing Corporation Computational and Mathematical Methods in Medicine Volume 2014, Article ID 761562, 7 pages http://dx.doi.org/10.1155/2014/761562