Mode-mismatched estimator design for Markov jump genetic regulatory networks with random time delays Zhenzong Zhu a , Yanzheng Zhu a , Lixian Zhang a,b,n , Maryam Al-Yami b , Elbaz Abouelmagd c , Bashir Ahmad b a School of Astronautics, Harbin Institute of Technology, Harbin 150080, China b Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia c Nonlinear Analysis and Applied Mathematics Research Group (NAAM) and Mathematics Department, Faculty of Science and Arts (Khulais), King Abdulaziz University, Jeddah 21589, Saudi Arabia article info Article history: Received 13 February 2015 Received in revised form 14 April 2015 Accepted 1 May 2015 Communicated by Yang Tang Available online 14 May 2015 keywords: Markov jump genetic regulatory networks (GRNs) Mismatching phenomenon Nonstationary Markov chain Random time delays abstract In this paper, the problem of H 1 state estimation is investigated for a class of discrete-time Markov jump genetic regulatory networks (GRNs) with random time delays. A mismatching characteristic of modes jumping between GRNs modes and desired mode-dependent estimators is recognized, and a nonsta- tionary mode transition among the estimators is used to model the mismatching characteristic of modes jumping to different degrees. The time delays are supposed to be time-varying and subject to another Markov chain. By using the linear matrix inequality techniques, sufficient conditions on the existence of the estimators with mismatching characteristic of modes jumping are first derived such that the resulting estimation error system is stochastically stable with a prescribed H 1 performance index. One interesting phenomenon is disclosed, i.e., the optimal performance index varies monotonously as changing the mismatching degrees of modes jumping. A numerical example is exploited to illustrate the effectiveness of the theoretical findings. & 2015 Elsevier B.V. All rights reserved. 1. Introduction Genetic regulatory networks (GRNs), structured by networks of regulatory interactions between DNA, mRNA and proteins, have been keeping as an important research topic in the biological and biomedical sciences, and a large number of results have been reported in the literature, see for example [1–5,31] and the references therein. Besides, in implementing the continuous- time network for computer simulation and experimental/compu- tational purposes, it is ubiquitous to discretize the continuous- time network to perform it conveniently on the computer. More- over, it is shown in [3,12–15] that some GRNs models can be described by discrete-time dynamical systems, and these models are much more convenient than their continuous-time counter- parts coping with the analysis/synthesis issues. On the other hand, the study on Markov jump linear systems has attracted a great deal of attention. This class of systems is normally used to model the systems that change from one mode to another randomly (according to some transition probabilities). The resulting Markov jump GRNs have also been intensively studied in recent years, e.g., [32]. In addition, the random time delays are a non-ignorable factor in dynamics of GRNs due to slow biochemical reaction such as actual regulation, transcription, translation, diffusion and trans- location, especially in that of a eukaryotic cell [7]. Besides, the random time delays can be modeled by Markov chains or Markov processes, see for example, [6]. So far, a growing number of results concerning GRNs with random time delays have been reported during the past decades, e.g., [8–11]. It should be noticed that state estimation is an important research issue in control discipline, especially in H 1 sense [35], and some attempts have been made on the topic in the area of GRNs over the past few years, see for example [17,25–27], for both discrete-time and continuous-time cases. In the case of GRNs with Markovian jumping parameters, the relevant state estimation approaches have been proposed with the aid of the techniques widely employed in the general Markov jump systems. In parti- cular, the designed estimators can be usually classified into two types, i.e., mode-dependent [18–24] and mode-independent [28,29] estimators relying on whether the modes belonging to the original system can be real-time detectable to designers or not. For the mode-dependent estimators, the Markov chain state of the underlying systems, often called system mode, is assumed to be Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/neucom Neurocomputing http://dx.doi.org/10.1016/j.neucom.2015.05.011 0925-2312/& 2015 Elsevier B.V. All rights reserved. n Corresponding author. E-mail addresses: zhenzongzhu@hit.edu.cn (Z. Zhu), yanzhengzhu@hit.edu.cn (Y. Zhu), lixianzhang@hit.edu.cn (L. Zhang), malyami@kau.edu.sa (M. Al-Yami), eabouelmagd@kau.edu.sa (E. Abouelmagd), bahmad@kau.edu.sa (B. Ahmad). Neurocomputing 168 (2015) 1121–1131