On Delayed Genetic Regulatory Networks With Polytopic Uncertainties: Robust Stability Analysis

Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time delays and uncertainties

Because of intracellular and extracellular noise perturbations and environment fluctuations, gene regulation is an intrinsically noisy process. In this paper, we present a hybrid genetic regulatory network (GRN) model which is based on the Markov chain. The GRNs are composed of N modes and the network switches from one mode to another according to a Markov chain with known transition probability. Time‐delays here are mode‐dependent. Based on the Lyapunov stability theory and the linear matrix inequality (LMI) technique, sufficient conditions are given to ensure the stochastic stability of the GRNs with polytopic uncertainties and disturbance attenuation. All the conditions are presented in terms of LMIs which are easily verified via the LMI toolbox. Examples are provided to illustrate the effectiveness of the theoretical results.Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

This study investigated the problem of robust finite-time stability (FTS) in uncertain genetic regulatory networks (GRNs) affected by delays that vary over time. The proposed work introduced new criteria for evaluating robust FTS, which accounted for parametric uncertainties, discrete interval time-varying delays, and distributed time-varying delays. These criteria were developed using the approach of selecting the appropriate Lyapunov–Krasovskii functionals (LKFs). The analysis considered cases where the time-varying delays were continuous and nondifferentiable. To enhance the robustness of the results, integral inequalities based on auxiliary functions were employed, along with convex and reciprocally convex techniques. The findings provided the FTS criteria for GRNs with delays that vary over time under uncertainties in the form of linear matrix inequalities (LMIs). Numerical examples illustrated the effectiveness of the proposed framework, validated the theoretical results, and enabled comparison with existing approaches. The results thus demonstrated the advantages of the developed criteria over previous methods.

Stability analysis of standard genetic regulatory networks with time-varying delays and stochastic perturbations

In this paper, we consider the problems of robust stability and stabilization of Lurpsilae control systems with time-varying delay. Firstly, by making use of an integral inequality, some delay-dependent robust stability conditions of the system are presented. Then, a sufficient stabilization criterion is established in the terms of linear matrix inequality (LMI), based on the above stability condition and matrix decomposition technique. Finally, numerical examples are given to illustrate the effectiveness of the proposed results.

Robust μ-stability analysis of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays

Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.

This paper presents a analysis approach to asymptotic stability of the genetic regulatory networks with interval time-varying delays, the delays play an important role in dynamics of genetic networks and cannot be ignored in the modeling of genetic regulation due to slow biochemical reactions. The free weighting matrices are employed to deal with cross product items, and the convexity of the matrix function is fully utilized in our proof to obtain less conservativeness, by utilizing some techniques, both the upper and lower bounds of the delays are brought into final stability conditions. Based on the Lyapunov stability theory and linear matrix inequality, sufficient conditions are given to ensure the asymptotic stability of the genetic regulatory networks, the obtained conditions are derived in terms of LMIs which are easy to be verified via the LMI toolbox. Illustrative examples are provided to show the effectiveness of our proposed method.

This paper considers the state estimation problem for Markovian jumping genetic regulatory networks (GRNs) with mode-dependent leakage and time-varying delays. In order to approximate the true concentrations of the mRNA and protein, the state estimator is designed using available measurement outputs. The GRNs are composed of N modes. The system switches from one mode to another according to a Markovian chain with known transition probabilities. Based on the Lyapunov functionals, including triple integral terms, some inequalities, and a time-varying delay partitioning approach, delay-dependent criteria which employ all upper bounds of time delays of each mode are obtained in terms of linear matrix inequalities (LMIs). This guarantees that the estimation error dynamics can be globally asymptotically stable from solutions of LMIs. Finally, a numerical example is presented to demonstrate the efficiency of the proposed estimation scheme.

In this paper, we consider generalized delay-dependent stability conditions of Takagi-Sugeno fuzzy systems with discrete and distributed time-delays. For fuzzy systems with discrete time-delays, both delay-independent stability conditions and delay-dependent stability conditions have already been obtained. However, no stability condition for fuzzy systems with distributed time-delays has appeared in the literature. This is also true in case of the robust stability for uncertain fuzzy systems with distributed time-delays. With the aid of the stability theorems of linear systems with distributed time-delays, we employ a similar Lyapunov functional for the stability of fuzzy time-delay systems. Then we introduce many free matrices to such a Lyapunov functional, in order to obtain delay-dependent stability conditions. These techniques lead to generalized and less conservative stability conditions. In fact, delay-dependent stability conditions thus obtained are given in linear matrix inequalities(LMIs) and guarantee a wide stability region. We also consider the robust stability of fuzzy time-delay systems with uncertain parameters. Applying the same techniques made on the stability conditions, we obtain delay-dependent sufficient conditions for the robust stability of uncertain fuzzy systems with discrete and distributed time-delays. Moreover, we consider the state feedback stabilization. Based on stability and robust stability conditions, we obtain conditions for the state feedback controller to stabilize the fuzzy time-delay systems. Finally, we take some examples to illustrate our results.

The model approximation problem is studied in this paper for switched genetic regulatory networks (GRNs) with time-varying delays. We focus on constructing a reduced-order model to approximate the high-order GRNs considered under the switching signal subject to certain constraints, such that the approximation error system between the original and reduced-order systems is exponentially stable with a disturbance attenuation performance. The stability conditions and the disturbance attenuation performance are established by utilizing two integral inequality bounding techniques and the average dwell-time method for the approximation error system. Then, the solvability conditions for the reduced-order models for the GRNs are also established using the projection method. Furthermore, the model approximation problem can be transferred into a sequential minimization problem that is subject to linear matrix inequality constraints by using the cone complementarity algorithm. Finally, several examples are provided to illustrate the effectiveness and the advantages of the proposed methods.

New results on robust exponential stability for discrete recurrent neural networks with time-varying delays

Stability analysis of the differential genetic regulatory networks model with time-varying delays and Markovian jumping parameters

SUMMARYThis paper investigates robust stability of time‐varying uncertain genetic regulatory networks (GRNs). In particular, the considered model includes, as special cases, SUM and PROD regulatory functions typically considered in the literature. It is supposed that the coefficients of the GRN are affine linear functions of an uncertain vector constrained in a polytope, and that the activation functions are uncertain into sector‐type regions. As the first problem, we consider to establish whether the GRN is robustly globally stable for all admissible uncertainties. It is shown that this problem can be addressed by solving a linear matrix inequality (LMI) feasibility test built by exploiting homogeneous polynomial Lyapunov functions. As the second problem, we consider to determine the slowest speed with which the concentrations of mRNAs and proteins reach their equilibrium values. It is shown that a guaranteed underestimate of such a speed can be provided by solving a generalized eigenvalue problem built from the proposed stability condition. Some numerical examples illustrate the proposed approaches. It is worth remarking that this paper proposes for the first time in the literature the use of nonquadratic Lyapunov functions for studying robust stability of uncertain GRNs, whereas existing works have addressed the problem only via quadratic Lyapunov functions (either common or parameter‐dependent), which are known to be conservative for time‐varying uncertainty. Copyright © 2011 John Wiley & Sons, Ltd.