Solvability Of Linear Matrix Inequalities Research Articles

Synchronization of Complex Dynamical Networks with Dynamical Behavior Links

AbstractIn this paper, synchronization of a complex dynamical network (CDN) is investigated while the coupling connections of the network exhibit dynamic behavior. As it is shown in simulations, the dynamic links can cause synchronization losing in the network whereas the links in many real‐world CDNs have dynamic behavior. To analyzing the effect of these links, two different CDN models are considered: without time delays and with delays in the coupling connections. By means of Lyapunov(‐Krasovskii) theory, stability analysis of the error dynamics between the nodes of these CDNs and an introduced individual node is investigated which yields some conditions in the form of linear matrix inequalities (LMIs). These LMIs can be solved easily by various existing LMI solvers. Moreover, the gain matrices of state feedback controllers will be obtained by solving the LMIs. In the end, illustrative numerical examples are given to specify the effectiveness of the proposed method.

Dissipativity-based non-fragile sampled-data control design of interval type-2 fuzzy systems subject to random delays

This paper investigates the β-dissipativity-based reliable non-fragile sampled-data control problem for a class of interval type-2 (IT2) fuzzy systems. In particular, it is allowed to have randomly occurring time-varying delays in the controller design, which are modeled by Bernoulli distributed white noise sequences. Precisely, the IT2 fuzzy model and the non-fragile sampled-data controller are formulated by considering the mismatched membership functions. By constructing an appropriate Lyapunov–Krasovskii functional, a set of delay-dependent conditions is derived to guarantee that the closed-loop IT2 fuzzy system is strictly <Q,S,R>-β-dissipative. Moreover, the gain matrices of feedback reliable non-fragile sampled-data controller are derived in terms of linear matrix inequalities (LMIs), which can be solved by using existing LMI solvers. Two numerical examples are eventually given to illustrate the applicability and effectiveness of the proposed controller design technique.

Design of sliding observers for Lipschitz nonlinear system using a new time-averaged Lyapunov function

ABSTRACTThis paper provides a procedure for designing a sliding mode observer for a nonlinear system in the presence of Gaussian input disturbances and sensor noises. The paper first proposes a novel candidate for Lyapunov stability, termed a Time-averaged Lyapunov (TAL) function. The TAL averages the Lyapunov analysis over a small finite time interval, allowing for intuitive analysis of noises and disturbance. The paper then provides the necessary and sufficient condition for the design of the sliding observer gainsusing the TAL in the form of Linear Matrix Inequality (LMI). The LMIs can then be explicitly solved offline using commercial LMI solvers. The paper compares the LMIs for the two observer designs to demonstrate the design of the sliding mode observer using TAL can greatly enhance the scope of observer design for nonlinear systems.

Adaptive observer design for a class of nonlinear systems with time delays

This paper deals with the design of adaptive observer for a class of nonlinear systems with time delays. Within this work, we develop an adaptive observer for a bilinear time delay system, then we extend those results in the presence of Lipschitz nonlinear functions in the system’s dynamics. In the stability analysis of the estimation errors, we combine a Linear Parameter Varying (LPV) approach in the presence of time delays with a polytopic approach. The obtained stability conditions are given in terms of the solvability of Linear Matrix Inequalities (LMIs) on the vertices of a convex polytope. Numerical examples are finally given to show the effectiveness and feasibility of our results.

On the Control Design for Linear Time–Invariant Systems with Moments Constraints of Disturbances in Anisotropy–based Theory

Abstract In this paper, the design procedure of anisotropic suboptimal dynamic output regulator for linear discrete time-invariant (LDTI) system is considered. The mean anisotropy level of exogenous disturbance is bounded by the certain positive value. Additional constraints of exogenous disturbance are connected with the first and the second moments. The anisotropic suboptimal regulator design problem is formulated in terms of solvability of linear matrix inequalities (LMI) under convex constraints.

A Polytopic Observer Design Approach for Landing Control of a Quadrotor UAV

In this paper, a constructive procedure to design functional unknown input observer for nonlinear continuous time systems under the Polytopic Takagi-Sugeno framework (also known as multiple models systems) is proposed. Applying the Lyapunov theory, Linear Matrix Inequalities (LMI)s conditions are deduced which are solved for feasibility to obtain observer design matrices. To reject the effect of unknown input, classical approach of decoupling the unknown input for the linear case is used. A comparative study between single and Polytopic Lyapunov function is made in order to prove the relaxation effect of the Multiple functions. A solver based solution is then proposed.It will be shown through applicative example (a Quadrotor Aerial Robots Landing) that even if the proposed LMIs solver based solution may look conservative, an adequate choice of the solver makes it suitable for the application of the proposed approach.

Input–output finite-region stability and stabilization for discrete 2-D Fornasini–Marchesini models

This paper establishes input–output finite-region stability (I–O FRS) and stabilization analysis methods to investigate the transient behavior of output signals for discrete two-dimensional (2-D) Fornasini–Marchesini (FM) models. For 2-D systems with two classes of input signals, by building special recursive formulas, sufficient I–O FRS conditions are formulated in terms of solvability of linear matrix inequalities (LMIs). Furthermore, aiming at designing static state feedback controllers, I–O finite-region stabilization issues are analyzed for the corresponding 2-D systems and sufficient conditions are reported to guarantee I–O FRS of the closed-loop systems. To conclude the paper, we provide some numerical examples to confirm the validity of the proposed methods.

Improved results on exponential stability of discrete-time switched delay systems

In this paper, we study the exponential stability problem of discrete-time switched delay systems. Combining a multiple Lyapunov function method with a mode-dependent average dwell time technique, we develop novel sufficient conditions for exponential stability of the switched delay systems expressed by a set of numerically solvable linear matrix inequalities. Finally, numerical examples are presented to illustrate less conservativeness of the obtained results.

Finite-region stability and boundedness for discrete 2D Fornasini–Marchesini second models

ABSTRACTIn this paper, the concepts of finite-region stability (FRS) and finite-region boundedness (FRB) are formulated for discrete two-dimensional (2D) Fornasini–Marchesini second (FMII) models, and then the analysis methods for FRS and FRB are proposed to investigate the transient behaviour of such discrete 2D FMII models. First, by building special recursive formulas, we develop a sufficient condition which guarantees the FRS of the system under solvable linear matrix inequalities (LMIs) conditions. Next, the FRB problem is addressed for the FMII model with exogenous disturbances and the corresponding criteria and LMIs conditions are reported. Finally, we apply the proposed FRS analysis method to consider the finite-region stabilisation problem of a chemical reactor thermal process, as well as some other numerical examples, to illustrate the validity of the proposed methods.

Finite‐region stability and finite‐region boundedness for 2D Roesser models

In this paper, we establish finite‐region stability (FRS) and finite‐region boundedness analysis methods to investigate the transient behavior of discrete two‐dimensional Roesser models. First, by building special recursive formulas, a sufficient FRS condition is built via solvable linear matrix inequalities constraints. Next, by designing state feedback controllers, the finite‐region stabilization issue is analyzed for the corresponding two‐dimensional closed‐loop system. Similar to FRS analysis, the finite‐region boundedness problem is addressed for Roesser models with exogenous disturbances and corresponding criteria, and linear matrix inequalities conditions are reported. To conclude the paper, we provide numerical examples to confirm the validity of the proposed methods. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Synchronization of chaotic Lur’e systems with state and transmission line time delay: a linear matrix inequality approach

This paper is concerned with the problem of the master–slave synchronization of chaotic Lur’e systems with multiple time delays in their states and transmission line. Based on the Lyapunov–Krasovskii functional, some delay-dependent synchronization criteria are obtained and formulated in the form of linear matrix inequalities (LMIs) to ascertain the global asymptotic stability of the error system such that the slave system is synchronized with the master. With the help of the LMI solvers, the time-delay feedback control law can easily be obtained. The effectiveness of the proposed method is illustrated using some numerical simulations performed on two chaotic systems.

SpecificD-Admissibility and Design Issues for Uncertain Descriptor Systems with Parametric Uncertainty in the Derivative Matrix

Stability analysis issues and controller synthesis for descriptor systems with parametric uncertainty in the derivative matrix are discussed in this paper. The proposed descriptor system can extend the system’s modeling extent of physical and engineering systems from the traditional state-space model. First, based on the extendedD-stability definitions for the descriptor model, necessary and sufficient admissibility andD-admissibility conditions for the unforced nominal descriptor system are derived and formulated by compact forms with strict linear matrix inequality (LMI) manner. In contrast, existing results need to involve nonstrict LMIs, which cannot be evaluated by current LMI solvers and need some extra treatments. Deducing from the obtained distinct results, the roust admissibility andD-admissibility of the descriptor system with uncertainties in both the derivative matrix and the system’s matrices thus can be coped. Furthermore, by involving a proportional and derivative state feedback (PDSF) control law, we further address the controller design for the resulting closed-loop systems. Since all the proposed criteria are explicitly expressed in terms of the strict LMIs, we can use applicable LMI solvers for evaluating the feasible solutions. Finally, the efficiency and practicability of the proposed approach are demonstrated by two illustrative examples.

Semi-periodically intermittent control for synchronization of switched complex networks: a mode-dependent average dwell time approach

In this paper, we study the synchronization problem for switched complex networks with delayed coupling via semi-periodically intermittent control technique and a mode-dependent average dwell time method. The mathematical model of switched complex networks is established, and the networks switch among a finite number of subnetworks in accordance with a switching rule. By applying multiple Lyapunov function method and the mode-dependent average dwell time (MDADT) approach, less conservative synchronization criteria are derived based on a sequence of solvable linear matrix inequalities. Particularly, the conventional synchronization conditions are improved by using semi-periodically intermittent control and MDADT approach. Moreover, delayed coupling may affect the stability of the switched complex networks. In this paper, novel synchronization criteria are derived to deal with the relations between time delay and other parameters appropriately. Finally, a numerical example is provided to demonstrate the effectiveness of the derived theoretical results.

Synchronization of fuzzy bidirectional associative memory neural networks with various time delays

The problem of synchronization for fuzzy bidirectional associative memory (BAM) neural networks (NNs) with various time delays is formulated and investigated. The various delays consist of discrete delays, unbounded distributed delays and constant delay in the leakage term (i.e. ‘leakage delay’) are considered. Then, some sufficient conditions are presented to guarantee the global asymptotic stability of the error dynamical system by using Lyapunov–Krasovskii functional and linear matrix inequality (LMI) method. As a result, we achieved synchronization of master-slave (drive-response) fuzzy BAM NNs systems. Even, if there is no leakage delay, the obtained results are less restrictive to some known sufficient conditions. Moreover, the proposed results do not require the boundedness, differentiability and monotonicity of the activation functions, which can also be easily checked via the LMI solver in robust control toolbox in Matlab. Furthermore, using a parameter-dependent Lyapunov function approach the synchronization problem for polytopic uncertain fuzzy BAM NNs with various time delays is considered. The parameter uncertainties under consideration are assumed to belong to a fixed convex polytope. Finally, two numerical examples and simulations are given to illustrate the effectiveness of the derived theoretical results.

Adaptive nonlinear observer for state and unknown parameter estimation in noisy systems

This paper proposes a novel adaptive observer for Lipschitz nonlinear systems and dissipative nonlinear systems in the presence of disturbances and sensor noise. The observer is based on an H∞ observer that can estimate both the system states and unknown parameters by minimising a cost function consisting of the sum of the square integrals of the estimation errors in the states and unknown parameters. The paper presents necessary and sufficient conditions for the existence of the observer, and the equations for determining observer gains are formulated as linear matrix inequalities (LMIs) that can be solved offline using commercially available LMI solvers. The observer design has also been extended to the case of time-varying unknown parameters. The use of the observer is demonstrated through illustrative examples and the performance is compared with extended Kalman filtering. Compared to previous results on nonlinear observers, the proposed observer is more computationally efficient, and guarantees state and parameter estimation for two very broad classes of nonlinear systems (Lipschitz and dissipative nonlinear systems) in the presence of input disturbances and sensor noise. In addition, the proposed observer does not require online computation of the observer gain.

H∞ Control for Markov Jump Systems with Nonlinear Noise Intensity Function and Uncertain Transition Rates

The problem of robust H∞ control is investigated for Markov jump systems with nonlinear noise intensity function and uncertain transition rates. A robust H∞ performance criterion is developed for the given systems for the first time. Based on the developed performance criterion, the desired H∞ state-feedback controller is also designed, which guarantees the robust H∞ performance of the closed-loop system. All the conditions are in terms of linear matrix inequalities (LMIs), and hence they can be readily solved by any LMI solver. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed methods.

Robust Discretization of LTI Systems with Polytopic Uncertainties and Aperiodic Sampling

In the previous work, the authors studied the problem of robust discretization of linear time-invariant systems with polytopic uncertainties, where linear matrix inequality (LMI) conditions were developed to find an approximate discrete-time (DT) model of a continuous-time (CT) system with uncertainties in polytopic domain. The system matrices of obtained DT model preserved the polytopic structures of the original CT system. In this paper, we extend the previous approach to solve the problem of robust discretization of polytopic uncertain systems with aperiodic sampling. In contrast with the previous work, the sampling period is assumed to be unknown, time-varying, but contained within a known interval. The solution procedures are presented in terms of unidimensional optimizations subject to LMI constraints which are numerically tractable via LMI solvers. Finally, an example is given to show the validity of the proposed techniques.

Continuous-Discrete Time Observer Design for Lipschitz Systems With Sampled Measurements

This technical note concerns observer design for Lipschitz nonlinear systems with sampled output. Using reachability analysis, an upper approximation of the attainable set is given. When this approximation is formulated in terms of a convex combination of linear mappings, a sufficient condition is given in terms of linear matrix inequalities (LMIs) which can be solved employing an LMIs solver. This novel approach seems to be an efficient tool to solve the problem of observer synthesis for a class of Lipschitz systems of small dimensions.

Non-fragile observer design for discrete-time genetic regulatory networks with randomly occurring uncertainties

This paper investigates the problem of non-fragile observer design for a class of discrete-time genetic regulatory networks (DGRNs) with time-varying delays and randomly occurring uncertainties. A non-fragile observer is designed, for estimating the true concentration of mRNAs and proteins from available measurement outputs. One important feature of the results obtained that are reported here is that the parameter uncertainties are assumed to be random and their probabilities of occurrence are known a priori. On the basis of the Lyapunov–Krasovskii functional approach and using a convex combination technique, a delay-dependent estimation criterion is established for DGRNs in terms of linear matrix inequalities (LMIs) that can be efficiently solved using any available LMI solver. Finally numerical examples are provided to substantiate the theoretical results.

Towards a global controller design for guaranteed synchronization of switched chaotic systems

In this paper, synchronization of identical switched chaotic systems is explored based on Lyapunov theory of guaranteed stability. Concepts from robust control principles and switched linear systems are merged together to derive a sufficient condition for synchronization of identical master–slave switched nonlinear chaotic systems and are expressed in the form of bilinear matrix inequalities (BMIs). The nonlinear controller design problem is then recast in the form of linear matrix inequalities (LMIs) to facilitate numerical computation by standard LMI solvers and is illustrated by appropriate examples.