Linear Lyapunov Function Research Articles

Robust α-exponential estimate for polytopic uncertain positive discrete-time systems with bounded time-varying delay

In this paper, the problem of finding a robust α-exponential estimate for a class of perturbed positive discrete-time time-delay systems with polytopic uncertainties will be addressed. By proposing a linear parameter-dependent Lyapunov function, a condition for the existence of robust α-exponential estimates will be introduced. Some optimisation techniques are also incorporated to obtain a more accurate estimate. In particular, a procedure to reformulate a class of fractional programming problems into a linear programming problem which can be solved efficiently. A numerical example is presented to demonstrate the effectiveness of the proposed method.

Global Stability Analysis of Onchocerciasis Transmission Dynamics with Vigilant Compartment in Two Interacting Populations

A deterministic compartmental model for the transmission dynamics of onchocerciasis with vigilant compartment in two interacting populations is studied. The model is qualitatively analyzed to investigate its global asymptotic behavior with respect to disease-free and endemic equilibria. It is shown, using a linear Lyapunov function, that the disease-free equilibrium is globally asymptotically stable when the associated basic reproduction number, R0<1. When the basic reproduction number R0>1, under some certain conditions on the model parameters, we prove that the endemic equilibrium is globally asymptotically stable with the aid of a suitable nonlinear Lyapunov function.

Event-triggered-based encoding–decoding consensus control of continuous-time multi-agent systems under DoS attacks

The security of multi-agent systems (MASs) is important particularly when considering the potential threat posed by malicious attackers. Specifically, network attacks and information theft caused by attackers have garnered significant attention due to their potential to destroy the consensus of the system. Therefore, this paper presents a novel approach to achieve leader-following consensus control of continuous-time MASs under DoS attacks, leveraging event-triggered encoding and decoding techniques. When the DoS attacks occur, the transmitted information between agents becomes unavailable. To address this challenge, a buffer is introduced to store the latest transmission data and then the stored data can be used when the DoS attacks exist. We also propose an event-triggered encoding–decoding strategy that can be employed for privacy protection in linear continuous-time MASs. The transmitted data is encoded as a codeword before transmitting, and the encoder updates it when event-triggered conditions are met. The receiving end decodes the data upon receipt. Furthermore, a distributed switching controller is designed using the encoded and stored state values to achieve leader-following consensus. The control gain is calculated to obtain sufficient conditions for DoS attacks through the linear matrix inequalities (LMI) and Lyapunov functions. The boundedness of transmitted data and the Zeno phenomenon are also analyzed. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed control algorithm.

Transient Stability Analysis of Grid-connected Converters in Wind Turbine Systems Based on Linear Lyapunov Function and Reverse-time Trajectory

Transient Stability Analysis of Grid-connected Converters in Wind Turbine Systems Based on Linear Lyapunov Function and Reverse-time Trajectory

Event-Triggered Positive Filter Design for Positive Polynomial Fuzzy Systems via Premise Integration and Membership-Function-Dependent Methods

This paper focuses on the design of the positive fuzzy filter with event-triggered mechanism for positive polynomial fuzzy-model-based (PPFMB) system. The main objective is to design a positive and stable fuzzy filter which makes the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> performance index of the estimated output signal as good as possible with limited networked bandwidth resources. Since the positive constraints exist in the positive systems, while the disturbance signal of the output signal and the transmission delay in the event-triggered mechanism are taken into account in this research, the augmented system is rebuilt by introducing the weight coefficient of disturbance signal, and a novel linear copositive Lyapunov function (LCLF) is proposed. In order to adapt to the analysis framework based on linear Lyapunov function, a novel linear event-triggered condition is proposed. Furthermore, to handle the mismatched premise variables caused by the event-triggered mechanism, the premise variable associated with event-triggered instants are integrated into the interval of continuous premise variable with the help of the novel event-triggered condition, which means that the original multi-dimensional type-1 membership functions (MFs) are transformed into single-dimensional interval type-2 MFs. Then, an interval type-2 membership-function-dependent (IT2MFD) method is employed to introduce MFs information into the resultant conditions, so that more optimized <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> performance index is obtained. Finally, an example with simulation results is given to verify the effectiveness of all the fuzzy filter design strategies proposed in this paper for optimizing performance index.

Counterexample-guided computation of polyhedral Lyapunov functions for piecewise linear systems

This paper presents a counterexample-guided iterative algorithm to compute convex, piecewise linear (polyhedral) Lyapunov functions for continuous-time piecewise linear systems. Polyhedral Lyapunov functions provide an alternative to commonly used polynomial Lyapunov functions. Our approach first characterizes intrinsic properties of a polyhedral Lyapunov function including its “eccentricity” and “robustness” to perturbations. We then derive an algorithm that either computes a polyhedral Lyapunov function proving that the system is asymptotically stable, or concludes that no polyhedral Lyapunov function exists whose eccentricity and robustness parameters satisfy some user-provided limits. Significantly, our approach places no a-priori bound on the number of linear pieces that make up the desired polyhedral Lyapunov function. The algorithm alternates between a learning step and a verification step, always maintaining a finite set of witness states. The learning step solves a linear program to compute a candidate Lyapunov function compatible with a finite set of witness states. In the verification step, our approach verifies whether the candidate Lyapunov function is a valid Lyapunov function for the system. If verification fails, we obtain a new witness. We prove a theoretical bound on the maximum number of iterations needed by our algorithm. We demonstrate the applicability of the algorithm on numerical examples.

Full Information Control for Switched Neural Networks Subject to Fault and Disturbance.

The article investigates full information control problem for switched neural networks subject to fault and disturbance. First, the main objective is realizing interval stability and zero tracking error under condition that neither of the neuron states' vectors including the plant and reference models is available. Second, the desired full information controller and neural networks' observer are designed to ensure observer-based dynamic error system mean-square exponentially stable with sufficient condition of strict weight H∞ /H- performance levels. Finally, we concentrate on stability analyses and fault tolerance for switched neural networks with fault accompanied by disturbance through linear matrix inequalities (LMIs), Lyapunov function, and average dwell time, discussing it according to different values of fault. Finally, simulation examples are listed to account for the availability and effectiveness of the research methodology.

Finite-Time Control for Discrete-Time Positive Systems Subject to Event-Triggered Scheme and Markov Jump Parameters

This brief investigates the finite-time control problem for discrete-time positive systems subject to event-triggered scheme and Markov jump parameters. Switching transition probability is used to extend the fixed transition probability, which is more suitable to describe the performance of Markov jump systems (MJSs). On the basis of the 1-norm, a developed event-triggered scheme is proposed to be associated with the sample signal and the actual signal. Furthermore, a linear Lyapunov function and an average dwell time approach are adopted to obtain sufficient conditions for guaranteeing stochastic finite-time stability of the underlying systems. Additionally, a matrix decomposition strategy is constructed to design a finite-time event-triggered control law such that the corresponding MJSs are positive and stochastically finite-time bounded. All conditions are proposed on the basis of linear programming. Finally, a local railway transportation model is provided to verify the practicability of the strategy.

Analysis of sliding-mode control systems with relative degree altering perturbations

We consider sliding-mode control systems subject to unmatched perturbations. Classical first-order sliding-mode techniques are capable to compensate unmatched perturbations if differentiations of the output of sufficiently high order are included in the sliding variable. For such perturbations it is commonly assumed that they do not affect the relative degree of the system. In this contribution we consider perturbations that alter the relative degree of the process and study their impact on the closed-loop control system with a classical first-order sliding-mode design. In particular we consider systems with full (nominal) relative degree subject to a perturbation reducing the relative degree by one and analyse the resulting closed-loop system. It turns out that the sliding-manifold is not of reduced dimension and the uniqueness of the solution may be lost. Also attractivity of the sliding-manifold and global stability of the origin may be lost whereas the disturbance rejection properties of the sliding-mode control are not impaired. We present a necessary and sufficient condition for the existence of unique solutions for the closed-loop system. The second-order case is studied in great detail and allows to parametrically specify the conditions obtained before. We derive a necessary condition for the global asymptotic stability of the closed-loop system. Further we present a constructive condition for the global asymptotic stability of the closed-loop system using a piece-wise linear Lyapunov function. Each of the prominent results is illustrated by a numerical example.

Stability analysis of switched ARX models and application to learning with guarantees

The main subject of this work is the stability analysis of Switched Auto-Regressive models with eXogenous inputs (SARX), which constitute a reference class for switched and hybrid system identification. The work introduces novel conditions for the arbitrary switching stability of multiple-input multiple-output SARX models which exploit the peculiar structure of their state-space realization. The analysis relies on the properties of block companion matrices, and partly leverages results from the theory of non-negative matrices, without nevertheless asking for an input–output positive behavior of the model. The novel stability conditions have a simple formulation in terms of linear co-positive common Lyapunov functions, and come at a remarkably low computational cost, being solvable by Linear Programming. The low computational burden is particularly attractive in an identification context, as it allows to efficiently constrain learning procedures in order to obtain SARX models with stability guarantees. The latter is itself a contribution of the work, as it fills a gap in the literature on the estimation of SARX models. The results are validated on a particular learning technique based on Regression Trees – a well known machine learning algorithm – which has shown remarkable accuracy in experimental environments.

Single Phase Second Order Sliding Mode Controller for Mismatched Uncertain Systems with Extended Disturbances and Unknown Time-Varying Delays

In this paper, a novel single phase second order sliding mode controller (SPSOSMC) is proposed for the mismatched uncertain systems with extended disturbances and unknown time-varying delays. The main achievements of this study consist of three tasks: 1) a reaching phase in conventional sliding mode control (CSMC) technique is removed to ensure the global stability of the system; 2) an influence of the undesired high-frequency oscillation phenomenon in control input is vanished; 3) an exogenous perturbation is generally extended to the k-order disturbance of state variable. Firstly, a single phase switching manifold function is defined to eliminate the reaching phase in CSMC. Secondly, an unmeasurable state variable is estimated by using the proposed reduced-order sliding mode observer (ROSMO) tool. Next, a SPSOSMC is built based on the help of ROSMO tool and output information only. Then, a sufficient condition is established by employing the linear matrix inequality (LMI) technique and Lyapunov function theory such that the resulting sliding mode dynamics is asymptotically stable. Finally, a numerical example is simulated via the well-known MATLAB software to validate the effectiveness of the proposed technique.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Event-Triggered Asynchronous Filter of Positive Switched Systems with State Saturation

This paper investigates the event-triggered asynchronous filter of positive switched systems with state saturation using linear programming and multiple Lyapunov functions. First, a filter is constructed for continuous-time positive switched systems. Under the asynchronous switching law, an error system is proposed with respect to positive switched systems and their filters, where the state saturation term is described in a polytopic form by virtue of the saturation property. A novel event-triggering condition is addressed based on a 1-norm inequality. Under the event-triggering condition, the error system is transformed into an interval system with lower and upper bounds. By using multiple Lyapunov functions and linear programming, the positivity and stability of the error system are achieved by considering the corresponding properties of the lower and upper bounds, respectively. Then, the event-triggered l1-gain filter and nonfragile filter are also proposed for the systems with disturbances. Moreover, the presented filter framework is extended to the discrete-time case. Finally, two examples are given to verify the effectiveness of the proposed filters.

Finite-time stability of positive switched time-delay systems based on linear time-varying copositive Lyapunov functional

In this paper, we deal with the finite-time stability of positive switched linear time-delay systems. By constructing a class of linear time-varying copositive Lyapunov functionals, we present new explicit criteria in terms of solvable linear inequalities for the finite-time stability of positive switched linear time-delay systems under arbitrary switching and average dwell-time switching. As an important application, we apply the method to finite-time stability of linear time-varying systems with time delay.

H∞ Control Observer-Based for Discrete-Time Singularly Perturbed Systems With Nonlinear Disturbances

This paper focuses on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula> observer-based control problem for discrete-time singularly perturbed systems (DTSPSs) with nonlinear disturbances. The main contributions of this paper include three aspects: First, a proper observer is constructed. A sufficient condition in terms of linear matrix inequality (LMI) and Lyapunov function is proposed such that the resulting observer error system is asymptotically stable with a prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> norm bound for sufficiently small values of the perturbation parameter. Second, based on the input-to-state stability (ISS) property, an observer-based feedback controller is designed such that the resulting closed-loop system is ISS with respect to the observer error. Meanwhile, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula> performance index is also satisfied. Then, a workable way for solving the exact upper bound is also given. Finally, two numerical examples are given to demonstrate the validity of the developed method.

Observer-based event-triggered control and application in active suspension vehicle systems

This paper focuses on event-trigger control of automotive suspension systems. Firstly, fuzzy T-S systems which suitable for automobile suspension systems are studied. Parameter uncertainties and disturbances are included in the fuzzy T-S systems. Secondly, based on linear matrix inequalities (LMIs) and Lyapunov function, the conditions for the stability of fuzzy T-S systems are given. In the meantime, the controller and observer of the fuzzy T-S systems are designed. Finally, the theory is applied to the automotive suspension systems in the simulation.

Stabilization of Variable Bit Rate Video Streams Using Linear Lyapunov Functions on Wireless Mobile Network

Stabilization of Variable Bit Rate Video Streams Using Linear Lyapunov Functions on Wireless Mobile Network

Lyapunov approach and global stability of Ebola virus infection model of an individual cells population

Ebola virus is among the most dangerous and devastating threats to human health, causing a large number of fatalities. In this paper, a mathematical modelling for the dynamics of Ebola virus infectious of an individual is presented as a system of nonlinear differential equations. The model has two equilibrium states namely, virus free equilibrium (VFE) and virus persistence equilibrium (VPE) states. The Effective reproduction number was obtained. The conditions under which the virus-free-equilibrium is globally asymptotically stable with the approach of linear Lyapunov function are shown when the effective reproduction numbers is less than unity. The nonlinear Lyapunov approach is employed to show the global stability of the endemic equilibrium only when the effective reproduction number is greater than unity. It was found that VFE is globally asymptotically stable if effective reproduction numbers is less than unity and VPE is globally asymptotically stable if M&lt;N, otherwise unstable if M&gt;N.

Dissipativity-Based Fault-Tolerant Control for Stochastic Switched Systems With Time-Varying Delay and Uncertainties.

This article investigates the fault-tolerant control problem for stochastic switched interval type-2 (IT2) fuzzy time-delayed uncertain systems based on unknown input observer synthesis, which can avoid uneasy measurement on the time derivative of output, and estimate unavailable or partially measurable states, including sensor and actuator faults accurately. First, a desired fuzzy observer is designed to ensure the observer-based dynamic error system mean-square exponentially stable with sufficient condition of a strict (l,ħ,℘) - [Formula: see text]-dissipative performance, which is a unified framework of passivity, and H∞ provides results with less conservativeness. Then, we concentrate on stability analyses on dissipativity-based switched IT2 fuzzy systems with stochastic perturbation through linear matrix inequalities, Lyapunov function, free-weighting matrices, and average dwell time, discussing it according to different values of disturbance. Finally, simulation examples are listed to account for availability and effectiveness of the research methodology.

Event-triggered positive l1-gain non-fragile filter design for positive Markov jump systems

This paper is concerned with the event-triggered positive l 1 -gain non-fragile filter design of positive Markov jump systems. By using the upper and lower bounds of the error vector of system measurement outputs, the positive l 1 -gain filtering system to be designed is transformed into interval uncertain systems. Combining linear Lyapunov functions and linear programming, the positivity and stability of such systems can be obtained. Then, the proposed approach is developed to investigate positive l 1 -gain additive and multiplicative non-fragile filters, respectively. Furthermore, two classes of non-fragile positive filters are proposed for positive Markov jump systems with incomplete measurements and sensors saturation, respectively. Finally, two examples are provided to verify the effectiveness of the proposed design.

Stability and stabilisation for switched impulsive positive systems

The current study is dedicated to addressing the stability and weighted L 1 -gain performance analysis of switched impulsive positive systems (SIPSs). Firstly, by designing a novel multiple piecewise-continuous copositive linear Lyapunov function and using the mode-dependent average dwell time (MDADT) switching method, improved stability conditions that are able to achieve a tighter dwell time bound are developed. It is shown that the system under study is stable and possesses an attenuation property under the designed switching signals. Secondly, with the above stability conditions, a more effective controller design strategy has been proposed. The solved controllers are both quasi-time dependent and mode-dependent. Moreover, the constraint on the rank of the controller and the computation burden caused by iteration algorithm used in literature are relaxed. Finally, the feasibility and superiority of the proposed strategies are verified by some simulations.