Lyapunov Function Research Articles

Global mathematical analysis of a patchy epidemic model

The dissemination of a disease within a homogeneous population can typically be modeled and managed in a uniform fashion. Conversely, in non-homogeneous populations, it is essential to account for variations among subpopulations to achieve more precise predictive modeling and efficacious intervention strategies. In this study, we introduce and examine the comprehensive behavior of a deterministic two-patch epidemic model alongside its stochastic counterpart to assess disease dynamics between two heterogeneous populations inhabiting distinct regions. First, utilizing a specific Lyapunov function, we demonstrate that the disease-free equilibrium of the deterministic model is globally asymptotically stable. For the stochastic model, we establish that it is well-posed, meaning it possesses a unique positive solution with probability one. Subsequently, we ascertain the conditions necessary to ensure the total extinction of the disease across both regions. Furthermore, we explicitly determine a threshold condition under which the disease persists in both areas. Additionally, we discuss a scenario wherein the disease persists in one region while simultaneously becoming extinct in the other. The article concludes with a series of numerical simulations that corroborate the theoretical findings.

Finite-time robust control of morphing aircraft based on time-varying observer

Morphable vehicles have strong nonlinear, strong coupling and strong time-varying characteristics in the process of variation, which brings great challenges to the design of their flight control systems. To address this problem, a time-varying gain extended state observer is proposed to accurately approximate the system coupling terms. Combining adaptive backstepping control technology and finite-time control theory, a finite-time bounded robust adaptive controller is designed. By designing the barrier Lyapunov function, the actual control law and the adaptive network update law are obtained step by step, ensuring that the system command tracking error can converge to a predetermined range within a finite time. Finally, the effectiveness of the designed control system is verified by attitude control simulation. The simulation results show that the morphable vehicle can track the command signal well in the process of variation and is basically not affected by the variation rate.

Reliable Finite‐Time Control for Nonlinear Chaotic Semi‐Markov Jump Systems With Incomplete Transition Rates

ABSTRACTThis article studies the finite‐time control problem for a class of nonlinear chaotic semi‐Markov jump systems with incomplete transition rates described by the T‐S fuzzy model approach. As a means to depict the dynamical properties pertaining to the examined system, parametric uncertainties, faults, external disturbances and input saturation are taken into consideration. The foremost objective of this study is to come up with a composite control mechanism that effectively rejects and attenuates the repercussions of faults and disturbances. In particular, we primarily built a disturbance observer in order to obtain a precise estimation of the external disturbances. After which, the fault diagnosis observer is designated to effectively estimate the faults. Specifically, a composite reliable control mechanism is developed by fusing the output of the constructed observers with the mode‐dependent fuzzy‐rule based state feedback controller. By employing suitable Lyapunov functions in conjunction with the linear matrix inequality technique, a set of mode‐dependent conditions is derived to ensure the finite‐time stochastic boundedness of the underlying closed‐loop and estimation error systems. Following this, the anticipated controller and observer gain matrices are elicited by resolving the established linear matrix inequalities. Thereafter, intending to ascertain the efficacy and usefulness of accrued theoretical findings, simulation results performed on Chua's circuit system is endowed.

Distributed Consensus Filtering Over Sensor Networks With Asynchronous Measurements

ABSTRACTIn practical sensor networks, sensor nodes may operate with different sampling periods and initial sampling time instants, and their observations may also be nonuniform. Unfortunately, the research on distributed state estimation problems over such asynchronous sensor networks is very limited. Thus, this article focuses on the distributed consensus filtering problem over sensor networks with asynchronous measurements. First, the asynchronous measurement from each sensor is synchronized by the continuous‐time state evolution equation to a unified filtering fusion time instant within a given filtering period. After measurement synchronization, the statistical characteristics of measurement noise change. The cross‐correlations between the converted measurement noises are analyzed, as well as one‐step correlations between the converted measurement noises and the process noise. Second, an optimal asynchronous distributed consensus filter is designed based on synchronization measurements under the criterion of minimum mean‐square error with the above correlations between various types of noises taken into account. Meanwhile, a suboptimal distributed consensus filtering algorithm is further proposed to reduce computational complexity. Finally, based on the Lyapunov function method, the stability of the estimation error is theoretically demonstrated with an appropriate selection of consensus filtering gain and validated through simulations.

Electromechanical coupling modeling and fractional-order control of the seeker stabilization platform

To address the electromechanical coupling and multi-source disturbance problems of the seeker stabilized platform, this paper constructs an electromechanical coupling model of the seeker stabilized platform based on the Lagrange-Maxwell equation. To mitigate the influence of electromechanical coupling on the control performance of the seeker, a super-twisting controller based on a fractional-order terminal sliding mode surface (FOSTSMC) is proposed. Additionally, to handle various disturbances in the system, this paper introduces a method that combines the extended state observer (ESO) with the proposed controller to enhance the system’s stability and anti-disturbance performance. The Lyapunov function is designed to prove that the proposed controller can reach a convergence state within finite time. Finally, the proposed control method is compared with PID control, fuzzy PID control, linear sliding mode control, and super spiral control combined with a disturbance observer (DOB). Multiple simulation experiments demonstrate that, under the influence of electromechanical coupling and multi-source disturbance, the FOSTSMC-ESO significantly improves the stability and anti-disturbance performance of the seeker stabilization platform.

Consensus for discrete-time second-order multi-agent systems in the presence of noises and semi-Markovian switching topologies

In this paper, we focus on the control problem of mean square consensus for second-order continuous-time multi-agent systems with multiplicative noises under Markovian switching topologies. A new Lyapunov function based on the Laplacian matrix of the corresponding union topology of all possible topologies is designed for the stochastic stability analysis of consensus. Applying matrix theory and stochastic stability for stochastic differential equations, we can analyze the consensus problem of the considered systems with the designed Lyapunov function. In addition, we present the sufficient conditions of the mean square consensus exponentially for the considered stochastic systems. Finally, we give an simulation example to numerically validate our theoretical results.

Dynamic behavior in a pursuit-evasion system with signaling mechanism

We mainly focus on the predator-prey problem associated with indirect predator taxis{ut=Δu+χ1∇⋅(u∇w)+u(1−μ1u−αv),vt=Δv−χ2∇⋅(ψ(v)∇u)+v(−1−μ2v+βu),0=Δw+v−w in a smooth bounded domain Ω⊂Rn(n≤3) with homogeneous Neumann boundary conditions, where the parameters χ1, χ2, μ1, μ2, α and β are positive. The chemotaxis sensitivity function with/without “volume-filling” effect ψ(v)=v1+σv, where σ≥0 is a constant. Compared with the well-known predator-prey system, the discussion of signaling mechanism w forms a novel feature of our system. For the case σ=0, we identify a positive constant χ0 such that the above system possesses global classical solutions with relative bound if χ2<χ0; For the case σ>0, we also show that the classical solutions of the system are globally bounded without any restrictions on system parameters. Furthermore, utilizing the Lyapunov functionals, we establish stability for the prey-only situation, the steady state of coexistence is also investigated.

Robust Fault Estimation and Fault‐Tolerant Control for a Class of Fuzzy Singularly Perturbed Systems With State Time Delay Based on Learning Observer

ABSTRACTThis article studies the problem of fault estimation (FE) and dynamic output fault‐tolerant control (DOFTC) for a class of Takagi–Sugeno (T–S) fuzzy singularly perturbed systems (SPSs) which subject to time delay, external disturbance and actuator fault. A new fault estimation and fault‐tolerant control scheme was proposed and designed for the influence of the change of perturbation parameter on the singularly perturbed system. This scheme adopts the Takagi–Sugeno (T–S) fuzzy model, learning observer and dynamic output feedback fault‐tolerant mechanism, so that the system has multi‐time‐scale dynamic stability. The results show that this method has more accurate estimation effect, faster convergence speed, and very fast steady‐state response of fault‐tolerant control when faults occur. Furthermore, when constructing the Lyapunov function, the improved matrix was selected to ensure that the closed‐loop system is stable for all , and at the same time, the conservativeness of the results was reduced. Finally, the feasibility and correctness of the proposed design method were illustrated through two numerical examples.

Analyzing Sumability in Mean Square for Stochastic Difference Equations via Lyapunov Methods

This paper investigates solutions to the second-kind stochastic volterra difference equation through two case studies. It focuses on, concentrates on the sumability of these solutions in the context of mean square and mean fourth criteria, utilizing the technique of constructing Lyapunov functionals.

Adaptive event-triggered stochastic estimator-based sampled-data fuzzy control for fractional-order permanent magnet synchronous generator-based wind energy systems

This study aims to develop an adaptive event-triggered estimation sampled-data control method for a fractional-order permanent magnet synchronous generator (PMSG)-based wind energy system (WES) using the Takagi–Sugeno (T–S) fuzzy approach. Unlike existing PMSG-based WES control schemes, we propose an aperiodic event-triggered communication scheme with an adaptive mechanism to reduce data transmission. To address the challenge of limited communication resources, we introduce an adaptive event-triggered (AET) estimation method with aperiodic sampling, significantly reducing communication overhead while maintaining stable WES performance. This method employs transmission techniques based on absolute errors, resulting in high data transmission efficiency. First, the fractional-order model for the PMSG-based WES is represented using linear sub-models through the T–S fuzzy approach. Next, a fuzzy aperiodic AET mechanism is proposed for the PMSG model with augmented estimation error systems. Then, the fractional Lyapunov function theory is employed to derive linear matrix inequalities (LMIs), ensuring bounded mean-square stability for the PMSG-based WES with the augmented estimation error systems. Additionally, the desired estimator control gains are determined through solvable LMIs. Finally, simulation studies are presented to demonstrate the superiority and feasibility of the proposed control scheme.

A Lyapunov Theory-Based SEIG–STATCOM Voltage Regulation Control Strategy

To improve the voltage regulation of asynchronous generators during load switching, a Lyapunov-based control strategy has been proposed to stabilize the generator’s voltage by connecting a static synchronous compensator. By constructing a Lyapunov function from the mathematical model, the error tracking problem is transformed into a global asymptotic stability problem of the Lyapunov function at the equilibrium point. The outer loop linearizes the direct current (DC) voltage control process, while the inner loop replaces integral terms with differential terms. The proposed Lyapunov method achieves linearized voltage control with a quadratic outer loop structure and the inner loop differential structure exhibits a shorter transient process, outperforming traditional methods. Simulation and experimental tests were then used, where the latter was a down-scale laboratory prototype experiment. Compared to traditional (voltage-oriented control) VOC, the outer loop (Lyapunov-function-based control) LBC reduces the DC voltage transient processes by approximately 9.4 milliseconds, while the inner loop LBC reduces both alternating current (AC) and DC voltage transient processes by approximately 2.6 ms and 8.7 ms, respectively.

ℒ2-gain performance of nonlinear sampled-data systems with gain-scheduled control

This paper presents new sufficient conditions for stabilisation of sampled-data nonlinear systems, recast as quasi-linear parameter-varying models. The adopted control strategy considers gain-scheduled state-feedback controllers subject to induced input delay. The L 2 -gain performance is investigated in the sense of Lyapunov. After the proposition of a Lyapunov function and with the application of Wirtinger's inequality, the derived conditions are rewritten in terms of linear matrix inequalities. Aiming at the enlargement of the aperiodic sampling time and at the minimisation of the L 2 -gain, the effectiveness of the developed approach is assessed through numerical simulations.

Exponential stability and non-weighted L 2-gain analysis for switched neutral systems with unstable subsystems

Exponential stability and non-weighted L 2 -gain are studied for switched neutral systems with unstable subsystems. Initially, a time scheduling strategy is introduced to formulate switching-signal-based multiple discontinuous Lyapunov functions. Multiple discontinuous Lyapunov functions can be applied to each subsystem, thereby relaxing the limitations of traditional Lyapunov functions. Meanwhile, mode-dependent average dwell times are utilised to design fast and slow switching signals, ensuring stability in switched neutral systems where the switching between unstable and stable subsystems is arbitrary. Moreover, in the presence of external disturbances, the non-weighted L 2 -gain performance is discussed. Sufficient conditions are proposed to achieve the stability of switched neutral systems and non-weighted L 2 -gain performance using the linear matrix inequality method. The validity of this study is ultimately demonstrated through simulation examples.

Lyapunov-based distributed secondary frequency and voltage control for distributed energy resources in islanded microgrids with expected dynamic performance improvement

Microgrids (MGs) can effectively integrate the high penetration distributed energy resources (DERs) for the energy and environmental crisis. Still, fluctuations in intermittent DERs may lead to severe frequency and voltage deviations or even the instability of islanded MGs. Secondary control has successfully compensated for the frequency and voltage deviations. Research on secondary control mainly focused on steady-state operating objectives, i.e., frequency and voltage recovery and power sharing. Still, improving the dynamic responses of secondary control is crucial for system-stable operation, especially in the presence of synchronous DERs. This is because these units can cause undesired oscillatory modes, leading to system instability. In addition, because of the line impedance effect, accurate reactive power sharing is usually ignored when voltage recovery is considered. This will lead to an overload of MGs. To improve the dynamics of secondary control while trading off voltage regulation and reactive power sharing, we propose a Lyapunov-Function (LF)–based distributed secondary control strategy. In the proposed LF-based control strategy, the frequency, voltage, and power information are introduced into feedback control based on the Lyapunov stability theorem. Therefore, the improved frequency and voltage deviations and accurate power sharing can be guaranteed when the Lyapunov function asymptotically attenuates to zero. The inherent conflict between voltage regulation and reactive power sharing is addressed by converging the average voltage to the rated reference. Besides, the improved dynamic performance of secondary control is achieved by considering global power variations, which are obtained by distribution-level phasor measurement units. Global power variations can establish control actions to dampen system oscillations and accelerate system restoration. Furthermore, the controller design method based on the Lyapunov stability theorem can naturally promise the stability of the proposed secondary frequency and voltage controllers. Numerical simulations on the IEEE 34-bus system validate that the proposed control strategy can ensure the significantly improved dynamic performance of secondary control while achieving steady-state operation objectives.

Robust finite-time output feedback stabilisation for a class of uncertain planar systems with asymmetric output constraints

This paper considers the finite-time output feedback stabilisation (FTOFS) problem for a class of uncertain planar systems with asymmetric output constraints. The presence of unknown control coefficients, unknown measurement sensitivity and asymmetric output constraints makes the system under consideration essentially different from those in the existing related works. By constructing a new barrier Lyapunov function and applying homogeneous system theory, a simple proportional-derivative like finite-time output feedback controller is explicitly constructed, which only utilises the output information. A rigorous proof demonstrates that not only the closed-loop system is finite time stable, but also the requirement of asymmetric output constraint can be achieved. The novelty of the control approach lies in that it is capable to handle the FTOFS problem for uncertain planar systems with or without asymmetric output constraints in a unified manner, without changing the control structure. The efficiency of the proposed theoretical results are confirmed by simulation results.

Event-based adaptive formation and tracking control with predetermined performance for nonlinear multi-agent systems

The article focuses on the event-based predetermined performance formation tracking control for uncertain nonlinear multi-agent systems. Each agent is subject to actuator saturation constraint and full-state constraints. To meet the full-state constraints, a barrier function-based nonlinear mapping method is employed instead of the barrier Lyapunov function, such that the undesirable “feasibility conditions” are eradicated. To overcome actuator saturation, the auxiliary systems are utilized to let the controller be designed under the framework of backstepping. The approximation capacity of RBF NNs and adaptive methods are adopted, which can solve unknown uncertainties. An event-triggered predetermined performance formation tracking control strategy is designed, which can guarantee that the agents form the prescribed formation shape in a predetermined settling time and have a fine transient and steady-state tracking performance, while reducing the communication burden from the controller to the actuator. At last, an example of unmanned aerial vehicles is given to test the availability of the results.

A stable model order reduction scheme for an acoustic finite element model with perfectly matched layers

Perfectly matched layers (PMLs) are often used in an exterior acoustic finite element model to simulate the Sommerfeld radiation condition. However, the time-domain model is often too large to efficiently handle the transient simulation due to the frequency-dependent stretching function and the number of elements needed per wavelength. Krylov subspace-based model order reduction can alleviate this problem by only matching the important moments of the original model. However, stability-preservation is not guaranteed, which is the key to perform time-domain simulation. This work proposes a stability-preserving model order reduction scheme for the time-domain PMLs. By assessing the stability conditions of time-domain PMLs, a two-step strategy is applied. Firstly, a one-sided split basis is used to preserve the form of Lyapunov function of the original model such that a stable intermediate reduced order model (ROM) results. Secondly, an unsplit basis is applied on the modal transformation of this intermediate ROM to reduce the size further. The proposed method is verified by numerical simulations.

Bounded Containment Maneuvering Protocols for Marine Surface Vehicles With Quantized Communications and Tracking Errors Constrained Guidance: Theory and Experiment.

A new type of containment maneuvering protocols for multiple marine surface vehicles (MSVs) is developed to follow a parameterized path in this work, where the tracking errors are constrained within finite time and the information needed to be transmitted is quantized during coordination. To achieve containment maneuvering of multiple MSVs, a two-objective coordinated control framework is proposed. For the geometric objective, by developing tan-type barrier Lyapunov functions (BLFs) and extended Lyapunov condition-based finite-time guidance laws, the performance of the parameterized line-of-sight guidance framework, including convergence speed and tracking error constraints, is improved. For the dynamic objective, based on quantized control strategy and smooth saturation functions, novel bounded containment maneuvering protocols are proposed to dramatically alleviate the burden of communications among MSVs and ensure more faster dynamic behavior on tracking the path updating speed. Both theoretical analysis and experimental tests with comparative studies illustrate the validity of the proposed containment maneuvering strategy.

Finite-Time Adaptive Control for Electro-Hydraulic Braking Gear Transmission Mechanism with Unilateral Dead Zone Nonlinearity

Autonomous vehicles require more precise and reliable braking control, and electro-hydraulic braking (EHB) systems are better adapted to the development of autonomous driving. However, EHB systems inevitably suffer from unilateral dead zone nonlinearity, which adversely affects the position tracking control. Therefore, a finite-time adaptive control strategy was designed for unilateral dead zone nonlinearity. Initially, the unilateral dead zone nonlinearity was reformulated into a matched disturbance term and an unmatched disturbance term to reduce the adverse effects of disturbances, thereby enhancing system controllability. Then, the “complexity explosion” in the design of the control strategy was avoided by command filtering, and the design process of the controller was simplified. Furthermore, the finite-time control theory was employed to boost the system’s convergence speed, thereby enhancing control performance. In order to ensure the stability of the system under the dead zone disturbance, the unknown disturbance terms were estimated. The stability of the control strategy was validated through the finite-time stability theorem and the Lyapunov function. Eventually, simulations and hardware-in-the-loop (HIL) experiments validated the feasibility and availability of the finite-time adaptive control strategy.

Improving extensions for the global uniform asymptotic stability of continuous-time nonlinear systems

This paper considers the classical problems of global asymptotic stability (GAS) of nonlinear autonomous (time-invariant) systems and global uniform asymptotic stability (GUAS) of nonlinear non-autonomous (time-varying) systems. By imposing new conditions on the Lyapunov function and its derivative, the state convergence to the origin from any initial condition with bounded norm in the classical setting is extended to the initial conditions with unbounded norms in the present work. The proposed new notions of stability are called extended global asymptotic stability (EGAS) and extended global uniform asymptotic stability (EGUAS) in this paper. The proposed Lyapunov-based criteria yield the global asymptotic stability of the origin and fixed-time/predefined-time attractiveness of any selected neighbourhood of the origin. The result is that the dynamical behaviour of the state trajectories from the convergence time point of view is improved considerably. The relationship between the smoothness and input-to-state stability (ISS) properties is also studied from a quantitative point of view. Under some new sufficient conditions, it is shown that dynamical systems with smaller Lipschitz constants represent smaller ultimate bounds, and as a result, are more robust. The desirable dynamical behaviour and robustness property of the proposed stability notion makes it comparable with the finite/fixed/predefined/prescribed (exact) stable systems that are naturally non-Lipschitz at the origin. Numerical results illustrate the effectiveness of the proposed theoretical results.