Stability Problem Research Articles

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.

A Uniqueness Theorem for Stability Problems of Functional Equations

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that it simplifies the proof by eliminating the need to repeatedly and cumbersomely prove uniqueness in stability studies.

On stability of iteration processes convergent to stationary points of semigroups of nonlinear operators in metric spaces

Let C be a closed subset of a complete metric space. In this paper we study the stability problem for some iteration processes that approximate stationary points of pointwise Lipschitzian semigroups of nonlinear mappings acting within C. We show that a large class of well-controlled processes is stable under summable errors. These results are then applied to the stability of some known processes including the generalized Krasnosel'skii-Mann and the two-step Ishikawa iteration processes.

Robust H-Infinity Dual Cascade MPC-Based Attitude Control Study of a Quadcopter UAV

Aimed at the stability problem of quadrotor Unmanned Aerial Vehicle (UAV) flight attitudes under random airflow disturbance conditions, a robust H-infinity-based dual cascade Model Predictive Control (MPC) attitude control method is proposed. Model Predictive Control itself has the capability to minimize the deviation between the prediction error and the control target by optimizing the control algorithm. The robust H-infinity controller can maintain stability in the face of system model uncertainty and external disturbances. The controller designed in this paper divides the attitude control loop into the following two parts: the angle loop and the angular velocity loop. The angle loop, serving as the main control loop of the attitude control, employs the robust H-infinity controller to process the angle of the quadrotor UAV and then transmits the processed value to the MPC controller. This approach reduces the computational load of the MPC controller. Simultaneously, by optimizing the algorithm, MPC minimizes the prediction error and the deviation from the control target. Simulation experiments confirm that the proposed algorithm improves the stability of the UAV attitude under random airflow disturbance conditions, while also achieving accurate tracking of the UAV’s position.

Basic Problems in Stability and Design of Switched Systems [25 Years Ago

Basic Problems in Stability and Design of Switched Systems [25 Years Ago

Parameter optimization control of multi-rotor unmanned aerial vehicles based on adaptive differential evolution

Abstract Aiming at the problems of instability and underactuation encountered in the control process of traditional multi-rotor unmanned aerial vehicles, an adaptive differential evolution algorithm is designed to optimize the parameters of the sliding mode controller. The advantages of the fast convergence of the sliding mode controller and the global search of the evolutionary algorithm are combined for parameter optimization. The simulation results show that the sliding mode controller, after parameter optimization, has good stability.

Dynamic Event-Triggered Control for Sensor–Controller–Actuator Networked Control Systems

We consider the problem of output feedback stabilization of LTI systems under event-triggering implementation. In particular, we assume that both the plant output and the control input are both transmitted over the network in an asynchronous manner. To that end, two independent event-triggering rules are constructed to generate the transmission instants of the submitted signals. The proposed approach is dynamic in the sense that the triggering rules involve internal dynamical variables to allow for further reduction in the communication load. Moreover, the inter-transmission times for both sides of the channel are lower bound by enforced dwell times to prevent the occurrence of Zeno phenomena. The problem is challenging due to mutual interactions between the sampling errors of the plant output and the control input, which requires careful handling to ensure closed-loop stability. The triggering mechanisms are designed by emulation as we first ignore the effect of the network and stabilize the plant in continuous-time. Then, the communication constraints are taken into account to derive the triggering conditions such that the stability of the networked control system is preserved. The required conditions are formulated in terms of a linear matrix inequality. The effectiveness of the technique is demonstrated by numerical simulations.

Calibration modeling of drilling fluid rheological parameters in variable temperature environment based on support vector machine.

Traditional measurement of drilling fluid rheological parameters suffers from significant lag due to the inability of the instruments to promptly capture real-time parameters of the drilling fluid. These measurement models are typically constructed based on fixed temperature conditions and empirical formulas, rendering them inadequate for complex temperature gradient environments. Consequently, this limitation results in increased prediction errors, severely compromising the precise monitoring of drilling fluid performance. Aiming at the problems of low accuracy and poor stability of drilling fluid measurements under variable temperature conditions, a support vector machine-based calibration model for drilling fluid rheological parameters in a variable temperature environment is proposed in this paper. First, the measurement principle of the double-tube differential pressure rheology real-time measurement device is analyzed. The relationship between shear stress and shear rate is then established using differential pressure sensor and flow rate data. Utilizing the gray wolf optimization algorithm to optimize the kernel function weights and parameters, an SVM-based calibration model for predicting drilling fluid rheology correction parameters is constructed. Finally, a real-time monitoring platform for drilling fluid is developed. Experimental results show that the maximum relative errors for the predictions of apparent viscosity, plastic viscosity, and yield point are within ±5%, with coefficients of determination (R2) all greater than 0.95. These results validate the effectiveness of the proposed method in accurately monitoring the rheological performance of drilling fluids.

Backward stability of the Schur decomposition under small perturbations

In the present paper, we show the backward stability of the Schur decomposition for a given matrix under small perturbations. We also present the forward stability problem for matrices with multiple eigenvalues and show conditions under which the forward stability fails.

Cooperative control of multi-vehicle formation based on cooperative game

In this paper, an optimization decision-making method of vehicle fleet queue based on game theory is proposed. By adopting the upper and lower controller structure, the longitudinal motion stability problem of the vehicle fleet is solved. The upper controller utilizes a game control algorithm specially developed for this study. The game payoff function is constructed by considering real-time vehicle information and comprehensively evaluating factors such as expected distance, vehicle safety, and stability. The optimal expected acceleration of the vehicle is determined. Subsequently, the lower controller adjusts the engine throttle opening or brake pressure to regulate the vehicle operation according to the desired acceleration. Simulation results show that the designed game strategy can effectively reduce vehicle spacing and speed errors, thus significantly improving the stability as well as safety of fleet operation.

On the asymptotic analysis of pulsating planar Poiseuille flow instabilities

The linear stability analysis of a pulsating flow between parallel plates at finite amplitudes and different frequencies demonstrates the presence of instability with a different nature to that initially reported in earlier studies at W o < 25 . The time-dependent stability problem is formulated using a Floquet expansion of the perturbation and solved in a matrix approach. At higher pulsation frequencies (i.e. W o > 30 ), pulsation amplitudes and sufficiently large Reynolds numbers, the time-dependent baseflow displays multiple inflexion points throughout the entire duration of the pulsation. A scaling analysis of the temporal growth rate reveals that the instability is of inviscid type and scales with R e − 1 . Freezing the time-dependent baseflow for a particular set of parameters, the growth rate is simultaneously controlled by viscosity solely acting on the perturbation, and the pulsating frequency coupling the harmonics. In the asymptotic limit ( R e → + ∞ ), we demonstrate that the temporal problem simplifies to a set of steady stability analyses dependent on the phase of the pulsation. In particular, we show that in this limit, Rayleigh and Fjørtoft’s criteria for instability become valid again, that sub-harmonic periodic orbits can be used to estimate the inviscid growth rate and match with the asymptotic values predicted from the viscous stability analysis.

Stability analysis of tunnel heading in clay with nonstationary random fields of undrained shear strength

The stability problem of a tunnel heading in clay remains a significant challenge in geotechnical engineering. Specifically, when considering the spatial variability of the soil, the stability factor may be influenced by geographically random fields. This study investigates the effect of random fields on a probabilistic analysis of a tunnel heading in undrained clay. The study assumes that the undrained shear strength of the clay increases linearly with depth due to a strength gradient factor. The random adaptive finite element limit analysis is employed to calculate the stability numbers for tunnel headings. Nonstationary random fields with varying vertical correlation lengths are simulated using Monte Carlo simulation technique. The stability analysis takes into account geometry parameters (i.e., cover depth ratio) and nonstationary random field of undrained shear strength parameters. (i.e., strength gradient, coefficient of variation, and vertical correlation length). The results of tunnel face stability using random adaptive finite element limit analysis have also been utilised to assess the probability of design failure over a practical range of deterministic factors of safety. In the context of probabilistic failure analysis, the failure mechanism resulting from varying vertical correlation lengths could influence the probability of design failure. The findings of this study can be of significant interest to tunnel engineering practitioners during the design phase of tunnel heading projects.

Pinning control design for the output regulation of Boolean networks

In previous studies, the output regulation problem (ORP) of Boolean networks (BNs) is usually studied by the augmented system method, however, the state transition matrix dimension of augmented system is growing exponentially as the number of network nodes increases. To reduce the computational complexity, a new method to solve the ORP is adopted in this paper. Firstly, the relationship among the attractors of two BNs and their augmented system is studied, based on which, the ORP is converted into the set stabilisation problem of the original BN, and some necessary and sufficient conditions are given to detect the solvability of the ORP. Secondly, a pinning control for the ORP is considered according to the attractors of the original BNs, and an algorithm to design this kind of controls is provided. Finally, a practical example of signal transduction network is given to illustrate the effectiveness of the method in this paper.

A Review of Convex Clustering From Multiple Perspectives: Models, Optimizations, Statistical Properties, Applications, and Connections.

Traditional partition-based clustering is very sensitive to the initialized centroids, which are easily stuck in the local minimum due to their nonconvex objectives. To this end, convex clustering is proposed by relaxing K -means clustering or hierarchical clustering. As an emerging and excellent clustering technology, convex clustering can solve the instability problems of partition-based clustering methods. Generally, convex clustering objective consists of the fidelity and the shrinkage terms. The fidelity term encourages the cluster centroids to estimate the observations and the shrinkage term shrinks the cluster centroids matrix so that their observations share the same cluster centroid in the same category. Regularized by the lpn -norm ( pn ∈ {1,2,+∞} ), the convex objective guarantees the global optimal solution of the cluster centroids. This survey conducts a comprehensive review of convex clustering. It starts with the convex clustering as well as its nonconvex variants and then concentrates on the optimization algorithms and the hyperparameters setting. In particular, the statistical properties, the applications, and the connections of convex clustering with other methods are reviewed and discussed thoroughly for a better understanding the convex clustering. Finally, we briefly summarize the development of convex clustering and present some potential directions for future research.

Path Optimization of Two-Posture Manipulator of Apple Packing Robots

Automated packing is urgently needed in apple production. This paper proposes an improved genetic algorithm fused with an optimal parameter selection algorithm to optimize the two-posture manipulator working path of packing robots. First, the structure and working principle of the packing robot were designed. Second, the kinematics and packing paths of the two-posture manipulator were analyzed. Finally, the path optimization method for the two-posture manipulator was introduced. The method was based on the improved genetic algorithm by using a two-level coding and region crossover operator. The parameter values can be automatically determined by the optimal parameter selection algorithm. Ten repeated comparative tests show that the total packing time is 23.86 s under the working conditions of four grasping points and fourteen placing points. The optimal performance of the proposed algorithm is better than that of the traditional genetic algorithm, and the average optimization amplitudes are 14.63%, 15.42%, 16.24%, and 13.82% for 9-groove, 12-groove, 14-groove, and 16-groove trays, respectively. The proposed algorithm can effectively prevent the premature convergence problem of the traditional genetic algorithm and the optimization process instability problem, improve the range of optimization, and reduce the manipulator working time during packing.

Electromechanical processes in electric power steering system based on permanent magnet synchronous motor

Relevance. Electric power steering of vehicles successfully competes with hydraulic boosters of similar purposes, showing high efficiency, energy saving and environmental friendliness. Aim. To study and research electromechanical processes of electric power steering system based on permanent magnet synchronous motor. Methods. Mathematical and numerical modeling using the Matlab/Simulink software package. Results. The article discusses the operating modes of an electric power steering based on an electric motor with permanent magnets. The authors analyzed two classic schemes for constructing steering systems with electric power steering – with the electric motor placed on the steering column and on the steering rack. Block diagrams of the mechanical and electrical parts of the steering system have been developed. Based on mathematical dependencies characterizing the operating modes of the steering system, a complete dynamic simulation model of the electric power amplifier was obtained on the MATLAB/Simulink platform. Based on a preliminary analysis of control methods and algorithms in the system to solve the problem of control stability, PI controllers with low-pass filters are implemented to regulate the current and voltage of the electric motor depending on the speed of the vehicle and the angle of rotation of the steering wheels. The simulation results were used to design a real electric power steering for a specific vehicle. Conclusions. The complete simulation model of an electric booster based on a synchronous motor with permanent magnets in a car steering system obtained by the authors adequately reflects the dynamic control processes and can be used in the design of automotive vehicles.

Collaborative design and construction analysis of building structure and foundation of subway overlying complex

With the acceleration of urbanization, land resources are becoming more and more precious, especially in densely populated urban centers. As an efficient land use model, the subway overlying complex can not only optimize the urban spatial layout, but also promote the integration of public transportation and urban development. This paper aims to discuss the collaborative design and construction technology of the building structure and foundation of the subway overcover complex, in order to provide theoretical support and technical guidance for the practice in related fields. Based on the literature review of domestic and foreign subway overcover development projects, this paper reviews the research status and development trend in this field, and identifies the main problems in structural safety, foundation stability and construction efficiency. On this basis, the paper puts forward the design principle of the subway overcover complex, and emphasizes the key role of the synergistic relationship between the structure and the foundation to the success of the whole project. The paper further analyzes the construction technology of subway overcover complex, including specific construction technology, reasonable construction sequence and project management measures. Through comparative analysis of success and failure cases, it reveals the problems that may be encountered in the actual operation and its coping strategies.

«5+1» COOPERATION FORMATS IN THE CENTRAL ASIAN REGION: ANALYSIS OF ECONOMIC INTERACTION WITH JAPAN

The article analyzes Japan’s economic interaction with Central Asia within the framework of the «5+1» format, which is a unique mechanism of multilateral cooperation that has attracted the attention of leading world powers in recent years due to the strategic importance of the region in the context of energy resources and transit routes. The research is based on a comprehensive analysis of investment, trade and economic relations between the countries of Central Asia and Japan. The main purpose was to identify key trends and factors that determine the dynamics of economic interaction and analyze the interrelations of economies. The hypotheses were tested using comparative statistics, dynamic analysis and graphical method, as well as using statistical methods, including calculation of correlation coefficients and multiple regression. Limitations of the study include the lack of data for some countries and periods. The analysis showed a moderate positive correlation between the GDP (Gross domestic product) of the countries of Central Asia and Japan, indicating economic interdependence. Japan’s investment activity is manifested in significant investments in Kazakhstan and Kyrgyzstan, with recent growth in Tajikistan. Trade turnover between the countries fluctuates due to global and regional factors. Despite the positive correlation of economies, Central Asia's influence on Japan remains limited. Japanese investment plays a key role in the region's development, but political instability and domestic problems in the countries reduce their effectiveness. Trade also faces challenges, including customs barriers and infrastructure limitations.

Existence theory for ψ -Caputo fractional differential equations

UDC 517.9 The target of this paper is to handle a nonlocal boundary-value problem for a specific kind of nonlinear fractional differential equations encapsuling a collective fractional derivative known as the ψ -Caputo fractional operator. The applied fractional operator generated by the kernel is of the following kind: k ( t , s ) = ψ ( t ) - ψ ( s ) . The existence of the solutions of the above-mentioned equations is tackled by using Mönch's fixed-point theorem combined with the technique of measures of noncompactness. In addition, we discuss the problem of stability within the scope of the Ulam–Hyers stability criteria for the main fractional system. Finally, an example is given to illustrate the viability of the reported results.

Discrete time stability of augmented Lagrangian formalism based underactuated inverse dynamics control method

Stability problems of robotic systems arise sometimes suddenly, seemingly for no reason. The digital time sampling is often the main cause of these instabilities. Discrete models, which are capable of the prediction of stability, are available for low-degree-of-freedom template models of linear position and force control. However, for the inverse dynamics control of underactuated systems, the literature has a lack of generally applicable results related to the effect of time discretization. Several control approaches are available in the literature out of which a widely used one, the augmented Lagrangian formalism and its stability properties are analysed in this work. Theoretical stability properties are obtained for a generally usable, linear, underactuated, two-degree-of-freedom constrained template model. The actuator dynamics, the finite difference approximation of the feedback velocity and the filtering of the feedback data are considered in the model. These phenomena strongly affects the stability properties. The theoretically obtained stability maps are experimentally validated on an underactuated crane-like indoor robot. The position and orientation accuracy of the robot were assessed: the absolute position error was below 30 mm and the orientation error was below 3°.