Stability Of Dynamical Systems Research Articles

Incipient Fault Detection and Reconstruction Using an Adaptive Sliding-Mode Observer for the Actuators of Fixed-Wing Aircraft

This paper proposes a new method of detection and reconstruction of the incipient fault of fixed-wing actuators based on an adaptive sliding-mode observer. First, a mathematical model of a fixed-wing aircraft is derived under certain assumptions, considering nonlinear terms and system disturbances. Second, by introducing a nonsingular coordinate transformation, the incipient faults are separated from the disturbances. For a subsystem with no disturbances, the Luenberger observer can estimate the incipient fault. For a subsystem with disturbances, the sliding-mode observer is robust against these disturbances. The Lyapunov stability theory guarantees dynamic error convergence and system stability. The evaluation function was designed to realize residual evaluation and threshold judgment. Third, based on the concept of equivalent output injection, an adaptive sliding-mode observer method is proposed to reconstruct actuator faults precisely under the condition of the uncertain system structure. The design steps of the proposed reconstruction method are introduced in the form of a linear matrix inequality problem, which provides an effective method for calculating the design parameters. Finally, the simulation results of the De Havilland DHC-2 “Beaver” aircraft demonstrate the correctness and effectiveness of the proposed method.

Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

Statistical study of dynamic behavior of triple star systems using available stability criteria

The present study investigates 141 dynamically stable triple-star systems. These systems are chosen from the latest multiple star systems catalog (MSC). The eccentricity distribution for the outer and inner orbits is obtained. According to the eccentricity diagram, the outer orbits are oval, which indicates higher eccentricity, while the inner orbits are almost circular. Therefore, the hierarchical configuration of triple-star systems is confirmed. Five different criteria are considered to study the dynamic stability of multiple star systems. Observation-based stability factor values and the corresponding critical values can be determined from mass and orbital values of the components. Moreover, the plot of stability margin versus the outer orbit eccentricity is drawn. According to this plot, an increase in the outer orbit eccentricity increases the distance from the stability boundary. Consequently, a high outer orbit eccentricity leads to a high system instability. Our investigations show that in some criteria, many systems are stabilized by considering the dependence on the inner orbit eccentricity as 1/(1−ein) so that the related criteria are modified. Additionally, nearly all triple-star systems have hierarchical configurations and are stable. However, in at least three criteria, there are only five unstable systems with the WDS indices 00247-2653, 08391-5557, 02022-2402, 06467+0822, and 18126-7340. The instability of these systems can be attributed, with high probability, to observation errors or the unrealistic theoretical criteria. At the end, the credibility and accuracy of the mentioned five criteria are considered as the basis for ordering them.

Mitigating Low-Frequency Oscillations and Enhancing the Dynamic Stability of Power System Using Optimal Coordination of Power System Stabilizer and Unified Power Flow Controller

The integration of a flexible alternating current transmission system (FACTS) and a power system stabilizer (PSS) can increase dynamic stability. This paper presents the enhancement of power system dynamic stability through the optimal design of a power system stabilizer and UPFC using an ant lion optimization (ALO) technique to enhance transmission line capacity. The gained damping ratio, eigenvalue and time domain results of the suggested ALO technique were compared with a base case system, ALO-based PSS and ALO-based PSS-UPFC to test the effectiveness of the proposed system in different loading cases. Eigenvalues gained from an ant lion approach-based UPFC with a PSS and a base case system are compared to examine the robustness of the ALO method for various loading conditions. Thus, this paper addresses the mechanism regarding the power system dynamic stability of transmission lines by integrating the optimal size of a PSS and UPFC into the power system. Therefore, the main contribution of this manuscript is the optimal coordination of a power system stabilizer, power oscillation damper and unified power flow using ant lion optimization for the mitigation of low-frequency oscillation.

New Energy Power System Dynamic Security and Stability Region Calculation Based on AVURPSO-RLS Hybrid Algorithm

Using a high proportion of new energy is becoming the development trend of the modern power industry, with broad application prospects and potential threats to power system operation safety. This paper proposes a hybrid adaptive velocity update relaxation particle swarm optimization algorithm (AVURPSO) and recursive least square (RLS) method to quickly estimate the DSSR boundary using hyper-plane expression. Firstly, the operating point data in the high-dimension nodal injection space are analyzed using the AVURPSO algorithm to identify the key generators, equivalent search space, and critical points, which have relatively great effects on transient angle stability. The hyper-plane expression of the DSSR boundary, which matches the critical points best, is finally fitted by the RLS approach. Hence, the adopted algorithm is applied to rapidly approximate the DSSR boundary by hyper-plane expression in power injection spaces. Finally, the proposed algorithm is validated using a simulation case study on three wind farm regions of the actual Hami Power Grid of China using the DIgSILENT/Power Factory software. Consequently, the mentioned method effectively captures the security stability boundary of the new energy power system and realizes the three-dimensional visualization space of DSSR. By leveraging the DSSR, the state analysis can be conducted rapidly on several parameters, including security and stability assessments in relation to various energy supply capabilities. Meanwhile, these indices are calculated offline and applied online. The findings of this investigation confirm the efficacy and accuracy of the suggested modeling used in the analyzed system, offering technical assistance ensuring the stability of the new energy power system. The DSSR allows the rapid analysis of several parameters, including security and stability assessments with various energy supply capabilities.

Fixed-time synchronization of delayed memristive neural networks with impulsive effects via novel fixed-time stability theorem

In this study, the fixed-time synchronization (FXTS) of delayed memristive neural networks (MNNs) with hybrid impulsive effects is explored. To investigate the FXTS mechanism, we first propose a novel theorem about the fixed-time stability (FTS) of impulsive dynamical systems, where the coefficients are extended to functions and the derivatives of Lyapunov function (LF) are allowed to be indefinite. After that, we obtain some new sufficient conditions for achieving FXTS of the system within a settling-time using three different controllers. At last, to verify the correctness and effectiveness of our results, a numerical simulation was conducted. Significantly, the impulse strength studied in this paper can take different values at different points, so it can be regarded as a time-varying function, unlike those in previous studies (the impulse strength takes the same value at different points). Hence, the mechanisms in this article are of more practical applicability.

Optimal consensus control for multi-satellite assembly in elliptic orbit with input saturation

This paper presents an optimal control design method for multi-satellite assembly in an elliptic orbit in close proximity operations considering input saturation constraints. Tschauner–Hempel (T–H) equations are utilized to model the relative motion dynamics of the satellites in the Local Vertical Local Horizontal (LVLH) frame of the target satellite, where every satellite in the assembly is modeled as a point mass. The communication topology between the satellites in the assembly is modeled using graph theory, where the graph has several nodes representing the satellites and several edges representing the data exchange direction. It is assumed that every satellite exchanges its states with a few neighbors in the team, where the neighbors of every satellite are prespecified and do not change during the mission. This graph is utilized to construct the global dynamic model of the assembly, in which every node is represented by the dynamic model of a single satellite. The local control law, which is applied to every satellite in the assembly, consists of a state feedback gain and a coupling gain. The state feedback gain is designed using the discrete-time periodic Riccati equation to guarantee the local stability of every satellite separately, while the coupling gain is selected to satisfy the Lyapunov condition to guarantee the global stability of the assembly. The invariant-set method is utilized to prove the stability of the global dynamic system that is subjected to symmetric input saturation. The precision and optimality of the proposed control law are successfully demonstrated by numerical nonlinear simulations in a MATLAB environment for a cubic formation satellite assembly.

Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems

The development issues of theories of robustness, roughness and bifurcations of dynamic systems are considered. In the modern theory of dynamic systems and automatic control systems, researches of the properties of roughness and robustness of systems are becoming more and more important. The work considers methods of research and ensuring robust stability of interval dynamic systems of both algebraic and frequency directions of robust stability. The main results of the original algebraic method of robust stability for continuous and discrete time are given. In the frequency direction of robust stability, the issues of a frequency-robust method to the analysis and synthesis of robust multidimensional control systems based on the use of the frequency condition number of the transfer matrix of the “input-output” ratio are considered. The main provisions of the theory and method of topological roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin are presented with the introduction of a measure of roughness of systems in the form of a condition number of matrices of reduction to a diagonal (quasi-diagonal) basis at special points of phase space. Criteria for dynamic systems bifurcations are formulated. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as Lorenz and Rössler attractors, Belousov-Jabotinsky, Chua systems, “predator-prey” and “predator-prey-food”, Hopf bifurcation, Schumpeter and Caldor economic systems, Henon mapping, and others. For research of weakly formalized and non-formalized systems, the use of the approach of analogies of theoretical-multiple topology and the abstract method to such systems is proposed. Further research suggests the development of roughness and bifurcation theories for complex nonlinear dynamical systems.

Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: One dimensional cases

The stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation parameter in the discrete dynamical systems, and the emergence condition of an additional bifurcation, flip bifurcation, is identified. The correspondence between the discrete dynamical systems with positivity and the ultradiscrete ones derived from them is discussed. It is found that the derived ultradiscrete max-plus dynamical systems can retain the bifurcations of the original continuous ones via tropical discretization and ultradiscretization.

Numerical simulation of stability and responses of dynamic systems under parametric excitation

The stability and responses of dynamic systems under parametric excitation are often encountered in many fields of science and engineering, such as slender columns and thin plates under axial loadings. This paper proposes a novel numerical simulation method to simultaneously construct stability diagrams and predict the responses of multiple degree-of-freedom (DOF) dynamic systems under arbitrary parametric loadings. The method divides an arbitrary load into discrete segments to approximate the variable excitation function by a step function and then accumulates the system responses of each segment using a matrix method. Dynamic stability and response analysis of undamped and damped 1DOF, 2DOF, 3DOF, and multiple DOF systems are conducted, and numerical examples are compared to conventional methods to show the accuracy and advantage of the proposed numerical simulation method. The instability diagrams are also substantiated by vibration response curves obtained from the same method. The method is applied to calibrate the validity and ranges of applicability of the Hill infinite determinants in the Floquet theory of 1DOF Mathieu-Hill equations. The second order approximation of the first instability region from the Hill determinant is acceptable. However, the fourth order approximation of the second instability region must be used to obtain a relatively accurate result. For 2DOF, 3DOF, and multi-parameter linear periodic systems, the method can simultaneously render four types of instability regions in a single procedure: no resonances, simple parametric resonances, combination additive resonances, and combination differential resonances. These results could provide a better understanding of stability and responses of dynamic systems under parametric excitation.

Hessian regularization of deep neural networks: A novel approach based on stochastic estimators of Hessian trace

In this paper, we develop a novel regularization method for deep neural networks by penalizing the trace of Hessian. This regularizer is motivated by a recent guarantee bound of the generalization error. We explain its benefits in finding flat minima and avoiding Lyapunov stability in dynamical systems. We adopt the Hutchinson method as a classical unbiased estimator for the trace of a matrix and further accelerate its calculation using a Dropout scheme. Experiments demonstrate that our method outperforms existing regularizers and data augmentation methods, such as Jacobian, Confidence Penalty, Label Smoothing, Cutout, and Mixup. The code is available at https://github.com/Dean-lyc/Hessian-Regularization.

Modelling and stability analysis of a high-speed train braking system

The stability of a braking system is crucial for the steady and safe operation of trains. To investigate the dynamic behaviors and stability of a train braking system as influenced by key parameters, a numerical model integrating mode-coupling, negative friction–velocity slope (NFVS), and stick–slip theories was established. Additionally, the wheelset connected with the braking system was contained in the model considering that mode-coupling and stick–slip characteristics were affected by system structures. Based on this model, the influence and mechanism of stiffness, damping, and NFVS characteristics on mode-coupling of the braking system were revealed. The results showed that when the tangential and normal contact stiffness values of the pad were close, the system experienced mode-coupling instability and vibrations in these two directions were coupled. When the tangential and normal damping values of the pad were unequal, the energy dissipation in the two directions became different and the system instability was enhanced. Additionally, the system complex eigenvalues always had positive real parts and the dynamic behaviors were more complicated by considering NFVS at relatively low speeds. This is because NFVS characteristics can be equivalent to negative damping values, which cause the difference in tangential and normal damping values and continuously input energy into the braking system, then weakening the system stability.

On robustness of neural ODEs image classifiers

Neural Ordinary Differential Equations (Neural ODEs), as a family of novel deep models, delicately link conventional neural networks and dynamical systems, which bridges the gap between theory and practice. However, they have not made substantial progress on activation functions, and ReLU is always utilized by default. Moreover, the dynamical behavior existing in them becomes more unclear and complicated as training progresses. Fortunately, existing studies have shown that activation functions are essential for Neural ODEs in governing intrinsic dynamics. Motivated by a family of weight functions used to enhance the stability of dynamical systems, we introduce a new activation function named half-Swish to match Neural ODEs. Besides, we explore the effect of evolution time and batch size on Neural ODEs, respectively. Experiments show that our model consistently outperforms Neural ODEs with basic activation functions on robustness both against stochastic noise images and adversarial examples across Fashion-MNIST, CIFAR-10, and CIFAR-100 datasets, which strongly validates the applicability of half-Swish and suggests that half-Swish function plays a positive role in regularizing the dynamic behavior to enhance stability. Meanwhile, our work theoretically provides a prospective framework to choose appropriate activation functions to match neural differential equations.

Comparative analysis of machine learning methods for active flow control

Machine learning frameworks such as genetic programming and reinforcement learning (RL) are gaining popularity in flow control. This work presents a comparative analysis of the two, benchmarking some of their most representative algorithms against global optimization techniques such as Bayesian optimization and Lipschitz global optimization. First, we review the general framework of the model-free control problem, bringing together all methods as black-box optimization problems. Then, we test the control algorithms on three test cases. These are (1) the stabilization of a nonlinear dynamical system featuring frequency cross-talk, (2) the wave cancellation from a Burgers’ flow and (3) the drag reduction in a cylinder wake flow. We present a comprehensive comparison to illustrate their differences in exploration versus exploitation and their balance between ‘model capacity’ in the control law definition versus ‘required complexity’. Indeed, we discovered that previous RL control attempts of controlling the cylinder wake were performing linear control and that the wide observation space was limiting their performances. We believe that such a comparison paves the way towards the hybridization of the various methods, and we offer some perspective on their future development in the literature of flow control problems.

Mittag-Leffler stability of random-order fractional nonlinear uncertain dynamic systems with impulsive effects

Mittag-Leffler stability of random-order fractional nonlinear uncertain dynamic systems with impulsive effects

Damping of Frequency and Power System Oscillations with DFIG Wind Turbine and DE Optimization

Wind power is one of the most promising renewable energy resources and could become a solution to contribute to the present energy and global warming crisis of the world. The commonly used doubly fed induction generator (DFIG) wind turbines have a general trend of increasing oscillation damping. Unless properly controlled, the high penetration of wind energy will increase the oscillation and affect the control and dynamic interaction of the interconnected generators. This paper discusses power oscillation damping control in the automatic generation control (AGC) of two-area power systems with DFIG wind turbines and Matlab code/Simulink interfacing optimization methods. The differential evolution (DE) optimization technique is used to obtain the controller gain parameters. In the optimization process, a step load perturbation (SLP) of 1% has been considered in Area 1 only, and the integral of time weighted absolute error (ITAE) cost function is used. Three different test studies have been examined on the same power system model with non-reheat turbine thermal power plants. In the first case, the power system model is simulated without a controller. In Case Study 2, the system is simulated with the presence of DFIG and without a controller. In Case Study 3, the system is simulated with a PID controller and DFIG. Most of the studies available in the literature do not optimize the appropriate wind penetrating speed gain parameters for the system and do not consider the ITAE as an objective function to reduce area control error. In this regard, the main contribution and result of this paper is—with the proposed PID+DFIG optimized DE—the ITAE objective function error value in the case study without a controller being 6.7865, which is reduced to 1.6008 in the case study with PID+DFIG-optimized DE. In addition, with the proposed controller methods, the dynamic system time responses such as rise time, settling time, overshoot, and undershoot are improved for system tie-line power, change in frequency, and system area controller error. Similarly, with the proposed controller, fast system convergence and fast system oscillation damping are achieved. Generally, it is inferred that the incorporation of DFIG wind turbines in both areas has appreciably improved the dynamic performance and system stability under consideration.

Delayed reset control design for uncertain fractional-order systems with actuator faults via dynamic output feedback scheme

In this work, the stabilization of uncertain fractional-order dynamical control system with actuator faults and external disturbances will be addressed. Specifically, a fractional-order reset control system consisting of two sub-systems namely flow and jump subsystems will be designed. Based on the triggering conditions of reset law, a particular subsystem of the control system will be invoked. Meanwhile, the uncertainty and time-delay will be incorporated in the reset control system. Then the output of the reset control system, that is the control input of the system is designed to be fault-tolerant due to the presence of actuator faults. Further, the fractional-order dependent linear matrix inequalities (LMIs) will be derived by an appropriate choice of a Lyapunov function and other matrix inequalities. Thus, by solving the obtained LMIs, the gain values can be obtained guaranteeing the stabilization of the closed loop system with prescribed disturbance attenuation level. Later, the theoretical results are authenticated by providing numerical simulation results.

Improving dynamic stability of wind integrated system by incorporating magnetic energy storage

AbstractThe need to incorporate renewable energy sources in the present grid due to energy demand and environmental issues has led the power industry to face new challenges and unexpected transformations. This new epitome to build a flexible power system with coordination of increasing intermittent renewable energy resources and delivering power efficiently with reliability demand new technologies. This work investigates the coupling of static synchronous compensator (STATCOM) with the most exclusive superconducting magnetic energy storage (SMES) device designed for the improved functioning of a grid network that is incorporated with a wind farm. The widespread integration of intermittent wind generators in the grid is disturbing its stability and reliability. The SMES is interlinked with the grid system via a power electronic interface and chopper for the energy exchange. This work suggests STATCOM as a power electronic interface and a three‐level chopper configuration with novel control scheme for enhancing the functioning of the test system. This integrated controller (STAT‐SMES) has been analyzed for grid‐connected doubly fed induction generator wind farm under different fault conditions and time durations. The results have been compared without any controller, with STATCOM only, and with the proposed STAT‐SMES controller using MATLAB. The simulation outcomes prove that coupled control scheme is effective and better in handling wind farm integration issues.

Intelligent control of convergence rate of impulsive dynamic systems affected by nonlinear disturbances under stabilizing impulses and its application in Chua’s circuit

This paper investigates the variable convergence rate stability and variable convergence rate stabilization of impulsive dynamic linear systems with stabilizing impulses affected by nonlinear disturbances. The eigenvalues (poles) of the system state are closely related to the convergence or divergence rate of the system. Sufficient conditions for the variable convergence rate stability are obtained by using the generalized pole placement idea and the method of system transformation. By designing a memoryless state feedback controller, sufficient conditions of variable convergence rate stabilization are obtained, ensuring the asymptotic stability of the target closed-loop system and accurately adjusting the system state’s convergence rate. An algorithm for adjusting the speed of system state convergence and its flow chart are designed by referring to the C programming language. Combined with the variable convergence rate stabilization method, the intelligent control of system state convergence speed is realized at the operation level. As a representative example of chaotic systems, Chua’s circuit affected by impulses verifies the effectiveness of the variable convergence rate stabilization method.

Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (Or: How to Prove Kabashima’s Replica Formula)

There has been a recent surge of interest in the study of asymptotic reconstruction performance in various cases of generalized linear estimation problems in the teacher-student setting, especially for the case of i.i.d standard normal matrices. Here, we go beyond these matrices, and prove an analytical formula for the reconstruction performance of convex generalized linear models with rotationally-invariant data matrices with arbitrary bounded spectrum, rigorously confirming, under suitable assumptions, a conjecture originally derived using the replica method from statistical physics. The proof is achieved by leveraging on message passing algorithms and the statistical properties of their iterates, allowing to characterize the asymptotic empirical distribution of the estimator. For sufficiently strongly convex problems, we show that the two-layer vector approximate message passing algorithm (2-MLVAMP) converges, where the convergence analysis is done by checking the stability of an equivalent dynamical system, which gives the result for such problems. We then show that, under a concentration assumption, an analytical continuation may be carried out to extend the result to convex (non-strongly) problems. We illustrate our claim with numerical examples on mainstream learning methods such as sparse logistic regression and linear support vector classifiers, showing excellent agreement between moderate size simulation and the asymptotic prediction.