Theoretical Stability Analysis Research Articles

Adaptive critic design for safety-optimal FTC of unknown nonlinear systems with asymmetric constrained-input

Safe fault tolerant control is one of the key technologies to improve the reliability of dynamic complex nonlinear systems with limited inputs, which is hard to solve and definitely a great challenge to tackle. Thus the paper presents a novel safety-optimal FTC (Fault Tolerant Control) approach for a category of completely unknown nonlinear systems incorporating actuator fault and asymmetric constrained-input, which can guarantee the system’s operation within a safe range while showcasing optimal performance. Firstly, a CBF (Control Barrier Function) is incorporated into the cost function to penalize unsafe behaviors, and then we translate the intractable safety-optimal FTC problem into a differential ZSG (Zero-Sum Game) problem by defining the control input and the actuator fault as two opposing sides. Secondly, a neural-network-based identifier is employed to reconstruct system dynamics using system data, and the resolution of handling asymmetric constrained-input with the introduced non-quadratic cost function is achieved through the design of an adaptive critic scheme, aiming to reduce computational expenses accordingly. Finally, through the theoretical stability analysis, it is demonstrated that all signals in the closed-loop system are consistently UUB (Uniformly Ultimately Bounded). Furthermore, the proposed method’s effectiveness is also verified in the simulation experiments conducted on a model of a single-link robotic arm system with actuator failure. The result shows that the algorithm can fulfill the safety-optimal demand of fault tolerant control in fault system with asymmetric constrained-input.

Comparative theoretical analysis of thermal stability of vinylheptafulvenes with fused cycles

A systematic TD-DFT/DFT study of dihydroazulenes (DHA) and vinylheptafulvenes (VHF) fused by cyclopentane (CP) and cyclohexane (CH) rings has been performed. The main result of the work is a theoretical explanation for the strikingly high stability of CP-VHF compared to CH-VHF. We showed that the reason for this is the greater bending rigidity of the CP ring than the CH ring, which prevents the recovery of the C1C8a bond. The potential barrier separating CP-DHA and CP-VHF turned out to be one and a half times higher than that between CH-DHA and CH-VHF. The corresponding lifetimes of VHF structures calculated using transition state theory are in good agreement with experimental data.

Robust 3-D Path Following Control Framework for Magnetic Helical Millirobots Subject to Fluid Flow and Input Saturation.

Precise trajectory control is imperative to ensure the safety and efficacy of in vivo therapy employing the magnetic helical millirobots. However, achieving accurate 3-D path following of helical millirobots under fluid flow conditions remains challenging due to the presence of the lumped disturbances, encompassing complex fluid dynamics and input frequency saturation. This study proposes a robust 3-D path following control framework that combines a disturbance observer for perturbation estimation with an adaptive finite-time sliding mode controller for autonomous navigation along the reference trajectories. First, a magnetic helical millirobot's kinematic model based on the 3-D hand position approach is established. Subsequently, a robust smooth differentiator is implemented as an observer to estimate disturbances within a finite time. We then investigate an adaptive finite-time sliding mode controller incorporating an auxiliary system to mitigate the estimated disturbance and achieve precise 3-D path tracking while respecting the input constraints. The adaptive mechanism of this controller ensures fast convergence of the system while alleviating the chattering effects. Finally, we provide a rigorous theoretical analysis of the finite-time stability of the closed-loop system based on the Lyapunov functions. Utilizing a robotically-actuated magnetic manipulation system, experimental results demonstrate the efficacy of the proposed approach in terms of the control accuracy and convergence time.

Effect of a uniform magnetic field on the nonlinear instability of double cylindrical interfaces concerning three magnetic liquids

The present manuscript relates to the movement of three immersed viscous magnetic liquids in porous media that are positioned between three concentric cylindrical interfaces. Every cylindrical layer has an axial extension that extends in both vertical directions to infinity. Moreover, the pressure gradients are the driving force behind the uniform motion of all fluids in a similar upward motion at varying velocities. Permanent magnetic fields are tangentially oriented exert stress on the system. The computations are rendered simpler by applying the viscous potential theory. The Maxwell equations are utilized for the magnetic field, while the Brinkman-Darcy equations define the fluid mobility. Moreover, the rise of the disturbance and the subsequent extraction of fuel from the tiny cracks of the reservoir stone can be managed through the implementation of engineering techniques such as electromagnetic field impacts and oil extraction engineering. Typically, the nonlinear strategy involves incorporating the applicable nonlinear boundary conditions and analyzing the linearized equations of motion. This research offers a framework for verifying theoretical models and simulations based on experimental observations. The inclusion of a homogeneous magnetic field introduces complexity to the system, rendering it a suitable candidate for validating and improving models in the field Magneto hydrodynamics. The originality of the problem lies in the dual nonlinear stability of cylindrical interfaces when subjected to uniform magnetic field. Accordingly, two nonlinear characteristic differential equations controlling the surface displacements are produced. The nonlinear stability prerequisite is met by applying the matrix concept and the multiple scale technique in conjunction with a theoretical analysis of stability. Additionally, the Routh-Hrutwitz criterion is encompassed to judge the stability cofiguration. A detailed examination of the associated nonlinear stability requirements is showed. Meanwhile, the estimated limited solutions of perturbed surfaces are achieved. For the cylindrical middle layer, it is found that the outer cylindrical interface has a more stabilizing effect than the inner one. The approximate solutions for displacements at the interface are calculated. The influence of Weber numeral of the problem on the stability profile is investigated.

Meta-heuristic optimization-based robust [formula omitted] controller design for active suspension systems subject to actuator saturation

This study focuses on designing robust H∞ controllers to mitigate actuator saturation-related issues of active suspension systems. Two design methods are presented: one utilizing full-state feedback and the other relying on measurable output feedback. The H∞ state-feedback control incorporates actuator saturation directly into the design, representing the control input with the saturation function in the state-space model. The resulting optimization-based design problem, derived from theoretical stability analyses, poses a non-convex NP-hard problem under bilinear matrix inequality (BMI) constraints. To address this challenge, the study proposes a meta-heuristic optimization-based technique, providing a straightforward yet effective method to manage the BMI problem. The scope extends to practical implementation considerations, where suspension systems may only utilize measurable output variables. The H∞ static output-feedback control case is explored by extending the design of a full-order state-feedback controller. Despite the BMI-related challenges entailed in this scenario, the proposed meta-heuristic optimization-based design technique demonstrates flexibility in managing BMI conditions, thus eliminating the need to modify the controller design methodology. A noteworthy contribution of this study is its consistent application of the same meta-heuristic optimization-based technique for designing both H∞ state-feedback and static output-feedback controllers. Experimental results validate the effectiveness of the proposed robust H∞ control strategies.

Visually guided swarm motion coordination via insect-inspired small target motion reactions

Despite progress developing experimentally-consistent models of insect in-flight sensing and feedback for individual agents, a lack of systematic understanding of the multi-agent and group performance of the resulting bio-inspired sensing and feedback approaches remains a barrier to robotic swarm implementations. This study introduces the small-target motion reactive (STMR) swarming approach by designing a concise engineering model of the small target motion detector (STMD) neurons found in insect lobula complexes. The STMD neuron model identifies the bearing angle at which peak optic flow magnitude occurs, and this angle is used to design an output feedback switched control system. A theoretical stability analysis provides bi-agent stability and state boundedness in group contexts. The approach is simulated and implemented on ground vehicles for validation and behavioral studies. The results indicate despite having the lowest connectivity of contemporary approaches (each agent instantaneously regards only a single neighbor), STMR achieves collective group motion. STMR group level metric analysis also highlights continuously varying polarization and decreasing heading variance.

Interpolated coefficients stabilizer-free weak Galerkin method for semilinear parabolic convection–diffusion problem

We continue our effort in Li et al. (2024) to explore an interpolated coefficients stabilizer-free weak Galerkin finite element method (IC SFWG-FEM) to solve a one-dimensional semilinear parabolic convection–diffusion equation. Due to the introduction of interpolated coefficients and the design without stabilizers, this method not only possesses the capability of approximating functions and sparsity in the stiffness matrix, but also reduces the complexity of analysis and programming. Theoretical analysis of stability for the semi-discrete IC SFWG finite element scheme is provided. Moreover, numerical experiments are carried out to demonstrate the effectivity and stability.

The Impact of Velocity Update Frequency on Time Accuracy for Mantle Convection Particle Methods

AbstractComputing the velocity field is an expensive process for mantle convection codes. This has implications for particle methods used to model the advection of quantities such as temperature or composition. A common choice for the numerical treatment of particle trajectories is classical fourth‐order explicit Runge–Kutta (ERK4) integration, which involves a velocity computation at each of its four stages. To reduce the cost per time step, it is possible to evaluate the velocity for a subset of the four time integration stages. We explore two such alternative schemes, in which velocities are only computed for: (a) stage 1 on odd‐numbered time steps and stages 2–4 for even‐numbered time steps, and (b) stage 1 for all time steps. A theoretical analysis of stability and accuracy is presented for all schemes. It was found that the alternative schemes are first‐order accurate with stability regions different from that of ERK4. The efficiency and accuracy of the alternate schemes were compared against ERK4 in four test problems covering isothermal, thermal, and thermochemical flows. Exact solutions were used as reference solutions when available. In agreement with theory, the alternate schemes were observed to be first‐order accurate for all test problems. Accordingly, they may be used to efficiently compute solutions to within modest error tolerances. For small error tolerances, however, ERK4 was the most efficient.

Swirl flow stability: thermodynamic analysis and experiments

The current paper presents a theoretical analysis of swirl flow stability, both inside a tube (vortex tube) and in a free annular swirl flow. The starting concept is the study of the evolution of velocity and temperature fluctuations. Methods of non-equilibrium thermodynamics are used to describe the magnitude of fluctuations and their properties. The important role of the total enthalpy follows from a variational analysis. Moreover, the thermodynamic criterion of the stability is formulated using the total enthalpy, and compared with experiments, numerical results and classical Rayleigh theory support its applicability. It was shown that the solid body vortex is at the margin of stability, which is experimentally observed. Analogously, the potential vortex is by the thermodynamic criterion stable; however, by the Rayleigh criteria it is on the onset of stability. The classical Taylor experiment of flow between two rotating cylinders is analysed from the point of view of this criterion. These results are underlined by swirl tube experiments at the Institute of Aerospace Thermodynamics at Stuttgart University and the annular nozzle experiments performed in the Institute of Thermomechanics CAS in Prague. Both independent experiments confirm the transformation of the initial annular vortex into a stable potential-type vortex. The results of this theory can also be used to explain the exceptional stability of tropical cyclones.

Viscous correction to the potential flow analysis of Rayleigh–Taylor instability in a Rivlin–Ericksen viscoelastic fluid layer with heat and mass transfer

AbstractThe current investigation focuses on examining viscous corrections for viscous potential flow (VCVPF) analysis concerning the Rayleigh–Taylor instability occurring at the interface of a Rivlin–Ericksen (R–E) viscoelastic fluid and a viscous fluid during the transfer of heat and mass between phases. The R–E model is a fundamental framework in the study of viscoelastic fluids, providing insights into their complex rheological behavior. It characterizes the material's response to both deformation and flow, offering valuable predictions for various industrial and biological applications. Within the framework of viscous potential flow (VPF) theory, viscosity is exclusively accounted for in the normal stress balance equation, disregarding the influence of shearing stress entirely. This study introduces a viscous pressure term into the normal stress balance equation alongside the irrotational pressure, presuming that this addition will improve the discontinuity of tangential stresses at the fluid interface. Through derivation of a dispersion relationship and subsequent theoretical and numerical stability analyses, the stability of the interface is investigated across various physical parameters. Multiple plots are generated using the dispersion relation, and a comparative analysis between VPF and VCVPF is conducted to establish improved stability criteria. The investigation highlights that the combined impact of heat/mass transport and shearing stress serves to delay the instability of the interface.

Minimum-Drag Fault-Tolerant Aircraft Control Allocation via Online Lifting Line Calculation

The minimization of drag at any given flight condition is necessary for the reduction of aircraft fuel consumption and is strongly linked to the way the different aerodynamic surfaces are deflected to control the flight trajectory. Current optimal control allocation methods calculate commands that minimize norm-based metrics that are only loosely related to aircraft drag. In this paper, using a novel real-time application of the lifting line concept, a new control allocation method for overactuated “biomorphic” fixed-wing aircraft is introduced, aiming at addressing the above limitation. The proposed technique outputs optimal, fault-tolerant minimum-drag control allocation solutions for vehicles with large numbers of aerodynamic surfaces, combined with angle-of-attack and angle-of-sideslip estimator functions that allow for direct, localized control of the lift force vectors. Owing to its close link to lifting line theory, which constitutes an integral part of the proposed allocation calculation, the method represents a low-computational-cost solution to the control allocation problem, easily adaptable to different aircraft configurations. Alongside its theoretical development and stability analysis, a series of simulated experiments are presented that demonstrate the proposed method’s characteristics and potential applications.

Bernstein polynomials method for solving multi-order fractional neutral pantograph equations with error and stability analysis

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series solution methods, outperforms others, exhibiting superior performance with higher accuracy orders compared to those obtained from Chebyshev spectral methods, Bernoulli wavelet method, and Spectral Tau method.

Analysis and application of MLPG7 for diffusion equations with nonlinear reaction terms

This paper extends the recently proposed variant of meshless local Petrov Galerkin (MLPG) method, i.e., MLPG7, for solving time dependent PDEs. As test function, the method uses a novel modification of fundamental solution of Laplace operator that not only the test function itself but also its derivative vanish on boundary of local subdomains. Therefore, more stable local integral equations are obtained by replacing a boundary integral of derivatives of field variables with a domain integral of field variables itself. MLPG7 formulation converts the nonlinear governing equation into a system of first order ordinary differential equations (ODEs). For this system of ODEs, a theoretical stability analysis is carried out for the case of regularly spaced nodal points. Stability and convergence conditions for fully discretized equations by Euler (forward and backward) and Crank–Nicolson methods are investigated. The nonlinear term is treated iteratively. A numerical study investigates the convergence and stability of the method. A comparison is also done with some well-known methods and the results reveal the excellence of MLPG7. The convergence of the iterative approach is numerically verified by another illustrative test example. As nonlinear test problems, Allen–Cahn equations are studied.

Subdivision collocation method for numerical treatment of regularized long wave (RLW) equation

This research project introduces a novel computational approach for solving the regularized long wave equation. The proposed method utilizes a subdivision scheme with appropriate basis functions to transform the equation into a system of linear algebraic equations. A suitable numerical technique is employed to compute the solution of the transformed equations. Theoretical analysis of stability and error for the proposed method is also conducted. Furthermore, the invariants of three physical properties, waves, mass (M), momentum (P), and energy (ɛ), are calculated. Additionally, numerical evidence is presented to demonstrate the effectiveness and accuracy of the method. The results of the numerical experiments confirm the efficiency and high accuracy of the proposed method. Moreover, the numerical results of the invariants validate the conservation laws and align with the theoretical results.

Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor’s series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

Efficient Prediction of Stability Boundaries in Milling Considering the Variation of Tool Features and Workpiece Materials.

Theoretical stability analysis is a significant approach to predicting chatter-free machining parameters. Accurate milling stability predictions highly depend on the dynamic properties of the process system. Therefore, variations in tool and workpiece attributes will require repeated and time-consuming experiments or simulations to update the tool tip dynamics and cutting force coefficients. Considering this problem, this paper proposes a transfer learning framework to efficiently predict the milling stabilities for different tool-workpiece assemblies through reducing the experiments or simulations. First, a source tool is selected to obtain the tool tip frequency response functions (FRFs) under different overhang lengths through impact tests and milling experiments on different workpiece materials conducted to identify the related cutting force coefficients. Then, theoretical milling stability analyses are developed to obtain sufficient source data to pre-train a multi-layer perceptron (MLP) for predicting the limiting axial cutting depth (aplim). For a new tool, the number of overhang lengths and workpiece materials are reduced to design and perform fewer experiments. Then, insufficient stability limits are predicted and further utilized to fine-tune the pre-trained MLP. Finally, a new regression model to predict the aplim values is obtained for target tool-workpiece assemblies. A detailed case study is developed on different tool-workpiece assemblies, and the experimental results validate that the proposed approach requires fewer training samples for obtaining an acceptable prediction accuracy compared with other previously proposed methods.

Theoretical unloading fracture mechanism and stability analysis of the slope rock masses in open-pit mines

Instability of the rock slopes in open pit mines during the excavation unloading is an increasingly serious problem in the field of slope engineering. To explain the mechanical mechanism of slope unloading damage from a theoretical point of view, a simplified mechanical model of slope excavation based on the theory of fracture mechanics and the rock strength damage criterion is established. The damage process of the slope under excavation disturbance is dynamically analyzed by combining the interstructural characteristics of the slope. The solution equations for the extent of the plastic zone at the end of the crack of the excavated slope and its propagation length are derived. Calculation method of unloaded slope stability coefficient is proposed based on the mechanical model of crack propagation. The results show that (1) the stress intensity factor (SIF) at crack end in the slope under the action of unloading was larger than that under the original condition. (2) The range of the plastic zone at crack end in the slope rock mass can be attributed to the slope height, inverse logarithmic function to the slope angle, positive proportional function to the crack length, and the periodic fluctuation function of the crack angle. (3) The slope safety factor (SF) was found to be negatively related with the slope angle, slope height, crack angle, and the crack length unloading factor and positively related with the friction factor. Finally, the reasonableness of the theoretical derivation is verified by an engineering case study.

An Efficient Meshfree Numerical Technique to Solve Fractional Schamel–KdV Equation for Ion-Acoustic Solitary Waves in Dusty Plasma

The Schamel–KdV equation plays a vital role in studying the effect of electron trapping on the nonlinear interaction of ion-acoustic waves in plasma and dusty plasma. This work presents a numerical scheme-based radial basis function-finite difference (RBF-FD) method to solve the time-fractional Schamel–KdV equation. The advantages of using this method are their meshfree nature and flexibility in dealing with complex geometries. In the discretization process, a finite difference (FD) technique is used to discretize the temporal derivative, while the multiquadric (MQ) RBF is used to approximate the spatial derivatives. The theoretical convergence and stability analysis of the time-discrete scheme are also established. In addition, numerical experiments are executed through some illustrated problems, and the obtained results are compared with that acquired by the tanh method and Kudryashov method solutions to show the high accuracy and plausibility of the proposed technique. Also, the graphical representations are given to demonstrate the physical interpretation of the resulting wave structures.

Theoretical and Experimental Analysis of Grinding Stability of Beam Workpiece

Grinding chatter is a kind of self-excited vibration in which the grinding system continuously absorbs energy from the grinding machine, increasing the mechanical energy of the system. Grinding chatter can damage the surface of the workpiece and accelerate the abrasion of the grinding wheel. The theoretical analysis of the grinding chatter for the beam surface was launched based on the behavior of a single abrasive grain, whose cutting thickness is a key factor affecting grinding stability. The dynamic grinding force model has been developed, which is the interaction interface between the grinding wheel and the workpiece. In this paper, rail beam grinding was taken as an example. The vibration performance of the rail beam was described with the Timoshenko beam. The characteristics of the frequency domain of the grinding wheel-workpiece system were observed, and the condition of the stability at any position in the longitudinal direction of the beam was gained, which could be quantitatively characterized with the stability limit curve. The grinding experiments of the rail beam surface demonstrated that as chatter developed, the chatter marks could be investigated on the surface of the rail, and the energy of the chatter signal was mainly concentrated around the chatter frequency, which was higher than the natural frequency of the grinding wheel.

On the stability of piston-driven planar shocks

We present a theoretical and numerical stability analysis for a piston-driven planar shock against two-dimensional perturbations. The results agree with the well-established theory for isolated planar shocks: in the range of $h_c&lt; h&lt;1+2 {\mathcal {M}}_2$ , where $h$ is the D'yakov–Kontorovich (DK) parameter related to the slope of the Rankine–Hugoniot curve, $h_c$ is its critical value corresponding to the onset of the spontaneous acoustic emission (SAE) and ${\mathcal {M}}_2$ is the downstream Mach number, non-decaying oscillations of shock-front ripples occur. The effect of the piston is manifested in the presence of additional frequencies occurring by the reflection of the sonic waves on the piston surface that can reach the shock. An unstable behaviour of the shock perturbation is found to be possible when there is an external excitation source affecting the shock, whose frequency coincides with the self-induced oscillation frequency in the SAE regime, thereby being limited to the range $h_c&lt; h&lt;1+2 {\mathcal {M}}_2$ . An unstable evolution of the shock is also observed for planar shocks restricted to one-dimensional perturbations within the range $1&lt; h&lt;1+2 {\mathcal {M}}_2$ . Both numerical integration of the Euler equations via the method of characteristics and theoretical analysis via Laplace transform are employed to cross-validate the results.