Proof Of Stability Research Articles

Unconditionally strong energy stable scheme for Cahn–Hilliard equation with second‐order temporal accuracy

We propose an unconditionally gradient stable scheme for solving the Cahn–Hilliard equation with second‐order accuracy in time using the effective time‐step analysis. The conventional convex splitting scheme is one of the most well‐known methods for solving gradient flows, and it guarantees both energy stability and first‐order accuracy in time. Recently, there are some researches, which are extended to the second‐order accuracy; however, most of results provide the proof of energy stability for a modified (pseudo) energy or in a weak sense. In this paper, we prove the energy stability of the proposed method with respect to the Ginzburg–Landau free energy functional. Moreover, the unique solvability, mass conservation, and accuracy of the proposed scheme are also proven based on the convex splitting approach. The numerical experiments are presented to show convergence rate, mass conservation, energy stability, and phase separation are in good agreement with the theory.

A Control Method of Mobile Manipulator Based on Null-Space Task Planning and Hybrid Control

The mobile manipulator is a floating base structure with wide space operability. An integrated mechanical device for mobile operation is formed through the organic combination of the mobile platform and multi-axis manipulator. This paper presents a general kinematic modeling method for mobile manipulators and gives the relevant derivation of the dynamic model. Secondly, the null-space composition of the mobile manipulator is analyzed, the task space is divided, and a variety of task-switching criteria are designed. Finally, a hybrid control model combining dynamic feedback and synovial control based on dynamic parameter identification is designed, and stability proof is given. The theoretical method is also verified by the experimental platform. The proposed method can effectively improve the control accuracy of the mobile manipulator, and the hybrid control method can effectively control the output torque to reach the ideal state.

Event-triggered fault-tolerant secure containment control of multi-agent systems through impulsive scheme

This paper studies a containment control problem of multi-agent systems. A two-layer distributed control framework is proposed to handle the external network attack and internal actuator failure. In the network layer, an event-triggered impulsive control protocol is designed to handle the deception attack, which can deal with stochastic attacks injected into the controller under limited energy bound. Compared with the traditional event-triggered-based continuous control method, the impulsive control mechanism can further reduce energy consumption with better robustness. In the physical layer, a fault-tolerant controller, which considers multiplicative failure, is investigated to maintain the system’s stability. Rigorous proofs for controller stability are given through Lyapunov theory, and Zeno behavior is also avoided within the designed control mechanism. Finally, a numerical simulation is demonstrated to test and verify the proposed control scheme.

UAV Formation Control under Fixed and Variable Adjacency Based Directed Network Topologies

The UAV formation control is one of the key aspects in several applications like surveillance, moving target tracking, load-transportation, and delivery systems etc. These situations demand the multiple UAVs to manoeuvre in a desired formation. To address this problem, a distributed formation control scheme is proposed incorporating the details about the state of the neighbouring UAVs. The communication network topology among the UAVs is considered to be directed with the constant and the weighted adjacency matrices. The nonholonomic constraints are considered while deriving the desired Euler angles. Satisfying the conditions of Lyapunov provides necessary proof of stability along the positional and the attitude subsystems. Simulation results demonstrate that the desired tetrahedron, octahedron, and cube shapes are attained and maintained by the UAVs successfully. Also, the designed formation paradigm works proficiently for both the constant and the weighted adjacency matrices based directed network topologies. The performance validation is done through extensive comparative analysis for varying network connections.

Robust Inverse Q-Learning for Continuous-Time Linear Systems in Adversarial Environments.

This article proposes robust inverse Q -learning algorithms for a learner to mimic an expert's states and control inputs in the imitation learning problem. These two agents have different adversarial disturbances. To do the imitation, the learner must reconstruct the unknown expert cost function. The learner only observes the expert's control inputs and uses inverse Q -learning algorithms to reconstruct the unknown expert cost function. The inverse Q -learning algorithms are robust in that they are independent of the system model and allow for the different cost function parameters and disturbances between two agents. We first propose an offline inverse Q -learning algorithm which consists of two iterative learning loops: 1) an inner Q -learning iteration loop and 2) an outer iteration loop based on inverse optimal control. Then, based on this offline algorithm, we further develop an online inverse Q -learning algorithm such that the learner mimics the expert behaviors online with the real-time observation of the expert control inputs. This online computational method has four functional approximators: a critic approximator, two actor approximators, and a state-reward neural network (NN). It simultaneously approximates the parameters of Q -function and the learner state reward online. Convergence and stability proofs are rigorously studied to guarantee the algorithm performance.

Fixed-time robust trajectory tracking control for quadrotor UAV with disturbances

Trajectory tracking control of nonlinear system with disturbances has been extensively investigated. A novel fixed-time robust trajectory tracking control scheme for quadrotor unmanned aerial vehicle (UAV) with disturbances is proposed in this paper. First, the quadrotor system is split into position and attitude loops. With the purpose of restraining coupling disturbances on position loop forced by imperfect attitude tracking, a novel fixed-time unknown input observer (UIO) is devised to accurately estimate the disturbances, with the unique feature that does not need the upper bound of disturbances or the derivatives. Second, taking unmeasurable velocity and unknown external disturbances into account for attitude loop, a fixed-time extended state observer (ESO) is proposed to precisely observe them within a fixed time. Third, considering control accuracy and convergence rate, robust fixed-time controllers are, respectively, designed for position and attitude loops, and the convergence time of overall system is independent of system initial states. Then, a comprehensive stability proof for the two-loop system is presented with a combination of bi-limit homogeneity and Lyapunov technique. Finally, simulations and comparisons are performed to demonstrate the effectiveness of the designed control scheme.

Robust nonlinear control of a power flow controller for meshed DC grids

The studied power flow controller is an electronic device capable of controlling the power flows at a node in a meshed low voltage DC grid. It is made of m buck-boost converters connected in parallel on their high voltage-side to a reservoir capacitor. This paper proposes a nonlinear control method achieving robust regulation of the power in each line and of the reservoir voltage inside the converter. This control law only requires measurements of local currents and voltages, and the knowledge of the maximum possible resistance of the grid, whereas prior control laws required the full knowledge of some nominal parameters. Proofs of stability are given for every possible uncertain parameters of the studied grid model whereas prior control laws were only valid at nominal parameters and their neighbourhood. The performance of this control method is asserted through a tenth-scale experimental set-up. Stochastic simulations are conducted to estimate the basin of attraction.

Neural-Network-Based Terminal Sliding Mode Control of Space Robot Actuated by Control Moment Gyros

This paper studies the trajectory tracking control of a space robot system (SRS) in the presence of the lumped uncertainties with no prior knowledge of their upper bound. Although some related control methods have been proposed, most of them have either not been applied to SRSs or lack rigorous stability proof. Therefore, it is still a challenge to achieve high accuracy and rigorous theoretical proof for tracking control of SRSs. This paper proposes a new integrated neural network- based control scheme for the trajectory tracking of a SRS actuated by control moment gyros (CMGs). A new adaptive non-singular terminal sliding mode (ANTSM) control method is developed based on an improved radial basis function neural network (RBFNN). In the control method, a new weight update law is proposed to learn the upper bound of the lumped uncertainties. With the advantages of RBFNN and ANTSM, the controller has high control accuracy, fast learning speed and finite-time convergence. Different from most on-ground robotic manipulator controllers, a kinematic controller with position and attitude control laws is also designed for the satellite platform to remain stable. The stability of the closed-loop system is proved by the Lyapunov method with a high mathematical standard. Comparative simulation results demonstrate the effectiveness of the proposed control scheme with preferable performance and robustness.

A new practical framework for the stability analysis of perturbed saddle-point problems and applications

In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babuška’s theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuška-Brezzi (LBB) condition, and the other standard assumptions in Brezzi’s theory, in a combined abstract norm. The construction suggests to form the latter from individualfittednorms that are composed from proper seminorms.This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biot’s equations.

Deep Deterministic Policy Gradient and Active Disturbance Rejection Controller based coordinated control for gearshift manipulator of driving robot

In order to improve the shift precision of the gearshift manipulator, a coordinated control method for the gearshift manipulator based on the Deep Deterministic Policy Gradient (DDPG) and Active Disturbance Rejection Controller (ADRC) is proposed. Firstly, the kinematic model and dynamics model of the gearshift manipulator are established. Secondly, a coordinated control strategy is proposed to solve the target rotation angle of the servo motor, so as to deal with the nonlinear trajectory problem caused by the mechanical decoupling strategy. Then, an ADRC-based controller and a DDPG-based parameter adjuster are proposed. A DDPG model is established to adaptively adjust the control parameters of ADRC through the target rotation angle, actual rotation angle and rotation angle error. Finally, the theoretical stability proof and experimental verification of the proposed method are conducted. The test results show that the mean absolute error of the proposed method is 34% and 18% lower than that of ADRC and fuzzy PID methods. It shows the effectiveness of the method.

Mode stability for massless scalars in five-dimensional black hole backgrounds

The mode stability of the Kerr black hole in four dimensions was demonstrated by Whiting in 1989, by separating the Teukolsky equation that describes gravitational perturbations and then transforming the radial and angular equations in such a way that the problem can be reformulated as a wave equation in an auxiliary spacetime in which the proof of stability is greatly simplified, owing to the absence of an ergoregion. As a preliminary step towards extending these ideas to higher-dimensional black holes, we study the mode stability of the massless scalar wave equation in the five-dimensional black hole solutions of Einstein gravity and supergravity. We show how the wave equation can again be mapped into one in an auxiliary spacetime in which there is no ergoregion, allowing us to give a proof of the mode stability of the solutions of the scalar wave equation.

Unknown system dynamics estimator based coordinated control of the ultra supercritical boiler‐turbine unit with coal pulverizing delay for large load accommodation

AbstractThe ultra‐supercritical boiler‐turbine unit is required to accommodate large‐range load following to maintain the stability of grid frequency. However, the input delay in the coal pulverizing process poses a challenge to the coordinated control of the boiler‐turbine unit, which is always neglected in existing works but dramatically affects the stability of steam pressure and temperature. In this paper, to handle the delay caused by coal pulverizing, a predictor‐based delay compensation structure is developed for producing a delay‐free control framework for the throttle steam pressure. Moreover, to overcome intrinsic nonlinearities in the dynamics of the boiler‐turbine unit, the unknown system dynamics estimator is employed in the controllers' design for the steam enthalpy in the separator and active power of the turbine. Thus, precise and fast regulation for the active power output can be achieved under wide load following task. The salient merit of the proposed control is that the overshot of the throttle steam pressure and the specific enthalpy in the separator can be significantly alleviated, which is necessary for turbine blade. A Lyapunov‐Krasovskii functional is introduced to deal with the time delay in the stability proof. Finally, the proposed coordinated control is validated in three typical working scenarios.

Nonlinear structural stability and linear dynamic instability of transonic steady-states to hydrodynamic model for semiconductors

For unipolar hydrodynamic model of semiconductor device represented by Euler-Poisson equations, when the doping profile is supersonic, and the boundary data are in subsonic region and supersonic region separately, the system possesses the shock transonic steady-states and the smooth transonic steady-states. In this paper we study the nonlinear structural stability and the linear dynamic instability of these steady transonic solutions. For any relaxation time: 0<τ≤+∞, by means of elaborate singularity analysis, we first investigate the structural stability of the C1-smooth transonic steady-states, once the perturbations of the initial data and the doping profiles are small enough. We note that, when the C1-smooth transonic steady-states pass through the sonic line, they produce singularities for the system, and cause some essential difficulty in the proof of structural stability. Moreover, when the relaxation time is large enough τ≫1, under the condition that the electric field is positive at the shock location, we prove that the transonic shock steady-states are structurally stable with respect to small perturbations of the supersonic doping profile. Furthermore, we show the linearly dynamic instability for these transonic shock steady-states provided that the electric field is suitable negative. The proofs for the structural stability results are based on singularity analysis, a monotonicity argument on the shock position and the downstream density, and the stability analysis of supersonic and subsonic solutions. The linear dynamic instability of the steady transonic shock for Euler-Poisson equations can be transformed to the ill-posedness of a free boundary problem for the Klein-Gordon equation. By using a nontrivial transformation and the shooting method, we prove that the linearized problem has a transonic shock solution with exponential growths. These results enrich and develop the existing studies.

Mathematical and experimental validation of an approach for simultaneously determining the binding parameters of two drugs to a receptor

Quantifying drug-protein interactions has a pivotal role in both early phase drug development and clinical processes. Diverse affinity chromatographic methods like nonlinear chromatography can realize such quantification, however, their throughputs are challenged due to the loading of a single ligand during each run. This work derivatized a new equation for simultaneously determining the bindings of two ligands to a protein relying on assumption that the retention factors of the ligands are dependent on their injection amounts. Experimental validation of the derivatization was performed on an immobilized endothelin A receptor (ETAR) column taking ambrisentan, bosentan, and macitentan as injecting solutes. All three ligands presented a decrease in retention times along with increasing moles of injection when they were singly injected into the column. Likewise, negative relationships between the retentions and the injection amounts were observed when co-injection of ambrisentan/bosentan or bosentan/macitentan was performed, thus confirming the assumption of the derivatization. The association constants of ambrisentan, bosentan, and macitentan binding to ETAR were (1.42 ± 0.78)×104, (1.81 ± 0.22)×104, and (1.71 ± 0.41)×104 L/mol when each of them was singly loaded on the column. Such data displayed insignificant changes in four weeks thereby providing a proof of good stability of the column during the period. Co-injections of the two ligand pairs resulted in the association constants of (2.97 ± 0.13)×104 for ambrisentan, (2.51 ± 0.87)×104 for bosentan, and (2.88 ± 0.34)×104 L/mol for macitentan. These results were in good agreement with the calculation when each of the ligands was injected alone into the column and demonstrated little differences from the data by nonlinear chromatography. Owning to the simultaneous analysis of two ligands, the throughput of the proposed method was twofold higher than the typical assays including frontal analysis, zonal elution, and nonlinear chromatography. It is possible to become an alternative for rapid analysis of drug-protein interaction.

Robust Neural Network Consensus for Multiagent UASs Based on Weights’ Estimation Error

We propose a neural network consensus strategy to solve the leader–follower problem for multiple-rotorcraft unmanned aircraft systems (UASs), where the goal of this work was to improve the learning based on a set of auxiliary variables and first-order filters to obtain the estimation error of the neural weights and to introduce this error information in the update laws. The stability proof was conducted based on Lyapunov’s theory, where we concluded that the formation errors and neural weights’ estimation error were uniformly ultimately bounded. A set of simulation results were conducted in the Gazebo environment to show the efficacy of the novel update laws for the altitude and translational dynamics of a group of UASs. The results showed the benefits and insights into the coordinated control for multiagent systems that considered the weights’ error information compared with the consensus strategy based on classical σ-modification. A comparative study with the performance index ITAE and ITSE showed that the tracking error was reduced by around 45%.

Command filter based adaptive DSC of uncertain stochastic nonlinear systems with input delay and state constraints

In this paper, a command filter based dynamic surface control (DSC) is developed for stochastic nonlinear systems with input delay, stochastic unmodeled dynamics and full state constraints. An error compensation system is designed to constrain the filtering error caused by the first-order filter in the traditional dynamic surface design. On this basis, the stability proof of DSC for stochastic nonlinear systems based on command filter is proposed. The definition of state constraints in probability is presented, and the problem of stochastic full state constraints is solved by constructing a group of coordinate transformations with nonlinear mappings. The Pade approximation is adopted to deal with input delay. The stochastic unmodeled dynamics is considered, which is processed by utilizing the property of stochastic input-to-state stability (SISS) and changing supply function. All the signals of the system are proved to be semi-globally uniformly ultimately bounded (SGUUB) in probability, and the full state constraints are not violated. The two simulation examples also verify the effectiveness of the proposed adaptive DSC scheme.

Topological and quantum stability of low-dimensional crystalline lattices with multiple nonequivalent sublattices* *Dedicated to the memory of our close friend, fellow colleague, former student (PA) and teacher (AK) Prof. A A Kuzubov (1974–2016).

The terms of topological and quantum stabilities of low-dimensional crystalline carbon lattices with multiple non-equivalent sublattices are coined using theoretical analysis, multilevel simulations, and available experimental structural data. It is demonstrated that complex low-dimensional lattices are prone to periodicity breakdown caused by structural deformations generated by linear periodic boundary conditions (PBC). To impose PBC mandatory limitations for complex low-dimensional lattices, the topology conservation theorem (TCT) is introduced, formulated and proved. It is shown that the lack of perfect filling of planar 2D crystalline space by structural units may cause the formation of (i) structure waves of either variable or constant wavelength; (ii) nanotubes or rolls; (iii) saddle structures; (iv) aperiodic ensembles of irregular asymmetric atomic clusters. In some cases the lattice can be stabilized by aromatic resonance, correlation effects, or van-der-Waals interactions. The effect of quantum instability and periodicity breakdown of infinite structural waves is studied using quasiparticle approach. It is found that both perfect finite-sized, or stabilized structural waves can exist and can be synthesized. It is shown that for low-dimensional lattices prone to breakdown of translation invariance (TI), complete active space of normal coordinates cannot be reduced to a subspace of TI normal coordinates. As a result, constrained TI subspace structural minimization may artificially return a regular point at the potential energy surface as either a global/local minimum/maximum. It is proved that for such lattices, phonon dispersion cannot be used as solid and final proof of either stability or metastability. It is shown that ab initio molecular dynamics (MD) PBC Nosé–Hoover thermostat algorithm constrains the linear dimensions of the periodic slabs in MD box preventing their thermostated equilibration. Based on rigorous TCT analysis, a flowchart algorithm for structural analysis of low-dimensional crystals is proposed and proved to be a powerful tool for theoretical design of advanced complex nanomaterials.

Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

A Decoupled Energy Stable Numerical Scheme for the Modified Cahn–Hilliard–Hele–Shaw System with Logarithmic Potential

A decoupled unconditionally stable numerical scheme for the modified Cahn–Hilliard–Hele–Shaw system with logarithmic potential is proposed in this paper. Based on the convex-splitting of the associated energy functional, the temporal discretization of the scheme is given. The fractional step method is used to decouple the nonlinear modified Cahn–Hilliard equation from the pressure gradient. Then, at each time step, one only needs to solve Poisson’s equation which is obtained by using an incremental pressure-stabilization technique. In terms of logarithmic potential, using the regularization procedure can make the domain extended from − 1,1 to − ∞ , ∞ for F ϕ . Finally, the proof of the energy stability and the calculation of the error estimate are presented. Numerical examples are recorded to illustrate the accuracy and performance for the proposed scheme.

Adaptive super-twisting sliding mode for DC-AC converters in very weak grids

ABSTRACT This paper presents the design of a direct-type adaptive control structure for current regulation of a single-phase grid-tied Voltage-Source Inverter (VSI) with an LCL filter under very weak grid condition. The initial content of this work was previously published, where the controller was presented and its discrete-time stability proof was provided. Power converters connected to weak grids bring challenges to the control system, since the grid can change abruptly from a strong to a very weak condition , degrading system power quality. This study evaluates the feasibility of a RMRAC-based robust adaptive Super-Twisting Sliding Mode controller for current control of VSI when the grid inductance changes substantially, from 0 mH (strong grid) to 10 mH (very weak grid). To show the benefits of the control strategy and its robustness, hardware-in-the-loop (HIL) simulation results are presented, comparing its performance with two other controllers..