Large Initial Deviations Research Articles

Global Stabilization of a Chain of Two Integrators by a Feedback in the Form of Nested Saturators

Stability of a switching system, which comes to existence when stabilizing a chain of two integrators by a feedback in the form of nested saturators, is studied. The use of the feedback in the form of nested saturators makes it possible to take into account boundedness of the control resource and to ensure the fulfillment of the phase constraint on the velocity of approaching the equilibrium state, which is especially important in the case of large initial deviations. A Lyapunov function is constructed by means of which global asymptotic stability of the closed-loop system is proved for any positive feedback coefficients.

Geometry and Force Guided Robotic Assembly With Large Initial Deviations for Electrical Connectors

Geometry and Force Guided Robotic Assembly With Large Initial Deviations for Electrical Connectors

Nonlinear trajectory estimation using extended state-space recursive least squares.

In a variety of filtering applications including target tracking, global positioning systems, and autonomous robots, the nonlinear nature of the system model makes estimation tasks challenging. This study presents an extended state-space recursive least squares (ESSRLS) filter for nonlinear trajectory estimation. The paper focuses on ESSRLS filter derivation for non-autonomous systems and investigates its performance against the extended Kalman filter and unscented Kalman filter for maneuvering aircraft trajectory estimation applications. The major accomplishment of the proposed approach is nonlinear filtering, independent of a priori information about noise statistics and the provision of the tuning parameter (forgetting factor). This makes the ESSRLS a more suited candidate for practical nonlinear filtering applications. The simulation results show that in the presence of model uncertainties, data outages (occlusion), and large initial condition deviations, the proposed method gives superior estimation performance as compared to the state-of-the-art.

Sliding mode control based impact angle constrained guidance with predefined convergence time

This paper proposes predefined-time convergent guidance schemes that can drive the pursuer on a collision course corresponding to the target interception from a pre-specified impact direction, irrespective of initial engagement geometries. The target interception at the desired impact angle is first transformed into a problem of controlling the line-of-sight angle and its rate. Then, guidance commands are derived using sliding mode control (SMC) to ensure the convergence of corresponding errors to zero within a predefined time. A nonlinear disturbance observer is used to estimate the target’s maneuver, and the estimation error is treated as an uncertainty while rejecting its effect by virtue of SMC. Guidance commands are also derived for the case where the target maneuver is completely unknown except for its upper bound. It was shown that the gain could be selected based on the maximum bound on target maneuver to enforce sliding mode on the chosen surface, and thus guarantees target interception at a pre-specified impact angle, provided the closing speed of the engagement is positive. Owing to the use of a nonlinear framework while deriving pursuer guidance commands, the proposed guidance strategies remain valid even for engagements with large initial deviations where schemes based on linearized dynamics may fail or have degraded performance. The efficacy of the proposed guidance methods is validated through simulations for the pursuers having constant speed and time-varying speed, for various engagement geometries. The results of proposed strategies are compared with those of existing guidance and shown to be superior.

Autonomous closed-loop guidance using reinforcement learning in a low-thrust, multi-body dynamical environment

Onboard autonomy is an essential component in enabling increasingly complex missions into deep space. In nonlinear dynamical environments, computationally efficient guidance strategies are challenging. Many traditional approaches rely on either simplifying assumptions in the dynamical model or on abundant computational resources. This research effort employs reinforcement learning, a subset of machine learning, to produce a ‘lightweight’ closed-loop controller that is potentially suitable for onboard low-thrust guidance in challenging dynamical regions of space. The results demonstrate the controller’s ability to directly guide a spacecraft despite large initial deviations and to augment a traditional targeting guidance approach. The proposed controller functions without direct knowledge of the dynamical model; direct interaction with the nonlinear equations of motion creates a flexible learning scheme that is not limited to a single force model, mission scenario, or spacecraft. The learning process leverages high-performance computing to train a closed-loop neural network controller. This controller may be employed onboard to autonomously generate low-thrust control profiles in real-time without imposing a heavy workload on a flight computer. Control feasibility is demonstrated through sample transfers between Lyapunov orbits in the Earth–Moon system. The sample low-thrust controller exhibits remarkable robustness to perturbations and generalizes effectively to nearby motion. Finally, the flexibility of the learning framework is demonstrated across a range of mission scenarios and low-thrust engine types.

Collision-Free Ecological Cooperative Robust Control for Uncertain Vehicular Platoons With Communication Delay

The varying real-world road conditions (eg. slopes, speed limits) have great effects on the vehicle dynamics and fuel characteristics. The lack of consideration for these effects may lead to deteriorative fuel economy and even collisions. This paper presents a hierarchical control framework for a connected and automated vehicular platoon subject to time-vary uncertain dynamics and uniform communication delays. This control method minimizes the fuel consumptions and ensures the tracking performance on the premise of collision-free property. Firstly, a centralized ecological speed planning (CESP) algorithm based on dynamic programing (DP) is designed as the upper layer. This layer produces the fuel-optimal and feasible speed profiles, which can be followed by all vehicles, for the entire platoon based on the preview information of road conditions (eg. slopes, speed limits). Secondly, the distributed collision-free speed tracking control (DCST) method is proposed in the lower real-time control layer. The collision-free property is guaranteed by the uniform boundedness performance of the well-designed potential function of the spacing error. The convergences of all speed tracking errors are proved through barbalat's lemma. By modifying the desired spacing model, the application range of DCST is extended to the conditions with large initial deviations. Finally, the main theoretical results are verified through numerical simulations.

SELF-OSCILLATIONS DESCRIBED BY THE GENERALIZED VAN DER POL EQUATION

Quasilinear self-oscillations, represented by the generalized Van der Pol equation, are considered. The generalization of the equation is carried out by replacing the second degree of the velocity by its arbitrary non-negative degree. An approximate analytical solution, describing the transition of the oscillatory system to the regime of steady-state self-oscillations, is constructed using the energy balance method. A compact formula is obtained for calculating the amplitude of this regime and it is proved that it does not depend on the initial conditions. The calculation of the indicated amplitude involves using the table of gamma functions. It is shown that in some cases the obtained approximate analytical solution generalizes the known results of the theory of oscillations. To identify the errors of this solution, we integrated the generalized differential equation numerically using a computer for specific numerical data. The satisfactory consistency of the numerical results obtained by applying two different methods confirmed the adequacy of the approximate formulas for engineering calculations. The oscillations described by the generalized equation with the dissipative force of opposite sign are also studied. In this case the motion of the oscillatory system depends on the initial conditions. For the deviations of the oscillator from the position of static equilibrium smaller than a threshold value, free damped oscillations occur. In the case of large initial deviations beyond the threshold, free oscillations build up and over time the oscillator sweeps tend to the infinity for a limited period of time. The threshold deviation formula is derived which makes it possible to draw a conclusion about the stability of a dynamic system for various nonlinearity indices in the equation of motion and various initial perturbations.

Maintenance of Libration Point Orbit in Elliptic Sun–Mercury Model

The maintenance of the nominal multirevolution elliptic halo orbit, whose special features can benefit mercurial explorations, is first investigated through Monte–Carlo simulations in the elliptic Sun–Mercury model, and then validated in the high-fidelity ephemeris model. The receding horizon control strategy solved by the indirect Radau pseudospectral method demonstrates that the orbit can be maintained robustly with respect to very large initial deviations. Moreover, the result proves that the elliptic Sun–Mercury model is an accurate approximation.

Full-time kinetics of self-assembly and disassembly in micellar solution via the generalized Smoluchowski equation with fusion and fission of surfactant aggregates.

Full-time kinetics of self-assembly and disassembly of spherical micelles with their fusion and fission in non-ionic micellar solutions has been considered in detail on the basis of direct numerical solutions of the generalized Smoluchowski equations describing the evolution of the time-dependent concentrations of molecular aggregates for every aggregation number. The cases of instant increase of the monomer concentration up or dilution of a surfactant solution below the critical micelle concentration at large initial deviations from the final equilibrium state have been studied. Different stages in assembly or disassembly of micelles have been described and compared with the results of the stepwise mechanism of monomer attachment-detachment described by the Becker-Döring kinetic equations. A relation of the full-time kinetics to micellar relaxation at small deviations from the equilibrium state has been checked.

Kinetics of micellisation and relaxation of cylindrical micelles described by the difference Becker–Döring equation

A numerical description of micellisation and relaxation to an aggregate equilibrium in a nonionic surfactant solution with spherical premicellar aggregates and stable polydisperse cylindrical micelles is presented for a wide interval of total surfactant concentrations and initial conditions. The Smoluchowsky-type model for the attachment-detachment rates of surfactant monomers to and from surfactant aggregates with matching rates for small spherical premicellar aggregates and the rates for larger cylindrical micelles have been used. The full discrete spectrum of characteristic times of micellar relaxation and the first three relaxation modes in their dependence on the equilibrium monomer concentration have been computed using the linearized form of the Becker-Döring difference equations. The overall time behavior of the surfactant monomer and aggregate concentrations in micellisation and relaxation at large initial deviations from the final equilibrium has been studied with the help of nonlinearized discrete Becker-Döring kinetic equations. The studies demonstrate a possibility for non-monotonic evolution of the monomer concentration at the initial stages. A comparison of the computed results with the analytical ones known from solutions of the linearized and nonlinearized differential Becker-Döring kinetic equations demonstrates a general agreement at higher concentrations of the surfactant above the critical micellar concentration.

Micellization and relaxation in solution with spherical micelles via the discrete Becker–Döring equations at different total surfactant concentrations

A numerical description of micellization and relaxation to an aggregate equilibrium in surfactant solution with nonionic spherical micelles has been developed on the basis of a discrete form of the Becker-Döring kinetic equations. Two different models for the monomer-aggregate attachment-detachment rates have been used, and it has been shown that the results are qualitatively the same. The full discrete spectrum of characteristic times of micellar relaxation and first relaxation modes in their dependence on equilibrium monomer concentration have been found with using the linearized form of the Becker-Döring kinetic equations. Overall time behavior of surfactant monomer and aggregate concentrations in micellization and relaxation at large initial deviations from final equilibrium has been studied with the help of nonlinearized discrete Becker-Döring kinetic equations. Comparison of the computed results with the analytical ones known in the limiting cases from solutions of the linearized and nonlinearized continuous Becker-Döring kinetic equation demonstrates general agreement.

Cyclostrophic adjustment and nonlinear oscillations in the core of an intense atmospheric vortex

Intense atmospheric vortices are characterized by a regime of cyclostrophic balance, i.e., the balance between the pressure gradient and centrifugal force. To describe motions in the core of an axisymmetrical vortex, a class of exact solutions to the equations of gas dynamics with a linear dependence on radius is derived for the velocity components and with a quadratic dependence for temperature. It is shown that small deviations from the balance state give rise to oscillations of the hydrothermodynamic fields in the vortex core with a frequency proportional to the angular velocity of the rotation of the core. For fairly large initial deviations, oscillations are clearly anharmonic and, under the conditions of the prevailing centrifugal force, result in a significant temperature decrease on the vortex axis. The application of this class of solutions to describing the Ranque vortex effect (the intense cooling of gas during rapid rotations) and the acoustic radiation from tornadoes is discussed.

European Financial Market Integration: A Closer Look at Government Bonds in Eurozone Countries

The European Union made a number of steps not least of them the introduction of a common currency to foster the integration of the European financial markets. A number of papers have tried to gauge the degree of integration for various financial markets looking at the convergence of interest rates. A common finding is that government bond markets are quite well integrated. In this paper stochastic Kernel density estimates are used to take a closer look at the dynamics that drive the process of interest rate convergence. The main finding is that countries with large initial deviations from the mean interest rate do indeed converge. Interestingly the candidates least suspected namely the countries initially with interest rates at the mean level show a pattern of slight divergence.

Filtering under nonrandom disturbances: the method of invariant ellipsoids

This paper is also concerned with the problem of filtering under bounded nonrandom disturbances. We consider only stationary problems, i.e., all the parameters of the model are time-independent. Moreover, we search for a state estimate whose error is guaranteed to lie in a uniform ellipsoid (invariant ellipsoid) for all times; i.e., the estimate is uniform. The filter itself is also sought in the class of linear stationary filters. In this narrowed class of problems and estimates, the problem is totally solvable, i.e., an optimal filter and a state estimate can be constructed. It is this point that differs our setting from those mentioned above. The latter considered more general models, but the resulting solutions were only suboptimal, and no uniform estimates were obtained. Technically, we follow the linear matrix inequality approach [9, 10], which proved to be an effective tool in the analysis and synthesis of control systems but was little used in filtering problems. An exception is [11], in contrast to which we (i) give simpler and more accurate estimates for the quality of filtering, (ii) extend them to the discrete case, and (iii) analyze the behavior of estimates for large initial deviations. An important new technical tool is the S -theorem for two constraints [12], while the same theorem for a single constraint was applied previously.

Congenitally blind individuals rapidly adapt to coriolis force perturbations of their reaching movements.

Reaching movements made to visual targets in a rotating room are initially deviated in path and endpoint in the direction of transient Coriolis forces generated by the motion of the arm relative to the rotating environment. With additional reaches, movements become progressively straighter and more accurate. Such adaptation can occur even in the absence of visual feedback about movement progression or terminus. Here we examined whether congenitally blind and sighted subjects without visual feedback would demonstrate adaptation to Coriolis forces when they pointed to a haptically specified target location. Subjects were tested pre-, per-, and postrotation at 10 rpm counterclockwise. Reaching to straight ahead targets prerotation, both groups exhibited slightly curved paths. Per-rotation, both groups showed large initial deviations of movement path and curvature but within 12 reaches on average had returned to prerotation curvature levels and endpoints. Postrotation, both groups showed mirror image patterns of curvature and endpoint to the per-rotation pattern. The groups did not differ significantly on any of the performance measures. These results provide compelling evidence that motor adaptation to Coriolis perturbations can be achieved on the basis of proprioceptive, somatosensory, and motor information in the complete absence of visual experience.

Adsorption Kinetics of Ionic Surfactants after a Large Initial Perturbation. Effect of Surface Elasticity

This theoretical study is devoted to the relaxation of surface tension of an ionic surfactant solution for submicellar concentrations. The effects of added nonamphiphilic electrolyte and counterion binding are taken into account. We consider a large initial deviation from equilibrium, which is defined as the formation of a new interface: there is no adsorbed surfactant and electric double layer at the initial moment. Next, the surfactant solution and its interface are allowed to relax without any subsequent perturbation. The electrodiffusion equations, which describe the adsorption kinetics, are nonlinear, and it is impossible to find a general analytical solution, especially in the case of large initial deviations. Nevertheless, the problem can be linearized in the asymptotic case of long times. The derived theoretical expressions show that the relaxation times in the cases of large and small initial perturbations are numerically close to each other. For that reason the relaxation time can be considered as a general kinetic property of the adsorption monolayer. The theory predicts also the slope of the experimental plot of dynamic surface tension vs inverse square root of time; this makes the theory useful for interpretation of experimental data. The theoretical expressions involve the surface (Gibbs) elasticity, whose definition for adsorption monolayers of soluble ionic surfactants is discussed in detail. The Gibbs elasticity of such monolayers is found to increase strongly with the rise of salt concentration. The derived asymptotic expressions are verified against an exact computer solution of the electrodiffusion problem, and excellent agreement is found.

OPTIMAL FEEDBACK GUIDANCE FOR LOW‐THRUST ORBIT INSERTION

SUMMARYAn optimal feedback guidance scheme is developed for the terminal orbit insertion phase of a low‐thrust planar transfer to geosynchronous orbit. The minimum fuel trajectory between specified energy levels is utilized as a reference trajectory and the guidance law is derived using linear quadratic regulator optimization. The feedback guidance scheme responds effectively to a wide range of initial conditions and results in very precise terminal orbit conditions with near‐optimal performance. The guidance scheme also successfully handles propulsion system uncertainties and extremely large initial state deviations. Numerical results are presented for a variety of orbit insertion manoeuvres.

A nonlinear flight controller design for aircraft

In this paper, a nonlinear flight controller design approach using the method of extended linearization is described and applied to an F-8 aircraft longitudinal model. Performances of both gain-scheduled and the nonlinear controllers are compared using simulation studies. The results indicate that the nonlinear controller is insensitive to the choice of scheduling variable, and returns all the state variables to their respective trim conditions under large initial deviations in angle of attack, while the gain-scheduled controllers fail to do so. In general, the proposed nonlinear control design gives better performance and removes some of the difficulties in the conventional gain-scheduling approach.

Space Station attitude control and momentum management - A nonlinearlook

Nonlinear design procedures are presented for obtaining attitude and angular momentum control laws for the Space Station. These are based on Lyapunov's second method for stability analysis. In the absence of disturbances, there exist four stable equilibrium points. Only one of these is desired and its stability boundary is estimated from the gravitational dynamic potential. Simulation results are presented for the current Space Station configuration for large initial deviations from the local-vertical-local-horizontal frame. The control laws obtained under the zero disturbance assumption are tested in the presence of constant disturbance torques. It is shown that the Space Station can be stabilized about a torque-equilibrium attitude and the momentum of the control moment gyroscopes is bounded, provided the disturbance is smaller than the maximum gravity gradient torque that can be produced by the Space Station.