Explicit Functions Of Time Research Articles

Dynamic SNR, Spectral Efficiency, and Rate Characterization in 5G/6G mmWave/sub-THz Systems with Macro- and Micro-Mobilities

The performance of 5G/6G cellular systems operating in millimeter wave (mmWave, 30–100 GHz) and sub-terahertz (sub-THz, 100–300 GHz) bands is conventionally assessed by utilizing the static distributions of user locations. The rationale is that the use of the beam tracking procedure allows for keeping the beams of a base station (BS) and user equipment (UE) aligned at all times. However, by introducing 3GPP Reduced Capability (RedCap) UEs utilizing the Radio Resource Management (RRM) Relaxation procedure, this may no longer be the case, as UEs are allowed to skip synchronization signal blocks (SSB) to improve energy efficiency. Thus, to characterize the performance of such UEs, methods explicitly accounting for UE mobility are needed. In this paper, we will utilize the tools of the stochastic geometry and random walk theory to derive signal-to-noise ratio (SNR), spectral efficiency, and rate as an explicit function of time by accounting for mmWave/sub-THZ specifics, including realistic directional antenna radiation patterns and micro- and macro-mobilities causing dynamic antenna misalignment. Different from other studies in the field that consider time-averaged performance measures, these metrics are obtained as an explicit function of time. Our numerical results illustrate that the macro-mobility specifies the overall trend of the time-dependent spectral efficiency, while local dynamics at 1–3 s scales are mainly governed by micro-mobility. The difference between spectral efficiency corresponding to perfectly synchronized UE and BS antennas and time-dependent spectral efficiency in a completely desynchronized system is rather negligible for realistic cell coverages and stays within approximately 5–10% for a wide range of system parameters. These conclusions are not affected by the utilized antenna array at the BS side. However, accounting for realistic radiation patterns is critical for a time-dependent performance analysis of 5G/6G mmWave/sub-THz systems.

Quasi-Analytical Solution of Kepler’s Equation as an Explicit Function of Time

Although Kepler’s laws can be empirically proven by applying Newton’s laws to the dynamics of two particles attracted by gravitational interaction, an explicit formula for the motion as a function of time remains undefined. This paper proposes a quasi-analytical solution to address this challenge. It approximates the real dynamics of celestial bodies with a satisfactory degree of accuracy and minimal computational cost. This problem is closely related to Kepler’s equation, as solving the equations of motion as a function of time also provides a solution to Kepler’s equation. The results are presented for each planet of the solar system, including Pluto, and the solution is compared against real orbits.

An Added Proof of the "Trace Anomaly Redefinition": Equivalent Wick Rotation Conditions

The Chiral Trace Anomaly cancellation in Gravitational Curved Spaces necessitated, due to the presence of the metric inside the action integral, that it is needed for both the “volume” and the “surface” terms, which can both be explicit functions of time, to be resolved and also sufficiently to be brought down by one more dimension. This leads into a description through orbits (close to Euclidean), allowing for a generalization of the Wick rotation and confirming the Trace anomaly redefinition already exposed.

Berry phase and the Mandel parameter of the non-degenerate parametric amplifier

We study the non-degenerate parametric amplifier problem from an algebraic approach of the SU(1,1) group. We write the Hamiltonian of this problem in terms of the boson generators of the SU(1,1) group and the difference operator. We apply the tilting transformation to our results to exactly solve this Hamiltonian and obtain its energy spectrum and eigenfunctions. Then, by assuming that our Hamiltonian is an explicit function of time, we calculate its Berry phase. Finally we obtain the Mandel Q-parameter of the photon numbers n a and n b .

Safe Control Design for Unknown Nonlinear Systems with Koopman-based Fixed-Time Identification

We consider the problem of safe control design for a class of nonlinear, control-affine systems subject to an unknown, additive, nonlinear disturbance. Leveraging recent advancements in the application of Koopman operator theory to the field of system identification and control, we introduce a novel fixed-time identification scheme for the infinitesimal generator of the infinite-dimensional, but notably linear, Koopman dynamical system analogous to the nonlinear system of interest. That is, we derive a parameter adaptation law that allows us to recover the unknown, residual nonlinear dynamics in the system within a finite-time, independent of an initial estimate. We then use properties of fixed-time stability to derive an estimation error bound on the unknown dynamics as an explicit function of time, which allows us to synthesize a safe controller using control barrier function based methods. We conduct a quadrotor-inspired case study in support of our proposed method, in which we show that safe trajectory tracking is achieved despite unknown, nonlinear dynamics.

Optimal Control of the Angular Momentum for a Solid (Spacecraft) Performing a Spatial Turn

A quaternion solution of the dynamic problem on the optimal turn of a solid (for example, a spacecraft) from a known initial to a given final angular position is presented. Optimization of the control program is carried out by using a combined indicator that combines a quadratic performance criterion and turn time; the minimized functional combines energy costs and maneuver duration in a given proportion. Based on the maximum principle, quaternion models, and methods for studying the controlled motion of a solid (spacecraft), a solution of the problem has been obtained. The construction of the optimal rotation is based on a differential equation relating the angular momentum and the orientation quaternion of a solid. The conditions of optimality are written in analytical form and the properties of the optimal motion are studied. Analytical equations and calculation formulas for finding the optimal control are presented. The control law is formulated as an explicit dependence of the control variables on the phase coordinates. Key relations that determine the optimal values of the parameters of the angular momentum control algorithm are given. In the case of a dynamically symmetric body, a complete solution of the turn problem in a closed form is obtained: analytical dependences as explicit functions of time for control variables and relations for calculating the parameters of the control law are given. A numerical example and the results of numeric simulation of the rotation of a spacecraft as a solid under optimal control that demonstrate the practical feasibility of the proposed control method are given.

Multiple waves of COVID-19: a pathway model approach.

A generalized pathway model, with time-dependent parameters, is applied to describe the mortality curves of the COVID-19 disease for several countries that exhibit multiple waves of infections. The pathway approach adopted here is formulated explicitly in time, in the sense that the model's growth rate for the number of deaths or infections is written as an explicit function of time, rather than in terms of the cumulative quantity itself. This allows for a direct fit of the model to daily data (new deaths or new cases) without the need of any integration. The model is applied to COVID-19 mortality curves for ten selected countries and found to be in very good agreement with the data for all cases considered. From the fitted theoretical curves for a given location, relevant epidemiological information can be extracted, such as the starting and peak dates for each successive wave. It is argued that obtaining reliable estimates for such characteristic points is important for studying the effectiveness of interventions and the possible negative impact of their relaxation, as it allows for a direct comparison of the time of adoption/relaxation of control measures with the peaks and troughs of the epidemic curve.

On Controller Design for Nonlinear Systems With Pure State Constraints

This brief is concerned with a new problem of designing a robust adaptive backstepping controller for nonlinear systems where state constraints are an explicit function of time and state variables, i.e., pure state constraints. Furthermore, a disturbance observer is designed to cope with the disturbances in the systems. Finally, a numerical example is demonstrated to show the efficacy of the proposed theoretical results.

Dynamic Problem of Optimal Control of Spacecraft Attitude under Restriction on Phase Variables

An analytical solution to the optimal control problem of spacecraft reorientation from an arbitrary initial angular position into a required final angular position under the restrictions on control functions and phase variables is presented (the controlling moment and angular velocity are restricted). Time of slew maneuver is minimized. The specific case was considered when maximum admissible kinetic energy of rotation is significant restriction. Constructing the optimal control of reorientation is based on Pontryagin’s maximum principle and the quaternionic variables and models. It is shown that optimal mode is piecewise-continuous control when a direction of spacecraft’s angular momentum is constant relative to the inertial coordinate system during rotation of a spacecraft; for a per forming an optimal turn, the moment of forces is parallel to a straight line fixed in inertial space. Two types of optimal control are possible depending on the given initial and final positions and spacecraft’s moments of inertia — relay control with one switching point when the controlling moment is maximal over the entire time interval of control (segments of acceleration and braking), and relay control with two switching point consisting of intensive acceleration, motion by inertia with the absented moment and an exit onto restriction of rotation energy, and then final braking with the maximum controlling moment. The analytical equations and relations for a finding the optimal control program are written down. The calculation formulas for determining the time characteristics of maneuver and computing a duration of acceleration and braking are given. The proposed algorithm of control provides maximally fast implementation of spacecraft reorientation under the limited kinetic energy of rotation. For an axially symmetric solid body (spacecraft), the optimal control problem, in dynamical statement, was solved completely — we obtained the dependences as explicit functions of time for the control variables, and relations for calculating the key parameters of the law of control are derived. The numerical example and results of mathematical simulation of spacecraft motion under the optimal control are presented, demonstrating the practical feasibility of the developed method for control of spacecraft attitude.

Development of a fourth-order compact finite difference scheme for simulation of simulated-moving-bed process

A fourth-order compact finite difference scheme was developed to solve the model equation of simulated moving bed, which has a boundary condition that is updated along the calculation process and cannot be described as an explicit function of time. Two different methods, direct method and pseudo grid point method, were proposed to deal with the boundary condition. The high accuracy of the two methods was confirmed by a case study of solving an advection-diffusion equation with exact solution. The developed compact finite difference scheme was then used to simulate the SMB processes for glucose-fructose separation and enantioseparation of 1,1′-bi-2-naphtol. It was found that the simulated results fit well with the experimental data. Furthermore, the developed method was further combined with the continuous prediction method to shorten the computational time and the results showed that, the computational time can be saved about 45%.

Analytical Solving the Optimal Control Problem of Spacecraft’s Slew Maneuver with Minimal Energy of Rotation

Dynamic problem of optimal reorientation from an arbitrary initial attitude into the given final angular position with restricted control which minimizes kinetic energy of spacecraft rotation was solved. Termination time of maneuver is known. Quadratic criterion of quality is applied for finding the optimal control program. Use of integral index in special form concerning angular velocity has helped solve the formulated problem by analytical way. Control law was written down in explicit form. Designing the optimal control is based on quaternion variables and models. It is shown that during optimal turn, the controlling moment is parallel to the straight line which is immobile in the inertial space, and direction of spacecraft’s angular momentum in the process of rotation is constant relative to the inertial coordinate system. Special control regime was studied in detail, and conditions of the impossibility of occurrence of this regime are formulated. It is proven that spacecraft rotates by inertia in special control regime if it exists. The formalized equations and computational expressions for determining the optimal rotation program and duration of acceleration and braking were written. A dependence of control variables on phase coordinates is presented also. The proposed control algorithm allows the spacecraft’s reorientation to be carried out within the fixed time period with minimal angular kinetic energy. Analytical expressions for computing the time characteristics of reorientation maneuver are given, and condition for determination of the moment of the beginning of the braking, based on factual kinematic parameters of motion judging by principles of terminal control is formulated, that provides high accuracy of orientation. A comprehensive solution to the control problem is presented for a dynamically symmetric spacecraft: the dependences as explicit functions of time for the control variables are obtained, and relations for calculating the key parameters of the turn maneuver’s control law are given also. A numerical example and the results of mathematical simulation of spacecraft’s motion with optimal control are presented, which demonstrate the practical feasibility of the designed method for controlling the spacecraft attitude. Presence of ready formulas for synthesis of optimal motion program during reorientation maneuver does the executed research as practically significant and suitable for direct use in practice of space flights.

Parametric Resonances of a Three-Blade-Rotor System With Reference to Wind Turbines

Abstract Coupled blade-hub dynamics of a coupled three-blade-rotor system with parametric stiffness, which is similar to a horizontal-axis wind turbine, is studied. Blade equations have parametric and direct excitation terms due to gravity and are coupled through the hub equation. For a single degree-of-freedom blade model with only in-plane transverse vibrations, the reduced-order model shows parametric resonances. A small parameter is established for large blades, which enables us to treat the effect of blade motion as a perturbation on the rotor motion. The rotor speed is not constant, and the cyclic variations cannot be expressed as explicit functions of time. Therefore, it is more convenient to use the rotor angle as the independent variable. By expressing the system dynamics in the rotor angle domain and assuming small variations in rotor speed, the blade equations are decoupled from the rotor equation. The interdependent blade equations constitute a three-degree-of-freedom system with periodic parametric and direct excitation. The response is analyzed by using a first-order method of multiple scales (MMS). The system has a superharmonic and a subharmonic resonances due to direct and parametric effects introduced by gravity. Amplitude-frequency relations and stabilities of these resonances are studied. The MMS solutions are compared with numerical simulations for verification.

Optimal Control of Kinetic Moment During the Spatial Rotation of a Rigid Body (Spacecraft)

The problems of optimal control of kinetic moment of a rigid body (for example, a spacecraft) during the reorientation maneuver from an arbitrary initial to a given final angular position, taking into account the requirements for the energy of rotation, are investigated. An analytical solution for the problem of optimal control of solid body reorientation is obtained. Formalized equations are presented and calculation expressions for the construction of an optimal control program are given. The task of controlling the turn is solved taking into account the restrictions on control moments. An analytical relationship is found between the turning time and the maximum rotational energy. The moment of the start of deceleration is determined by the actual parameters of movement (the mismatch quaternion and the kinetic moment), based on the principles of terminal control (using information about the angular position and measuring the angular velocity). Control algorithms created make it possible to make turns in a given time with a minimum rotational energy. For a dynamically symmetric solid body, the control problem is solved to the end — dependencies are obtained, as explicit functions of time, for control variables and relations for calculating the key parameters of the kinetic moment control law. A numerical example and the results of mathematical modeling of the motion of a spacecraft with optimal control, which demonstrate the practical feasibility of the developed orientation control algorithms, are presented.

Chaos analysis in attitude dynamics of a satellite with two flexible panels

We investigate chaos in attitude dynamics of a satellite consisting of a rigid body and two flexible panels. We use the assumed mode method for the vibration of the panels and attempt to find an appropriate methodology to define the internal perturbation due to the flexibility of the panels, followed by the derivation of the Hamiltonian of the system. A canonical transformation on the entire flexible satellite is then applied, reducing the dimensionality of the system by two. The new Hamiltonian will have 2N +1 degrees of freedom with only two main variables. These two variables are preserved, while the rest of the variables are approximated by explicit functions of time due to the rotation of the main body. This will lead to the proof of the existence of a chaotic behavior using the Melnikov–Wiggins’ method, which in turn produces an analytical relation for the width of chaotic layers. This relation provides a means to make a proper comparison of the analytical and numerical results. The results obtained indicate that this approach is useful in the analysis of the effect of the parameters on the chaotic motion of the satellite. The results also show that one or at most two mode shapes suffice to reveal the vibration behavior of the panels and the effect of the panels on the chaotic behavior of the satellite.

Variable Gain Predictive PN Guidance for Interception of High Speed Re-entry Targets

A new guidance philosophy by introducing a trajectory dependent variable guidance gain in the predictive PN guidance has been proposed in this paper. The predictive PN guidance uses nonlinear models of interceptor and target to compute zero effort miss and time to go, while guidance gain computation utilizes a formula obtained by application of optimal control theory for a weighted control minimization of a linearized geometry PN guidance problem, where weight in the cost function could be any positive definite explicit function of time. The specific problem considered in this paper is interception of high speed re-entry ballistic target at various altitudes, this guidance logic shapes the guidance demand according to the maximum capability of the vehicle in such a fashion that almost full guidance correction happens during initial phase of the interceptor flight where it has sufficient aerodynamic capability. Hence the guidance demands in latter phase of the trajectory is almost negligible, where the interceptor capabilities are very low. The simulation has been done and improvement of this guidance against PN and Predictive PN guidance are shown. This guidance is named as “variable gain predictive PN guidance (VGPRPN)”.

Fast phase gates with trapped ions

We implement faster-than-adiabatic two-qubit phase gates using smooth state-dependent forces. The forces are designed to leave no final motional excitation, independently of the initial motional state in the harmonic, small-oscillations limit. They are simple, explicit functions of time and the desired logical phase of the gate, and are based on quadratic invariants of motion and Lewis-Riesenfeld phases of the normal modes.

Second-order theory for the two-body problem with varying mass including periastron effect

The model of varying mass function, including periastron effect, in terms of Delaunay variables will be expanded. The Hamiltonian of the problem is developed in the extended phase space by introducing a new canonical pair of variable (\(q_4, Q_4\)). The first “\(q_4 \)” is defined as explicit function of time and the initial mass of the system. The conjugate momenta “\(Q_4\)” is assigned as the momenta raises from the varying mass. The short-period analytical solution through a second-order canonical transformation using “Hori’s” method developed by “Kamel” is obtained. The variation equations for the orbital elements are obtained too. The results of the effect of the varying mass and the periastron effect in the case \(n = 2\) are analyzed.

Exact solutions to a class of feedback systems on [formula omitted

This paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on SO(n). The closed-loop systems are solved exactly for the rotation matrices as explicit functions of time, the initial conditions, and the gain parameters of the control laws. The exact solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix. Applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance. Knowledge of the future trajectory of the states can also be used as an alternative to the zero-order hold in systems where the attitude is sampled at discrete time instances.

Modeling and prediction of surface roughness for running-in wear using Gauss-Newton algorithm and ANN

In this paper, surface roughness model for running-in and steady state of the wear process is proposed. In this work monotonously decreasing trend of surface roughness during running-in was assumed. The model was developed by considering the surface roughness as an explicit function of time during running-in, keeping other system parameters (velocity, load, hardness, etc.) constant. The proposed model being non-linear, optimal values of the model parameters were evaluated by Gauss–Newton (GN) algorithm. The experimental results adopted from the literature, for steel and Cu–Zn alloy specimens, were used for validation of the model. Artificial neural network (ANN) based model was also developed and was compared with the proposed model on the basis of statistical methods (coefficient of determination (R2), mean square error (MSE) and mean absolute percentage error (MAPE)).

Minimum time optimal synthesis for two level quantum systems

For the time optimal problem of an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the Pontryagin maximum principle are explicit functions of time. We use this fact here to perform the optimal synthesis for these systems, i.e., to find all time optimal trajectories. Although the Lie group SU(2) is three dimensional, time optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle, another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. The results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system.