Time Optimal Control Problem Research Articles

Hamilton-Jacobi-Bellman equations for Rydberg-blockade processes

We discuss time-optimal control problems for two setups involving globally driven Rydberg atoms in the blockade limit by deriving the associated Hamilton-Jacobi-Bellman equations. From these equations, we extract the globally optimal trajectories and the corresponding controls for several target processes of the atomic system, using a generalized method of characteristics. We apply this method to retrieve known results for CZ and C-phase gates, and to find new optimal pulses for all elementary processes involved in the universal quantum computation scheme introduced in []. Published by the American Physical Society 2024

Reduction in Optimal Time in Systems with Input Redundancy

This paper discusses a reduction in the optimal time due to the presence of input redundancy in time-optimal control problems. By introducing a non-idle channel to represent an active input channel, we establish the necessary and sufficient conditions that ensure a strict reduction in the optimal time for affine nonlinear systems. In cases of identical input redundancy, its impact varies according to the type of input constraint, and certain types may not lead to a reduction in the optimal time. Ultimately, in linear time-invariant (LTI) systems, the extent of the optimal time reduction depends on the system’s controllability.

The turnpike property for high‐dimensional interacting agent systems in discrete time

AbstractWe investigate the interior turnpike phenomenon for discrete‐time multi‐agent optimal control problems. While for continuous systems the turnpike property has been established, we focus here on first‐order discretizations of such systems. It is shown that the resulting time‐discrete system inherits the turnpike property with estimates of the same type as in the continuous case. In particular, we prove that the discrete time optimal control problem is strictly dissipative and the cheap control assumption holds.

On the time-optimal control problem for a fourth order parabolic equation in a two-dimensional domain

On the time-optimal control problem for a fourth order parabolic equation in a two-dimensional domain

Time Optimal and PID Controller for Armed Manipulator Robots

This paper is fourfold. First, three different time-optimal control problems for simulating the manipulator robots with one, two, and three arms are mathematically formulated. Corresponding to the related dynamical systems, the nonlinear system of ordinary differential equations is derived. It has been found that these problems are time-free and have distinct initial and boundary conditions, making them hard to solve. To find the minimum time with different payloads, a successful numerical method based on the finite difference and the three-stage Lobatto formula is proposed. Secondly, the related torquecontrolling problems are simulated, and then for one, two, and three armed manipulator robots, they are solved using the PID controller method. Thirdly, it is shown that, compared to the time-optimal controlling problem, the optimal PID torque controller solution takes more time to do the job than was expected. However, the solution in the PID controller method shows less oscillation than the time-optimal control problem. Fourthly, mathematical theories are used, and the numerical results for both methods and different payloads are compared.

Fast generation of collision-free low-thrust trajectories for asteroid landing using deep neural networks

This study presents a fast generation method of time-optimal low-thrust trajectories based on deep neural networks (DNN) for the collision-free asteroid landings. An equivalent unconstrained differentiable problem is transformed via the introduction of smooth penalty function, so that it can be addressed with the optimal control framework. To efficiently solve the equivalent problem to generate sufficient dataset for training DNNs, a double homotopy method is developed. Specifically, the equivalent problem is connected to the time-optimal control problem without anti-collision through two consecutive simplifications, and a solving process with double backward continuation is also formulated. Besides, two DNN models are constructed to provide high-quality predictions for the minimum glide-slope constraint angle of an initial state, and the initial costates and landing time for time-optimal low-thrust trajectories with anti-collision constraints, respectively. As such, the efficiency of collision-free low-thrust landing trajectory generation is improved significantly. For the cases within the range of training dataset or within a certain extension, the low-thrust landing trajectories with a very small terminal landing error can be generated using the predicted initial costates. Meanwhile, with the high-quality initial costates, the high-precision landing trajectories can be obtained by the double homotopy method in only a few iterative steps. Finally, the promising results in a Bennu collision-free asteroid landing scenario validate the efficiency of the proposed method.

Necessary Extremum Conditions and the Neustadt–Eaton Method in the Time-Optimal Control Problem for a Group of Nonsynchronous Oscillators

Necessary Extremum Conditions and the Neustadt–Eaton Method in the Time-Optimal Control Problem for a Group of Nonsynchronous Oscillators

Influence of the Road Model on the Optimal Maneuver of a Racing Motorcycle

Motorcycle motion is largely influenced by the road geometry, which alters the allowable accelerations in longitudinal and lateral directions and influences the vertical wheel loads. Recently, a method for three-dimensional road reconstruction and its incorporation into transient and quasi-steady-state (QSS) minimum lap time simulations (MLTSs) has been proposed. The main purpose of this work is to demonstrate how significantly different results from a minimum lap time optimal control problem can be obtained when using inappropriate elevation data sources in the track reconstruction problem, and how the road model reconstructed using poor input data can lead to misleading conclusions when analyzing real vehicle and driver performances. Two road models derived from high- and low-resolution digital elevation models (DEMs) are compared and their impact on the optimal maneuver of a racing motorcycle is examined. The essentials of track identification are presented, as well as vehicle positioning on the 3D road and the generalized QSS motorcycle model. Obtained 3D and 2D road models are analyzed in detail, on a case example of the Road Atlanta racetrack, and used in minimum lap time simulations, which are validated by the experimental data recorded on the Supersport motorcycle. The comparative analysis showed that great care should be taken when selecting the elevation dataset in the track reconstruction process, and that the 1 m resolution local DEMs seem to be sufficient to obtain MLTS results close to the measured ones. The example of using the 3D free-trajectory QSS minimum lap time problem to localize the track segments where real driver actions can be improved is also presented. The differences between simulation results on different road models of the same racetrack can be large and influence the interpretation of optimal maneuver.

Investigation of near-rectilinear halo orbit search and rescue using staging [formula omitted]/[formula omitted] Lyapunov and distant retrograde orbit families

Cislunar space is a region of growing interest with nations investing resources to cultivate long presence habitations on the lunar surface. With this increased attention and expansion of missions, both crewed and uncrewed, the likelihood of a mishap or a spacecraft becoming impaired and unable to continue its mission will also increase. The present research adds to the field of cislunar mission operations and trajectory analysis by investigating search and rescue (SAR) operations via rendezvous and proximity operations (RPO) with an impaired notional spacecraft located in a Near-Rectlinear Halo Orbit (NRHO). This research compares the response times of rescuer spacecraft located in sample distant retrograde orbits (DROs) and L1/L2 Lyapunov orbits for the timely far rendezvous with the impaired spacecraft located in the NRHO. This will simulate a variety of far rendezvous whereby the impaired spacecraft’s location within the NRHO and the rescuer spacecraft in the L1/L2 Lyapunov and DRO orbit families are varied. A series of minimum time optimal control problems are posed using the circular restricted three-body problem (CR3BP) dynamics, and pseudospectral methods are used to find solutions given an example maximum ΔV constraint of 3 km/s. The results reinforces our intuition that rendezvous time of flight (TOF) between orbits within the L1, L2, and DRO families and the targeted NRHO correlate with proximity to the NRHO, with the shortest far rendezvous times in each family found to be approximately 6 hours, 4.5 hours, and 10 hours respectively. The results further show that a constellation of two rescue spacecraft could be positioned within the three orbit families to achieve far rendezvous with the chosen NRHO in under one day.

Enhanced transfer performance of sun-facing diffractive sails in solar polar imager missions

A diffractive sail represents an advancement in solar sailing technology, capable of generating tangential radiation pressure force when oriented towards the sun. This unique property allows for varied radiation pressure distributions, making it a potential candidate for complex space missions. This paper delves into the transfer performance of the sun-facing diffractive solar sail in a solar polar imager mission.We establish both the radiation pressure force model and the time-optimal control problem specific to the sun-facing diffractive solar sail in interplanetary transfers. Applying an indirect method and a combined approach of parameter optimization with the direct method, we explore how different diffractive angles and acceleration capacities affect transfer times. This research offers insights to the design and fabrication of thin novel optical metamaterial films that efficiently diffract light at appropriate angles for potential solar sails in the solar polar imager mission. Compared with reflective sails, the diffractive sail demonstrates remarkable transfer efficiency and simplicity in attitude control, making it a more suitable choice in the solar polar imager mission.

On the Use of Ellipsoidal Estimation Techniques in the RRT* Suboptimal Pathfinding Algorithm

The article is devoted to the development of an algorithm for the approximate solution of the time-optimal control problem for a system of ordinary differential equations, under the condition of avoiding collisions with stationary obstacles and subject to the specified pointwise constraints on the possible values of the control parameters. The main idea is to use a modification of the algorithm for finding suboptimal paths using rapidly growing random trees (RRT*). The most difficult part of this algorithm is to find the optimal trajectories for the problems of transferring the system from one fixed position to another, close to it, without taking into account state constraints. This subproblem is proposed to be solved using the methods of ellipsoidal calculus. This approach makes it possible to efficiently search suboptimal trajectories both for linear systems with large state space dimension and for systems with nonlinear dynamics. Algorithms for the linear and non-linear cases are sequentially analyzed in the paper, and the corresponding examples of calculations are presented.

Bang-Bang Property and Time-Optimal Control for Caputo Fractional Differential Systems

In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results.

Minimal time sampled-data controls for the ordinary differential system

In this paper, we design a minimal time optimal control problem for an ordinary differential system with sampled-data controls, and use it to approximate a time optimal control problem for the ordinary differential system with distributed controls. We find connections among this problem, a minimal norm sampled-data control problem and a minimization problem, and obtain some properties on these problems. Based on these, we not only build up error estimates for optimal time and optimal controls between the minimal time sampled-data control problem and the minimal time distributed control problem, in terms of the sampling period, but we also prove that such estimates are optimal in some sense.

High-Order Guidance for Time-Optimal Low-Thrust Trajectories with Accuracy Control

This paper develops a high-order guidance scheme based on the expansion of the solution to a time-optimal low-thrust optimal control problem by using differential algebraic techniques. The result allows new optimal trajectories to be autonomously computed from the simple evaluation of polynomials, including updated times of flight. Furthermore, a differential algebraic technique known as automatic domain splitting is implemented to improve the guidance scheme by adaptively and automatically splitting the uncertainty domain into smaller subdomains, in which new guidance polynomials are then generated which are better equipped to handle deviations in their respective subdomains. The result is not only a more effective handling of deviations in the trajectory compared to previous algorithms which use just a single set of polynomials but one that can provide the control and time-of-flight updates to a desired accuracy across the entire domain. Combined with its high-order nature, the guidance scheme is capable of dealing with nonlinearity and large deviations with respect to the reference trajectory, thus being robust to mismodeling and unplanned events. The guidance scheme is evaluated in three scenarios: simplified rendezvous using Clohessy–Wiltshire equations to demonstrate automatic domain splitting in the guidance problem, an Earth–Mars transfer, and an asteroid landing trajectory. The results show the ability of the differential algebraic-based guidance to provide optimal control and time-of-flight updates and improvements of several orders of magnitude over previous differential algebraic-based schemes in handling large uncertainty experienced in spaceflight by leveraging automatic domain splitting.

Параметрическая идентификация и оптимальное управление процессом индукционного нагрева

In the paper the experimental induction heating installation for through heating of steel cylindrical workpieces is considered. It is well known that the use of an industrial heating system presupposes incomplete information about its main characteristics due to the complex nature of changes in some parameters both from the ambient temperature and from the temperature of the heated part. These parameters primarily include the coefficients of convective and emissive heat transfer from the surface of the workpiece. At the first step, it is necessary to obtain the values of unknown parameters, which also include the voltage of the power source, and then solve the optimal control problem. The problem of identifying these parameters is solved using the alternance method of parametric optimization of systems with distributed parameters, using experimental data from thermocouples. The obtained values of unknown parameters are then used in a numerical finite-element FLUX model of the considered process. Based on the developed model, the problem of time-optimal control with an additional phase restriction on the maximum temperature is formulated. This problem is solved using the alternance method. The solution of the problem made it possible to obtain a emperature field with a maximum deviation of 32 °C from the required temperature T* = 1 200 °C. This value is less than 3% of the desired temperature value and fully meets the technological requirements for induction through heating processes prior subsequent plastic deformation. The maximum temperature in this case does not exceed the maximum permissible value of 1 300 °C during the entire heating process.

Approximate time optimal control by deep neural networks trained with numerically obtained optimal trajectories

AbstractThis paper focuses on online time optimal control of nonlinear systems. This is achieved by approximating the results of time optimal control problems (TOCP) with deep neural networks (DNN) depending on the initial and terminal system state. In general, solving a TOCP for nonlinear systems is a computationally challenging task. Especially in the context of time optimal nonlinear model predictive control (TMPC) with hard real time constraints successful termination of a TOCP within sample times suitable for controlling mechanical systems cannot be guaranteed. Therefore, our approach is to train three DNNs with different aspects of numerical solutions of TOCPs with random initial and terminal state. These networks can then be used to approximate the TMPC by a one step model predictive control scheme with a significantly simpler structure and decreased calculation time. In order to verify this procedure by simulation, it is applied to a prove of concept example as well as the model of an industrial robot.

On the time-optimal control problem for a heat equation

In previous works, we have considered some control problems for parabolic type equations, namely, control problems for parabolic type equations were studied as boundary value problems of the first type, and the weight function was expanded into a Fourier series by sines. In this paper, we consider boundary control problem for a heat equation on the interval. In the part of the bound of the given domain it is given value of a solution and it is required to find a control to get the average value of the solution. By the mathematical-physics methods it is proved that like this control exists and the estimate of a minimal time for achieving the given average temperature over some domain is found.

An Analytic Method to Determine the Optimal Time for the Induction Phase of Anesthesia

We obtain an analytical solution for the time-optimal control problem in the induction phase of anesthesia. Our solution is shown to align numerically with the results obtained from the conventional shooting method. The induction phase of anesthesia relies on a pharmacokinetic/pharmacodynamic (PK/PD) model proposed by Bailey and Haddad in 2005 to regulate the infusion of propofol. In order to evaluate our approach and compare it with existing results in the literature, we examine a minimum-time problem for anesthetizing a patient. By applying the Pontryagin minimum principle, we introduce the shooting method as a means to solve the problem at hand. Additionally, we conducted numerical simulations using the MATLAB computing environment. We solve the time-optimal control problem using our newly proposed analytical method and discover that the optimal continuous infusion rate of the anesthetic and the minimum required time for transition from the awake state to an anesthetized state exhibit similarity between the two methods. However, the advantage of our new analytic method lies in its independence from unknown initial conditions for the adjoint variables.

One New Property of a Class of Linear Time-Optimal Control Problems

The following paper deals with a new property of linear time-optimal control problems with real eigenvalues of the system. This property unveils the possibility of synthesizing the time-optimal control without describing the switching hyper-surfaces. Furthermore, the novel technique offers an alternative solution to the classic example of the time-optimal control of a double integrator system.

Perturbations of Time Optimal Control Problems for Parabolic Integro-Differential Equations

Perturbations of Time Optimal Control Problems for Parabolic Integro-Differential Equations