Multiphase Optimal Control Problem Research Articles

Determining Fast Battery Charging Profiles Using an Equivalent Circuit Model and a Direct Optimal Control Approach

This work used an electrical equivalent circuit model combined with a temperature model and computational optimal control methods to determine minimum time charging profiles for a lithium–ion battery. To effectively address the problem, an optimal control problem formulation and direct solution approach were adopted. The results showed that, in most cases studied, the solution to the battery’s fast-charging problem resembled the constant current–constant voltage (CC-CV) charging protocol, with the advantage being that our proposed approach optimally determined the switching time between the CC and CV phases, as well as the final time of the charging process. Considering path constraints related to the terminal voltage and temperature gradient between the cell core and case, the results also showed that additional rules could be incorporated into the protocol to protect the battery against under/over voltage-related damage and high-temperature differences between the core and its case. This work addressed several challenges and knowledge gaps, including emulating the CC-CV protocol using a multi-phase optimal control approach and direct collocation methods, and improving it by including efficiency and degradation terms in the objective function and safety constraints. To the authors’ knowledge, this is the first time the CC-CV protocol has been represented as the solution to a multi-phase optimal control problem.

Multi-Phase Vertical Take-Off and Landing Trajectory Optimization with Feasible Initial Guesses

The advancement of electric vertical take-off and landing (eVTOL) aircraft has expanded the horizon of urban air mobility. However, the challenge of generating precise vertical take-off and landing (VTOL) trajectories that comply with airworthiness requirements remains. This paper presents an approach for optimizing VTOL trajectories considering six degrees of freedom (6DOF) dynamics and operational constraints. Multi-phase optimal control problems are formulated to address specific constraints in various flight stages. The incremental nonlinear dynamic inversion (INDI) controller is employed to execute the flight mission in each phase. Controlled flight simulations yield dynamically feasible trajectories that serve as initial guesses for generating sub-optimal trajectories within individual phases. A feasible and sub-optimal initial guess for the holistic multi-phase problem is established by concatenating these single-phase trajectories. Focusing on a tilt-wing eVTOL aircraft, this paper computes VTOL trajectories leveraging the proposed initial guess generation procedure. These trajectories account for complex flight dynamics, align with various operation constraints, and minimize electric energy consumption.

Train Control and Schedule Integrated Optimization With Reversible Substations

The Carbon Neutralization is a grand strategy to deal with global climate change. As a major energy consumer, urban rail transit (URT) system should contribute corresponding solutions. This paper proposes an energy-efficient train control and schedule integrated optimization method under DC traction power supply system (TPSS) with reversible substation (RSS). The dynamic coupling mechanism between the train movement process and the RSS working mode is analyzed and modeled. Aiming to minimize the total RSS energy consumption for the train traveling over multi-interstation with a given total trip time, the train trajectory and schedule optimization problem is integrated in the same level, and is formulated as a multi-phase optimal control problem (MOCP) to be solved by a pseudo-spectral method (PSM). As a supplement, a customized dynamic programming (DP) method for the problem is introduced to verify the validity of the proposed solution framework. Simulation results under the models of the RSS energy together with conventional rectifier substation energy and vehicle mechanical energy are presented and compared, indicating that the proposed methodology has great energy saving performance for the new-type DC railway system.

Fuel-Optimal Guidance for End-to-End Human-Mars Entry, Powered-Descent, and Landing Mission

This article investigates the fuel-optimal guidance problem of the end-to-end human-Mars entry, powered-descent, and landing (EDL) mission. It applies a unified modeling scheme and develops a computationally efficient new optimization algorithm to solve the multiphase optimal guidance problem. The end-to-end EDL guidance problem is first modeled as a multiphase optimal control problem with different dynamics and constraints at each phase. Via polynomial approximation and discretization techniques, this multiphase optimal control problem is then reformulated as a polynomial programming problem. By introducing intermediate variables and quadratic equality constraints, a polynomial program is equivalently converted into a nonconvex quadratically constrained quadratic program (QCQP). Then, a novel customized alternating direction method of multipliers (ADMM) is proposed to efficiently solve the large-scale QCQP with convergence proof to a local optimum under certain conditions on the algorithmic parameters. The fuel savings under the end-to-end human-Mars EDL guidance are verified by comparing to the fuel consumption using the separate phase guidance approach. Furthermore, the computational efficiency of the customized ADMM algorithm is validated by comparing to the state-of-the-art nonlinear programming method. The robustness of the customized ADMM algorithm is verified via extensive simulation cases with random initial conditions.

Onboard fuel-optimal guidance for human-Mars entry, powered-descent, and landing mission based on feature learning

This paper investigates the end-to-end human-Mars entry, powered-descent, and landing (EDL) guidance problem by developing an on-board learning-based optimal control method (L-OCM) to achieve the goal of precise and fuel-efficient planetary landing. First, the end-to-end EDL guidance problem is formulated as a multi-phase optimal control problem with hybrid dynamics and constraints. Then a customized alternating direction method of multipliers is applied to solve the end-to-end EDL guidance problem with varying initial states off-line. After that, the L-OCM is developed to generate real-time optimal guidance commands. To be specific, supported by the optimal control theory, the necessary conditions of optimality for optimal control of the entry phase and powered-descent phase are derived, respectively, which leads to two two-point-boundary-value-problems (TPBVPs). Then, critical parameters are identified to approximate the complete solutions of the TPBVPs. To find the implicit relationship between the initial states and these critical parameters, deep neural networks are constructed to learn the values of these critical parameters in real-time with training data obtained from the off-line solutions. Furthermore, when random disturbances are considered during the EDL process, the single stage L-OCM is extended to the multi-stage L-OCM to regenerate real-time optimal guidance commands. Finally, the proposed L-OCM is implemented in extensive simulation cases to verify the effectiveness and efficiency of the new method.

On-Line Train Speed Profile Generation of High-Speed Railway With Energy-Saving: A Model Predictive Control Method

By considering dynamic operational conditions in high-speed railway, this paper focuses on the on-line generation problem of train speed profile with energy-saving. This problem is formulated via the model predictive control framework in a moving-horizon manner, such that the real-time running conditions (e.g., temporary speed restrictions) can be involved in the on-line scheduling process of the train. At each control step, a hybrid scheme combining the energy-efficient and time-optimal train control strategies is proposed to ensure the feasibility of the optimal train control problem within the prediction horizon. The optimal control problem in the horizon is transformed into a multi-phase optimal control problem, which is then solved efficiently on-line using the pseudospectral method. By repeatedly solving the train control problem at each step, the energy-efficient train speed trajectory for the whole trip involving dynamic operational conditions can be obtained on-line. In addition, a delay recovery process is designed to re-schedule the train operation if the delay time during the trip exceeds a given threshold value. Finally, numerical examples using data for a real high-speed railway line are given to demonstrate the effectiveness and robustness of the proposed approach.

Multiphase mixed-integer nonlinear optimal control of hybrid electric vehicles

This study considers the problem of computing a non-causal minimum-fuel energy management strategy for a hybrid electric vehicle on a given driving cycle. Specifically, we address the multiphase mixed-integer nonlinear optimal control problem that arises when the optimal gear choice, torque split and engine on/off controls are sought in off-line evaluations. We propose an efficient model by introducing vanishing constraints and a phase specific right-hand side function that accounts for the different powertrain operating modes. The gearbox and driveability requirements translate into combinatorial constraints. These constraints have not been included in previous research; however, they are part of the algorithmic framework for this investigation. We devise a tailored algorithm to solve this problem by extending the combinatorial integral approximation (CIA) technique that breaks down the original mixed-integer nonlinear program into a sequence of nonlinear programs and mixed-integer linear programs, followed by a discussion of its approximation error. Finally, numerical results illustrate the proposed algorithm in terms of solution quality and run time.

Trajectory optimization with constraints for alpine skiers based on multi-phase nonlinear optimal control

The super giant slalom (Super-G) is a speed event in alpine skiing, in which the skier trajectory has a significant influence on the athletes’ performances. It is a challenging task to determine an optimal trajectory for the skiers along the entire course because of the complexity and difficulty in the convergence of the optimization model. In this study, a trajectory optimization model for alpine skiers competing in the Super-G is established based on the optimal control theory, in which the objective is to minimize the runtime between the starting point and the finish line. The original trajectory optimization problem is converted into a multi-phase nonlinear optimal control problem solved with a pseudospectral method, and the trajectory parameters are optimized to discover the time-optimal trajectory. Using numerical solution carried out by the MATLAB optimization toolbox, the optimal trajectory is obtained under several equality and inequality constraints. Simulation results reveal the effectiveness and rationality of the trajectory optimization model. A test is carried out to show that our code works properly. In addition, several practical proposals are provided to help alpine skiers improve their training and skiing performance.

Geostationary debris mitigation using minimum time solar sail trajectories with eclipse constraints

SummaryMinimum time solar sailing trajectories are introduced using a combination of indirect and direct optimal control techniques. Here, large‐scale, multiphase optimal control problems are solved using a pseudospectral collocation technique applied to an orbital debris mitigation concept. These solutions are obtained for realistic sail dimensions, producing multirevolution, Earth‐centered trajectories while accounting for uncontrolled spacecraft dynamics in the eclipse regions. Specifically, minimum time solutions for orbit transfer and phasing maneuvers are obtained using only solar radiation pressure for propulsion and control. First, an optimal primer vector steering history is obtained through numerical approximation. Locally optimal, closed‐form solutions are then implemented based on the primer vector direction, resulting in minimum time satisfaction of desired terminal orbital conditions. In addition, a novel solution strategy is presented for multiphase optimal control problems characterized by uncontrollable dynamics with flexible phase boundaries. The maneuvers presented will be shown to enable efficient orbital debris mitigation for large‐scale debris in geostationary orbits.

ISRO’s Unprecedented Journey to the Moon

Chandrayaan-2, being the sequel of Indian Lunar Mission, differs with Chandrayaan-1 in targeting a specific lunar orbit. The primary ground trajectory design objective is to achieve the desired lunar orbit, from which the lander attempts soft-landing on the pre-selected landing site. The target lunar orbit is designed such that while landing, the Sun elevation rise angle is 6° on the desired site. This is to ensure the lander touchdown is close to sunrise, which in-turn will maximize the lander mission life to one lunar solar day (approximately 14 Earth days). This makes the Chandrayaan-2 trajectory design unique and challenging in itself. The maneuver strategy has been worked out to acquire the specific target lunar orbit. The objective of the powered descent phase is soft-landing over the desired site satisfying all the sensor constraints. This multi-phase lunar landing optimal control problem has been solved for trajectory design of the powered descent phase. Based on this design, the Chandrayaan-2 spacecraft has been launched on 22nd July, 2019 UT and successfully rendezvoused with the Moon on 20th August, 2019 UT and has attempted a soft-landing on 6th September, 2019 UT.

Energy-Efficient Arrival with RTA Constraint for Multirotor eVTOL in Urban Air Mobility

The electric vertical takeoff and landing (eVTOL) aircraft can alleviate transportation congestion on the ground by using three-dimensional airspace efficiently. However, the endurance of lithium-ion polymer (Li-Po) batteries imposes critical constraints on the operational time span of an eVTOL aircraft on urban air mobility (UAM) passenger transportation mission. This research focuses on the formulation of fixed-final-time multiphase optimal control problem with energy consumption as the performance index for a multirotor eVTOL aircraft. The proposed multiphase optimal control problem formulation and the numerical solution enables a multirotor eVTOL aircraft to meet the assigned required time of arrival and achieve an energy-efficient arrival trajectory for a given concept of operation (CONOP), which is a critical enabler for the safe and efficient future eVTOL operations for passenger transportation and cargo delivery. The problem formulation is applied to a UAM passenger transportation use case with EHang 184 eVTOL aircraft, and an Uber-proposed vertiport for five different types of CONOPs. Finally, the energy consumed for all the CONOPs is compared to propose the most energy-efficient CONOP for a multirotor eVTOL on UAM passenger transportation mission. The proposed framework can also be used to address an energy-efficient cargo delivery application in a UAS traffic management context.

Direct Solution of Multi-Objective Optimal Control Problems Applied to Spaceplane Mission Design

This paper presents a novel approach to the solution of multiphase multi-objective optimal control problems. The proposed solution strategy is based on the transcription of the optimal control problem with Finite Elements in Time and the solution of the resulting multi-objective nonlinear programming (MONLP) problem with a memetic strategy that extends the Multi Agent Collaborative Search algorithm. The MONLP problem is reformulated as two nonlinear programming problems: a bilevel and a single-level problem. The bilevel formulation is used to globally explore the search space and generate a well spread set of nondominated decision vectors, whereas the single-level formulation is used to locally converge to Pareto efficient solutions. Within the bilevel formulation, the outer level selects trial decision vectors that satisfy an improvement condition based on Chebyshev weighted norm, whereas the inner level restores the feasibility of the trial vectors generated by the outer level. The single-level refinement implements a Pascoletti-Serafini scalarization of the MONLP problem to optimize the objectives while satisfying the constraints. The approach is applied to the solution of three test cases of increasing complexity: an atmospheric reentry problem, an ascent and abort trajectory scenario, and a three-objective system and trajectory optimization problem for spaceplanes.

Time-Optimal Attitude Scheduling of a Spacecraft Equipped with Reaction Wheels

The time-optimal control problem of a spacecraft equipped with reaction wheels has been studied, in which the spacecraft is constrained to sequentially assume a set of attitudes, whose order is not specified. This attitude scheduling problem has been solved as a multiphase mixed-integer optimal control problem in which binary functions have been introduced to model the choice of the optimal sequence of target attitudes and to enforce the constraint of adopting once and only once each attitude. Given the dynamic model of the spacecraft, the initial and final attitudes, and a set of target attitudes, solving this problem consists in finding the control inputs, the sequence of attitudes with the corresponding passage times, and the resulting trajectory of the spacecraft that minimize the time of the maneuver. The multiphase mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the target attitudes part of the state, then introducing binary variables to discretize the binary functions, and finally applying a fifth-degree Gauss-Lobatto direct collocation method to tackle the dynamic constraints. The resulting problem has been solved using a nonlinear programming-based branch-and-bound algorithm.

Conflict-Free Trajectory Optimization with Target Tracking and Conformance Monitoring

This paper proposes an optimization framework that computes conflict-free optimal trajectories in dense terminal airspace, while continuously monitoring trajectory conformance in an effort to improve predictability. The objective is to allow, as much as possible, continuous vertical trajectory profiles without impacting negatively on airspace capacity. Given automatic dependent surveillance–broadcast intent information, the future state of potential intruder aircraft are predicted, and this nominal trajectory is used as a constraint in the ownship trajectory optimization process. In it, a continuous multiphase optimal control problem is solved, taking into account spatial and temporal constraints. Additionally, a linearized Kalman filter keeps track of the target by estimating the deviations of its actual trajectory from its nominal trajectory, issuing a warning when an appropriate threshold is exceeded. This may be due to unexpected events, biases in the performance and weather models, wrong parameter assumptions, etc. An illustrative example is given, based on a computer simulation of two hypothetical trajectories in the Barcelona terminal maneuvering area. The results show how this framework resolves the problem of uncertainties in the trajectory predictions and results in a more efficient conflict resolution.

Combined Speed and Steering Control in High-Speed Autonomous Ground Vehicles for Obstacle Avoidance Using Model Predictive Control

This paper presents a model predictive control-based obstacle avoidance algorithm for autonomous ground vehicles at high speed in unstructured environments. The novelty of the algorithm is its capability to control the vehicle to avoid obstacles at high speed taking into account dynamical safety constraints through a simultaneous optimization of reference speed and steering angle without a priori knowledge about the environment and without a reference trajectory to follow. Previous work in this specific context optimized only the steering command. In this paper, obstacles are detected using a planar light detection and ranging sensor. A multi-phase optimal control problem is then formulated to simultaneously optimize the reference speed and steering angle within the detection range. Vehicle acceleration capability as a function of speed, as well as stability and handling concerns such as preventing wheel lift-off, are included as constraints in the optimization problem, whereas the cost function is formulated to navigate the vehicle as quickly as possible with smooth control commands. Simulation results show that the proposed algorithm is capable of safely exploiting the dynamic limits of the vehicle while navigating the vehicle through sensed obstacles of different sizes and numbers. It is also shown that the proposed variable speed formulation can significantly improve performance by allowing navigation of obstacle fields that would otherwise not be cleared with steering control alone.

A multiphase optimal control method for multi-train control and scheduling on railway lines

We consider a combined train control and scheduling problem involving multiple trains in a railway line with a predetermined departure/arrival sequence of the trains at stations and meeting points along the line. The problem is formulated as a multiphase optimal control problem while incorporating complex train running conditions (including undulating track, variable speed restrictions, running resistances, speed-dependent maximum tractive/braking forces) and practical train operation constraints on departure/arrival/running/dwell times. Two case studies are conducted. The first case illustrates the control and scheduling problem of two trains in a small artificial network with three nodes, where one train follows and overtakes the other. The second case optimises the control and timetable of a single train in a subway line. The case studies demonstrate that the proposed framework can provide an effective approach in solving the combined train scheduling and control problem for reducing energy consumption in railway operations.

Multiphase Return Trajectory Optimization Based on Hybrid Algorithm

A hybrid trajectory optimization method consisting of Gauss pseudospectral method (GPM) and natural computation algorithm has been developed and utilized to solve multiphase return trajectory optimization problem, where a phase is defined as a subinterval in which the right-hand side of the differential equation is continuous. GPM converts the optimal control problem to a nonlinear programming problem (NLP), which helps to improve calculation accuracy and speed of natural computation algorithm. Through numerical simulations, it is found that the multiphase optimal control problem could be solved perfectly.

Optimal Push Recovery for Periodic Walking Motions

We study how humans performing periodic walking motions react to strong perturbations that are applied in form of pushes from behind. We propose a computational method that allows to generate optimal recovery motions based on a dynamic multi-body model of the human walking process. The assumption is that humans in such a situation would aim to pursue the walking motions while at the same time limiting the effort for the motion. For a given position and velocity in the middle of a periodic walking motion and a given pushing force profile, magnitude and contact point we determine the motion of the human model that minimizes a combined criterion joint torques and a deviation from periodicity. The recovery horizon considered in the multiphase optimal control problem is one step. The optimal control problem is solved using a direct boundary value problem approach based on multiple shooting. We present resulting optimal recovery motions for pushing forces between 150 N and 600 N which all look realistic. The proposed method has potential applications in the computation of push recovery motions for humanoid robots or for elderly humans with physical assistive devices, in each case applied to the respective model.

Optimal Control Based Vertical Trajectory Determination for Continuous Descent Arrival Procedures

Continuous descent arrival (CDA) procedures provide time and fuel savings, and reduce the noise impact of aircraft operations near airports. To date, these procedures have been designed to limit the performance envelopes to achieve trajectory commonality. In this paper, the performance bounds of CDA procedure via multiphase optimal vertical trajectory generation problems with respect to two performance indices—flight time and fuel consumption—have been investigated. The optimal CDA vertical profile is first divided into two segments: the cruise segment before the top of descent (TOD) and the descent segment from the TOD. Then, the second segment is further subdivided based on flap setting and speed constraints. Finally, the resulting multiphase optimal control problem is solved using a pseudospectral method. Two different types of aircraft, B737-500 (light) and B767-400 (heavy), are used in a numerical study, and the optimal CDA trajectories of the two types are compared with a typical vertical navigation (VNAV) CDA trajectory used in CDA applications at several airports. Then, the utility of suboptimal VNAV CDA trajectories that can be created with known flight management system (FMS) VNAV descent modes is explored, and thereby enabling onboard calculation in the FMS via the sensitivity analysis that is performed with various wind conditions.

Fuel and time savings by flying continuous cruise climbs: Estimating the benefit pools for maximum range operations

This paper estimates the benefits, in terms of fuel and time, which continuous climb operations can save during the cruise phase of the flights, assuming maximum range operations. Based on previous works, a multiphase optimal control problem is solved by means of numerical optimization and using accurate aircraft performance data from the manufacturer. Optimal conventional trajectories (subject to current air traffic management practices and constraints) are computed and compared with ideal continuous operations only subject to aircraft performance constraints. Trip fuel and time for both concepts of operations are quantified for two aircraft types (a narrow-body and a wide-body airplane) and a representative set of different trip distances and landing masses. Results show that the continuous cruise phase can lead to fuel savings ranging from 0.5% to 2% for the Airbus A320, while for an Airbus A340 the dispersion is lower and savings lie in between 1% and 2%. Interestingly, trip time is also reduced between 1% and 5%.