Neighboring Optimal Control Research Articles

Variable-time-domain neighboring optimal guidance and attitude control of low-thrust lunar orbit transfers

Lunar orbit dynamics and transfers at low altitudes are subject to considerable perturbations related to the gravitational harmonics associated with the irregular lunar mass distribution. This research proposes the combination of two techniques applied to low-thrust lunar orbit transfers, i.e. (i) the variable-time-domain neighboring optimal guidance (VTD-NOG), and (ii) a proportional-derivative attitude control algorithm based on rotation matrices (PD-RM). VTD-NOG belongs to the class of feedback implicit guidance approaches, aimed at maintaining the spacecraft sufficiently close to the reference trajectory. This is an optimal path that satisfies the second-order sufficient conditions for optimality. A fundamental original feature of VTD-NOG is the use of a normalized time scale, with the favorable consequence that the gain matrices remain finite for the entire time of flight. VTD-NOG identifies the trajectory corrections by assuming the thrust direction as the control input. Because the thrust direction is fixed with respect to the spacecraft, VTD-NOG generates the desired orientation pursued by the attitude control system. A proportional-derivative approach using rotation matrices (PD-RM) is employed in order to drive the actual spacecraft orientation toward the desired one. Reaction wheels are considered as the actuators that perform attitude control. Extensive Monte Carlo simulations are performed, in the presence of nonnominal flight conditions related to (i) lunar gravitational harmonics, (ii) gravitational pull of the Earth and the Sun as third bodies, (iii) unpredictable propulsive fluctuations, and (iv) errors on initial attitude. The numerical results unequivocally demonstrate that the joint use of VTD-NOG and PD-RM control represents an accurate and effective methodology for guidance and control of low-thrust lunar orbit transfers.

Near space interceptor on-line correction research based on predictive intercept point

Aiming at the problem of large maneuvering range and difficult target detection of hypersonic target in the near space, according to the maneuvering range of predictive intercept point (PIP), this paper proposes a method to select different correction algorithms. First, The Gauss pseudospectral method is used to solve the trajectory optimization problem and the off-line optimal trajectory is generated. Then, aiming at the situation that the range of predictive intercept point is small, based on neighboring optimal control, the on-line correction method is used to find out the optimal trajectory near the off-line optimal trajectory. Finally, when the interceptor is located in a high airspace, aiming at the situation that the range of target motion is wide and the range of predictive intercept point is large, a direct force correction method based on velocity gain guidance is adopted. Simulation results verify the effectiveness of the proposed methods.

Neighboring optimal control for open-time multiburn orbital transfers

While a parametric approach has been proposed recently to establish the neighboring optimal control (NOC) for multiburn orbital transfer problems with fixed final time, it is not applicable once the final time is free. Indeed, it is quite common for orbital transfer missions to leave their final time free. In this paper, we continue developing the parametric approach and extend it to establishing the NOC for such problems with free final time. The key idea is to construct a new parameterized family of extremals such that every neighboring extremal around a nominal one is totally determined by its final time and a vector. Then, the NOC for free-time scenarios is readily established through deriving the Taylor expansion of the parameterized neighboring extremals; as a by-product, some prerequisite conditions, fundamental for establishing the NOC, are formulated. Moreover, a scheme for performing the NOC is proposed accordingly. Finally, to illustrate the improvement of this paper, a free-time multiburn orbital transfer problem is computed and some Monte Carlo campaigns are tested in the presence of various perturbations, showing that the final condition errors are significantly reduced by employing the NOC developed in the paper.

High-fidelity quantum state preparation using neighboring optimal control

We present an approach to single-shot high-fidelity preparation of an $n$-qubit state based on neighboring optimal control theory. This represents a new application of the neighboring optimal control formalism which was originally developed to produce single-shot high-fidelity quantum gates. To illustrate the approach, and to provide a proof-of-principle, we use it to prepare the two qubit Bell state $|\beta_{01}\rangle = (1/\sqrt{2})\left[\, |01\rangle + |10\rangle\,\right]$ with an error probability $\epsilon\sim 10^{-6}$ ($10^{-5}$) for ideal (non-ideal) control. Using standard methods in the literature, these high-fidelity Bell states can be leveraged to fault-tolerantly prepare the logical state $|\overline{\beta}_{01}\rangle$.

Variable-Time-Domain Online Neighboring Optimal Trajectory Modification for Hypersonic Interceptors

The predicted impact point (PIP) of hypersonic interception changes continually; therefore the midcourse guidance law must have the ability of online trajectory optimization. In this paper, an online trajectory generation algorithm is designed based on neighboring optimal control (NOC) theory and improved indirect Radau pseudospectral method (IRPM). A trajectory optimization model is designed according to the features of operations in near space. Two-point boundary value problems (TPBVPs) are obtained based on NOC theory. The second-order linear form of transversality conditions is deduced backward to express the modifications of terminal states, costates, and flight time in terms of current state errors and terminal constraints modifications. By treating the current states and the optimal costates modifications as initial constraints and perturbations, the feedback control variables are obtained based on improved IRPM and nominal trajectory information. The simulation results show that when the changes of terminal constraints are not relatively large, this method can generate a modified trajectory effectively with high precision of terminal modifications. The design concept can provide a reference for the design of the online trajectory generation system of hypersonic vehicles.

Neighboring optimal control for fixed-time multi-burn orbital transfers

Unlike the classical variational method to design the neighboring optimal control (NOC) by minimizing the second variation of cost functional, this paper, from a geometric point of view, presents a parametric approach to establish the NOC for the problem of fixed-time multi-burn orbital transfers. The key idea of the proposed approach is to construct a parameterized family of neighboring extremals around a reference one. By analyzing the canonical projection of the parameterized family, a geometric interpretation is given to show that, even though the gain matrix for the NOC is not explosive, it is impossible to construct the NOC for the multi-burn problem if conjugate points occur at switching times. Then, through deriving the Taylor series expansion of the parameterized extremals, the neighboring optimal feedback not only on thrust direction but also on switching times is readily devised and the procedure for computing the gain matrix is presented. Moreover, a scheme for performing the NOC is proposed such that the typical singularity when approaching the final time is avoided. Finally, the development of this paper is illustrated by two optimal multi-burn orbital transfer problems.

Improving quantum gate performance through neighboring optimal control

Successful implementation of a fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold ${P}_{a}$ exists for any quantum gate that is to be used in such a computation. Specifically, the error probability ${P}_{e}$ for such a gate must fall below the accuracy threshold: ${P}_{e}<{P}_{a}$. Estimates of ${P}_{a}$ vary widely, although ${P}_{a}\ensuremath{\sim}{10}^{\ensuremath{-}4}$ has emerged as a challenging target for hardware designers. In this paper, we present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. We illustrate this approach by applying it to all gates in a universal set of quantum gates produced using nonadiabatic rapid passage that has appeared in the literature. Performance improvements are substantial, both for ideal and nonideal controls. Under suitable conditions detailed in this paper, all gate error probabilities fall well below the target threshold of ${10}^{\ensuremath{-}4}$.

RFID-Based Mobile Robot Trajectory Tracking and Point Stabilization Through On-line Neighboring Optimal Control

In this manuscript, we propose an on-line trajectory-tracking algorithm for nonholonomic Differential-Drive Mobile Robots (DDMRs) in the presence of possibly large parametric and measurement uncertainties. Most mobile robot tracking techniques that depend on reference RF beacons rely on approximating line-of-sight (LOS) distances between these beacons and the robot. The approximation of LOS is mostly performed using Received Signal Strength (RSS) measurements of signals propagating between the robot and RF beacons. However, an accurate mapping between RSS measurements and LOS distance remains a significant challenge and is almost impossible to achieve in an indoor reverberant environment. This paper contributes to the development of a neighboring optimal control strategy where the two major control tasks, trajectory tracking and point stabilization, are solved and treated as a unified manner using RSS measurements emitted from Radio Frequency IDentification (RFID) tags. The proposed control scheme is divided into two cascaded phases. The first phase provides the robot's nominal control inputs (speeds) and its trajectory using full-state feedback. In the second phase, we design the neighboring optimal controller, where RSS measurements are used to better estimate the robot's pose by employing an optimal filter. Simulation and experimental results are presented to demonstrate the performance of the proposed optimal feedback controller for solving the stabilization and trajectory tracking problems using a DDMR.

Analytical mathematical feedback guidance scheme for low-thrust orbital plane change maneuvers

This paper which is based on neighboring optimal control investigates the application of an analytical feedback guidance scheme for the problems of low-thrust orbital transfer. Neighboring optimal control is one of the methods that has been proposed for the closed-loop optimal guidance. This method focuses on expanding the cost function to second order and examining the robustness of the system by the feedback law gains. At first, the open-loop analytical guidance policy is considered as the optimal thrust steering program that will transfer the vehicle from the inclined low earth orbits to the high earth orbits. Secondly, proper feedback optimal guidance laws are analytically obtained to maintain the trajectories around their optimum with assessed disturbances. The proposed guidance scheme is distinguished for two desired performance indices as minimum-time and minimum-effort. Finally performance indices are compared and the best policy is obtained regarding the robustness of the two performance indices against disturbances.

Methods for Computing Minimum-Time Paths in Strong Winds

This paper describes methods for computing minimum-time flight paths at high altitudes in the presence of strong horizontal wind s. The first method shows how to calculate nonlinear feedback (“dynamic programming”) solutions for minimum-time flight paths. To do this the Zermelo Problem for arbitrary winds is extended from a flat earth model to a spherical earth model as a two-state problem (latit ude and longitude) with one control (heading angle). Many minimum-time paths are calculated backwards from New York JFK Airport and San Francisco International Airport to points in the continental U.S. Then the optimal heading angle and the minimum-time can be found by interpolation given the current latitude and longitude using an interpolant created by Delaunay triangulation. The second method is based on an analytical neighboring optimal control solution that computes neighboring optimal heading commands as a function of the winds along a nominal flight path. As an example, minimum-time paths to New York and San Francisco are determined for the actual jet stream winds on 14 February 2001 at 36 kft.

Robust Nonlinear Optimal Solution to the Lunar Landing Guidance by Using Neighboring Optimal Control

A closed-loop time-optimal control strategy for the highly nonlinear problem of the lunar landing mission by using the perturbation technique is developed in this study. The first part of the study considers analytical solution for an optimal control policy of variable mass spacecraft, while it descents on the surface of the moon in the variable gravitational field of it. To validate the accuracy of perturbation solution, a numerical approach based on steepest descent method is employed. The second part considers analytical derivation of an optimal feedback guidance solution by employing the neighboring optimal control (NOC) law when effects of imperfection in the dynamic model or disturbing noises have been taken into account. The technique of NOC produces time-varying feedback gains that minimize the performance index to the second order for perturbations from a nominal optimal path. The robustness of the designed NOC law is examined with applying sinusoidal noises. From the study of the simulation results, it may be concluded that the developed optimal guidance laws may be used in real world spacecraft applications.

On-line robust trajectory generation on approach and landing for reusable launch vehicles

A major objective of next generation reusable launch vehicle (RLV) programs includes significant improvements in vehicle safety, reliability, and operational costs. In this paper, novel approaches that can deliver an RLV to its landing site safely and reliably are proposed. Trajectory generation on approach/landing (A&L) for RLVs using motion primitives (MPs) and neighboring optimal control (NOC) is first discussed. In this stage, the proposed trajectory generation approach is based on an MP scheme that consists of trims and maneuvers. From an initial point to a given touchdown point, all feasible trajectories that satisfy certain constraints are generated and saved into a trajectory database. An optimal trajectory can then be found off-line by using Dijkstra’s algorithm. If a vehicle failure occurs, perturbations are imposed on the initial states of the off-line optimal trajectory, and it is reshaped into a neighboring feasible trajectory on-line by using an NOC approach. If the perturbations are small enough, a neighboring feasible trajectory existence theorem (NFTET) is then investigated and its proof is provided as well. The approach given in the NFTET shows that a vehicle with stuck effectors can be recovered from failures in real time. However, when the perturbations become large, for example, in severe failure scenarios, the NFTET is no longer applicable and often the vehicle cannot be recovered from such failures. A new method is then used to deal with this situation. The NFTET is now extended to the trajectory robustness theorem (TRT). According to the TRT and its proof, a robustifying term is introduced to compensate for the effects of the linear approximation in the NFTET. The upper bounds with respect to input deviation are adaptively adjusted to eliminate their uncertainty. In order to obtain best performance, σ -modification is employed. The simulation results verify the excellent robust performance of this method.

Determination of Nonlinear Optimal Feedback Law for Satellite Injection Problem Using Neighboring Optimal Control

An optimal trajectory design of a nonlinear satellite injection problem for transfer to a final target orbit by minimizing the time was investigated. Indeed, this design was an exact solution to the nonlinear two-point boundary value problem which determined optimal control history as well as optimal state trajectories in the open-loop form. Furthermore, the obtained optimal guidance strategy was exerted in the closed-loop form against the environment disturbances using neighboring optimal control method in the exact solution. Neighboring Optimal Control (NOC) law could produce time-variant feedback gains minimizing the performance measure to second order for perturbations from a nominal optimal path. Generally, this law was a function of perturbations appeared in the states and constraints and could be computed utilizing the backward sweep method. The simulation results indicated that the presented methodology was successfully utilized in the real world applications with good robustness to each noise or disturbance produced in each state variable.

Startup of a distillation column using nonlinear analytical model predictive control

We discuss the application of nonlinear analytical model predictive control to distillation column startup between the time when the trays are hydraulically sealed and the time steady-state operation is reached. A novel startup method is proposed whereby the liquid feed is initially used as reflux to seal the trays by establishing the plate holdups. The control algorithm utilizes a nonlinear analytical process model to generate an optimal control vector trajectory, which is implemented by cascade controllers. The analytical model, based on material and energy balances, additionally utilizes the secondary process measurements resulting in improved controller performance. As the steady state is approached, a neighboring optimal control signal is superimposed on the output from the supervisory level controller to eliminate fast disturbances entering the column. In the hierarchical control structure, the adaptive online optimizer resides in the PC/AT which is networked with PC/XTs for neighboring optimal and cascade controls. A PID type controller with periodic update of set points and a manual startup have been compared with the present algorithm. Experimental results showed a significant reduction in time as well as cost in the case of the proposed scheme

On singular arcs and surfaces in a class of quadratic minimization problems

A class of quadratic minimization problems is studied whose optimal control functions are partially singular. An explicit expression is obtained for the ( n − 1)-dimensional singular surface and it is shown that the optimal value function is twice continuously differentiable across this surface; this allows the piecing together of known sufficiency conditions for totally singular and totally bang-bang arcs to obtain sufficient conditions for optimality for this class of partially singular problems. A neighboring optimal feedback control law is suggested by these results. The closing sections of the paper are concerned with sufficient conditions for nonexistence of optimal singular controls in quadratic minimization problems.

A neighboring optimal feedback control scheme for systems using discontinuous control

The calculation and implementation of the neighboring optimal feedback control law for multiinput, nonlinear dynamical systems, using discontinuous control, is the subject of this paper. The concept of neighboring optimal feedback control of systems with continuous, unbounded control functions has been investigated by others. The differentiating features between this class of problems and that considered here are the control discontinuities and the inherent system uncontrollability during the latter stages of the control-law operating time.

Optimal control of multivariable systems with pure time delays

A brief review of the development of the theory and computational approaches for the optimal control of multivariable systems having pure time delays is given. Following this is a detailed discussion of the optimal feedback control of both linear and non-linear systems having pure time delays. A neighboring optimal feedback control scheme is developed for non-linear problems, and a chemical reactor control problem is worked to illustrate the performance of the controller.