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This paper proposes the use of Legendre pseudospectral method (PSM) to obtain the optimal guidance strategy for a two-dimensional interceptor problem. An optimal control problem is formulated that addresses the conflicting objective of minimizing the energy usage, along with minimizing the time taken by missile to capture the target. The PSM-based guidance strategy is compared with other conventional guidance laws such as pure proportional navigation (PPN) guidance law and also evolutionary algorithm inspired differential evolution tuned proportional navigation (DEPN) guidance law. A scheme by which the PSM guidance strategy can be applied online is also included in this paper. The cost function value and the interception time indicates the superiority of the PSM-based guidance strategy.

In this paper, the attitude controller of spacecraft tracks the time-energy near-optimal angular velocity generated by the Legendre Pseudospectral method (LPSM). The near-optimal reference trajectory is obtained offline by applying the LPSM on the nominal spacecraft dynamics, which is without noises and disturbances. The proposed tracking control scheme works online, and along with the tracking of the near-optimal path, it also rejects the external disturbances and noises. The composite control law is developed by combining the sliding mode control (SMC) and the output of a finite-time disturbance observer. The SMC ensures the finite tie results whereas the disturbance observer helps in avoiding the chattering in control. The stability analysis of relative dynamics under the proposed scheme guarantees the convergence of sliding surface and relative states in finite-time. The simulation analysis further illustrates the effective performance of the proposed strategy.

Effective formation reconfiguration of satellite flying is increasingly becoming important. This paper presents an alternative algorithm for satellite formation reconfiguration based on the hybridization of Legendre pseudospectral method (LPM) and co-evolutionary particle swarm optimization (CPSO), which aims to give a fuel-optimal maneuver strategy and avoid collision among satellites. The proposed method is not only able to combine the advantages of two algorithms, but also overcomes the limitations of each algorithm to some extent. More specifically, LPM is used to solve formation reconfiguration with feasible solution in the case of that the distance between any two satellites is far away. The computation time is greatly shortened because there is no collision avoidance constraint. Meanwhile, the CPSO algorithm is applied to enhance solving precision, as well as avoid collision when the satellites stay at a relatively short distance from each other. In order to fully demonstrate the efficiency of proposed method, an array of experiments with comparison to multi-phase LPM are carried out on three satellites. The numerical results verify that proposed method performs well in designing fuel-optimal strategy and ensuring the safety of satellite. It is a feasible and effective algorithm for addressing satellite formation reconfiguration problems.

This paper presents a two-stage cooperative path planner for multiple autonomous underwater vehicles operating in dynamic environment. In case of static environment, global Legendre pseudospectral method is employed for collision-free paths of vehicles for the purpose of minimum time consumption and simultaneous arrival. Moreover, in order to keep the multiple autonomous underwater vehicles safe from collisions on the path segments connecting two adjacent control nodes, an adaptive intermediate knots insertion algorithm is introduced. In the on-line planning stage, the local re-planning strategy aims at avoiding collisions with unexpected dynamic obstacles by two consecutive avoidance maneuvers, and the differential flatness property of autonomous underwater vehicle is utilized, which can help the vehicles react fast enough to avoid moving obstacles.

This paper proposes an optimal integral sliding mode control (ISMC) scheme for attitude regulation of the rigid spacecraft. This control technique is capable of handling inertial matrix uncertainties as well as external disturbances. To incorporate optimality into the robust control law, the ISMC is integrated with Legendre pseudospectral method (LPSM). The minimization of the cost function and constraint handling of the spacecraft is obtained by LPSM, while the ISMC provides disturbance rejection. LPSM is chosen for its relatively high rate of convergence and its capability of solving a wide range of challenging optimal control problems. Theoretical stability analysis of closed loop system using Lyapunov theorem guarantees the convergence of attitude states. A comparative analysis between the proposed LPSM-ISMC and Chebyshev Pseudospectral Method (CPSM) based ISMC, is also presented in this paper. The effectiveness of the proposed robust-optimal control strategy is established through simulation results.

This paper proposes the use of pseudospectral methods to solve the nonlinear trajectory planning problem for free-floating robots. Specifically, three different optimization tools are analyzed. Usi...

The main aim presented in this article is to provide an efficient transferred Legendre pseudospectral method for solving pantograph delay differential equations. At the first step, we transform the problem into a continuous-time optimization problem and then utilize a transferred Legendre pseudospectral method to discretize the problem. By solving this discrete problem, we can attain the pointwise and continuous estimated solutions for the major pantograph delay differential equation. The convergence of method has been considered. Also, numerical experiments are described to show the performance and precision of the presented technique. Moreover, the obtained results are compared with those from other techniques.

This paper develops a robust controller for the attitude tracking of time-energy near-optimal angular velocity obtained by the Legendre Pseudospectral method (LPSM). The information of near-optimal reference trajectory is realized offline by employing the LPSM for the nominal spacecraft model, which is without any uncertainties. Then, the proposed robust controller is employed online to track that near-optimal path while compensating for the inertial uncertainties, external disturbances, and noises under input saturation. The robust scheme is developed by combining the finite-time robust non-linear disturbance observer (RNDO) and the adaptive non-singular fast terminal sliding mode control (NSFTSMC). The estimated output of RNDO directly attenuates lumped disturbance of moderate frequency and thus significantly helps in alleviating the chattering problem. Whereas NSFTSMC ensures the finite-time convergence of relative system states, and adaptive law solves the overestimation problem of controller gains while rejecting high-frequency disturbances. The simulation analysis is carried out to validate the proposed strategy.

This paper discusses the use of optimal control strategies for ascertaining efficient usage of infection eliminating methodologies. The mathematical model of tuberculosis is considered which is incorporated with three control terms. The preventive controls used for containing the spread of tuberculosis, include vaccinations along with treatment control for the latent, susceptible and infected population. By implementing Pseudospectral method on the Tuberculosis model, the original continuous time optimal control problem is converted into its equivalent non-linear programming problem and then solved. On application of the proposed control scheme, a minimization in the intervention cost and a significant reduction in the disease burden is expected. A comparative analysis between the widely used Legendre Pseudospectral method and Chebyshev Pseudospectral method when applied to the mathematical model of tuberculosis is included in this work. The response of the states in absence of any control effort is also included to demonstrate the effect of optimal control strategy on the disease.

The pseudospectra of a linear time-invariant system are the sets in the complex plane consisting of all the roots of the characteristic equation when the system matrices are subjected to all possible perturbations with a given upper bound. The pseudospectral abscissa are defined as the maximum real part of the characteristic roots in the pseudospectra and, therefore, they are for instance important from a robust stability point of view. In this paper we present a numerical method for the computation of the pseudospectral abscissa of retarded delay differential equations with discrete pointwise delays. Our approach is based on the connections between the pseudospectra and the level sets of an appropriately defined complex function. These connections lead us to a bisection algorithm for the computation of the pseudospectral abscissa, where each step relies on checking the presence of imaginary axis eigenvalues of an appropriately defined operator. Because this operator is infinite-dimensional a predictor-corrector approach is taken. In the predictor step the bisection algorithm is applied where the operator is discretized into a matrix, yielding approximations for the pseudospectral abscissa. The effect of the discretization is fully characterized in the paper. In the corrector step, the approximate pseudospectral abscissa are corrected to any given accuracy, by solving a set of nonlinear equations that characterize extreme points in the pseudospectra contours., Comment: arXiv admin note: text overlap with arXiv:2003.03248

A control strategy combining feedforward control and feedback control is presented for the optimal deployment of a spacecraft solar array system with the initial state uncertainty. A dynamic equation of the spacecraft solar array system is established under the assumption that the initial linear momentum and angular momentum of the system are zero. In the design of feedforward control, the dissipation energy of each revolute joint is selected as the performance index of the system. A Legendre pseudospectral method (LPM) is used to transform the optimal control problem into a nonlinear programming problem. Then, a sequential quadratic programming algorithm is used to solve the nonlinear programming problem and offline generate the optimal reference trajectory of the system. In the design of feedback control, the dynamic equation is linearized along the reference trajectory in the presence of initial state errors. A trajectory tracking problem is converted to a two-point boundary value problem based on Pontryagin’s minimum principle. The LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations which can be easily calculated. Then, the closed-loop state feedback control law is designed based on the resulting optimal feedback control and achieves good performance in real time. Numerical simulations demonstrate the feasibility and effectiveness of the proposed control strategy.

The main purpose of this work is to develop an integral pseudospectral scheme for solving integro-differential equations. We provide new pseudospectral integration matrices (PIMs) for the Legendre–Gauss and the flipped Legendre–Gauss–Radau points, respectively, and present an efficient and stable approach to computing the PIMs via the recursive calculation of Legendre integration matrices. Furthermore, we provide a rigorous convergence analysis for the proposed pseudospectral scheme in both L ∞ and L 2 spaces via a linear integral equation, and the spectral rate of convergence is demonstrated by numerical results.

Intelligent planning and accurate execution of connected automated vehicles (CAVs) enable not only improved traffic safety but also better fuel economy. This paper presents two longitudinal speed planning algorithms for fuel-saving driving on highways with varying road slopes. One is designed on the top of the model predictive control (MPC) and the other is called equivalent kinetic-energy and fuel conversion method. The MPC algorithm solves the optimal speed profile in a receding finite horizon with repeated optimization, which is numerically solved by the Legendre pseudospectral method. The latter is designed based on an instantaneous optimization, which considers vehicle kinetic energy an admissible power source, and then minimizes a weighted sum of fuel energy and kinetic energy. This strategy is capable of generating analytical rules to get the economical speed as well as the corresponding commands of the engine, transmission, and brake. The two algorithms are featured by near-global optimization and local optimization, respectively. Their performances in fuel economy and computational load are quantitatively explored and compared in order to distinguish the potential of real implementation in CAVs.

In this paper, we show that the complementarity dynamical systems can be reformulated as optimal control problems. By using this reformulation, we present a pseudospectral scheme to discretize the complementarity dynamical systems. Applying this discretization, the complementarity dynamical system is reduced to a sequence of nonlinear programming problems. Numerical examples and comparison with two other methods are included to demonstrate the capability of the proposed method.

A new trajectory generation for heat load test is proposed based on gauss pseudospectral method within limit range. Firstly, with multiple path constraints and flight task requirements taken into consideration, heat load parameters are introduced into the dynamics equations. In order to solve the problem of generating such a trajectory within limit range rapidly, the dynamics equations have been normalized by Earth related parameters. Secondly, since the gauss pseudospectral method is just employed to solve the discrete nonlinear programming problem, transformations are developed, which can relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. In addtion, another approach of trajectory generation by tracking the given heat rate is also presented. Finally, simulation results with common aero vehicle (CAV-H) show that the trajectories obtained by both methods can well perform the heat load test with high stagnation heating rate and the large total aeroheating amount; meanwhile, gauss pseudospectral method is better than the compared one in the given range. Furthermore, the 3-D trajectory states and control variables, angle of attack and bank, which are generated by gauss pseudospectral method, can change smoothly.

Lunar landing problem has been formulated as desensitized optimal control problem and solved by Legendre Pseudospectral method. The problem has been split into three phases to account for mission constraints, namely the braking with rough navigation stage (from 18km to 7km), attitude hold stage (holding the attitude for 35sec) for a typical mission scenario. In presence of uncertainties, following the open loop reference trajectory using the closed loop linear quadratic regulator contribute greatly in trajectory dispersions. The goal is to desensitize this multi-phase optimal trajectory with reduced error in presence of initial state error, thrust error, Moon's gravity uncertainty. To achieve this, the fuel minimization cost function is augmented with closed loop covariance to generate the open loop reference trajectory. Following this reference trajectory using closed loop linear quadratic regulator, shows significant reduction in landing error. The amount of extra fuel consumed by desensitizing optimal trajectory is less significant when compared to improvement in landing accuracy to meet mission constraints assuring the desensitized optimal control a viable technique for lunar landing trajectory design. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Nonlinear state-delayed optimal control problems have complex nonlinear characters. To solve this complex nonlinear problem, an iterative symplectic pseudospectral method based on quasilinearization techniques, the dual variational principle and pseudospectral methods is proposed in this paper. First, the proposed method transforms the original nonlinear optimal control problem into a series of linear quadratic optimal control problems. Then, a symplectic pseudospectral method is developed to solve these converted linear quadratic state-delayed optimal control problems. Coefficient matrices in the proposed method are sparse and symmetric since the dual variational principle is used, which makes the proposed method highly efficient. Converged numerical solutions with high precision can be obtained after a few iterations due to the benefit of the local pseudospectral method and quasilinearization techniques. In the numerical simulations, other numerical methods were used for comparisons. The numerical simulation results show that the proposed method is highly accurate, efficient and robust.

The main purpose of this work is to provide new fractional pseudospectral methods for solving fractional optimal control problems (FOCPs). We develop differential and integral fractional pseudospectral methods and prove the equivalence between them from the distinctive perspective of Caputo fractional Birkhoff interpolation. As a result, the present work establishes a new unified framework for solving fractional optimal control problems using fractional pseudospectral methods, which can be viewed as an extension of existing frameworks. Furthermore, we provide exact, efficient, and stable approaches to compute the associated fractional pseudospectral differentiation/integration matrices even at millions of Jacobi-type points. Numerical results on two benchmark FOCPs including a fractional bang–bang problem demonstrate the performance of the proposed methods. MOE (Min. of Education, S’pore)

In this paper, Legendre wavelet collocation method is applied for numerical solutions of the fractional-order differential equations subject to multi-point boundary conditions. The explicit formula of fractional integral of a single Legendre wavelet is derived from the definition by means of the shifted Legendre polynomial. The proposed method is very convenient for solving fractional-order multi-point boundary conditions, since the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces equations to those of solving a system of algebraic equations which greatly simplifies the problem. Several numerical examples are solved to demonstrate the validity and applicability of the presented method.

An adaptive symplectic pseudospectral method based on the dual variational principle is proposed and is successfully applied to solving nonlinear optimal control problems in this paper. The proposed method satisfies the first order necessary conditions of continuous optimal control problems, also the symplectic property of the original continuous Hamiltonian system is preserved. The original optimal control problem is transferred into a set of nonlinear equations which can be solved easily by Newton–Raphson iterations, and the Jacobian matrix is found to be sparse and symmetric. The proposed method, on one hand, exhibits exponent convergence rates when the number of collocation points are increasing with the fixed number of sub-intervals; on the other hand, exhibits linear convergence rates when the number of sub-intervals is increasing with the fixed number of collocation points. Furthermore, combining with the hp method based on the residual error of dynamic constraints, the proposed method can achieve given precisions in a few iterations. Five examples highlight the high precision and high computational efficiency of the proposed method.

The main purpose of this work is to provide new fractional pseudospectral schemes with equivalence for solving fractional differential equations (FDEs). We develop differential and integral fractional pseudospectral schemes, and prove their equivalence from the distinctive perspective of the Caputo fractional Birkhoff interpolation with zero initial condition. We introduce the notion of fractional pseudospectral differentiation/integration matrices in terms of weighted Lagrange interpolating functions, and provide exact, efficient, and stable approaches to computing these matrices via the Lagrange interpolation and the Jacobi--Gauss quadrature. Numerical results on benchmark FDEs demonstrate the performance of the proposed pseudospectral schemes.

Pathogen suppressing and immune boosting drugs, are used as control measure for regulating the infections due to water borne disease. A mathematical SIWR (susceptible-infectious-recovered) model is considered in which the $W$ compartment tracks the concentration of pathogen in water. An optimal scheme for disease control is proposed in this paper by converting the original optimal control problem to a nonlinear programming problem (NLP). This transformation is done by Legendre pseudospectral method (LPSM) and the resulting NLP problem is solved. The objective of minimizing the count of infected population, along with drug dosage cost is addressed. The simulation results reported illustrates the effectiveness of the proposed optimal drug use strategy.

An optimal control scheme for attitude control of rigid body spacecraft is proposed in this paper. The dynamic optimal control problem is first converted into a nonlinear programming problem (NLP) by Legendre pseudospectral method (LPM) and then this NLP problem is solved. The time as well as energy is optimized for the rigid body spacecraft using LPM. The re-orientation problem of the rigid spacecraft is solved both as a regulation problem and then as a tracking problem. The simulation is carried out using a Matlab© toolbox called DIDO and the simulation results illustrates the desired performances of spacecraft attitude regulation and tracking.

To fulfil the high precision requirements of the manoeuvre imaging mode of an agile satellite and solve the shortcomings of existing methods, a segmented attitude planning method based on the Pseudospectral method is proposed. First, a state equation was built by integrating the attitude constraint, time window and other restrictions. Furthermore, to guarantee the accuracy and stability of objects, a segmented strategy of an attitude trajectory approaching the imaging target using a constant angular velocity was proposed. The segmented time, three-axis position and velocity were also determined. Second, taking the global performance into account, the first section of the attitude trajectory was planned by the Legendre Pseudospectral method in order to acquire the optimal state variables and control variables. The second section of the attitude trajectory was then obtained by the constant angular velocity planning method. Next, in order to track the planning trajectory efficiently, a PD controller was designed that was suitable for a segmented planning method. High precision control parameters were obtained by the optimized method. Finally, numerical simulation results show that the proposed segmented attitude planning method successfully meets the high precision requirements of the manoeuvre imaging mode, and the feasibility and effectiveness of this method is verified.

In this paper, Galerkin method has been introduced using Legendre polynomials as basis functions over the interval \([-1, 1]\) to solve the eighth-order linear boundary value problems with two-point boundary conditions. Legendre Galerkin method is an effective tool in numerically solving such problems. The performance and applicability of the method is illustrated through some examples that reveal the method presents much better results. The obtained numerical results are convincing and very close to the analytical ones.

This paper presents an efficient path-planner based on a hybrid optimization algorithm for autonomous underwater vehicles (AUVs) operating in cluttered and uncertain environments. The algorithm integrates particle swarm optimization (PSO) algorithm with Legendre pseudospectral method (LPM), which is named as hybrid PSO-LPM algorithm. PSO is first employed as an initialization generator with its strong global searching ability and robustness to random initial values. Then, the searching algorithm is switched to LPM with the initialization obtained by PSO algorithm to accelerate the following searching process. The flatness property of AUV is also utilized to reduce the computational cost for planning, making the optimization algorithm valid for local re-planning to efficiently solve the collision avoidance problem. Simulation results show that the hybrid PSO-LPM algorithm is able to find a better trajectory than standard PSO algorithm and with the re-planning scheme it also succeeds in real-time collision avoidance from both static obstacles and moving obstacles with varying levels of position uncertainty. Finally, 100-run Monte Carlo simulations are carried out to check robustness of the proposed re-planner. The results demonstrate that the hybrid optimization algorithm is robust to random initializations and it is effective and efficient for collision-free path planning.

This paper studies the fuel-optimal cruising strategies of parallel hybrid electric vehicles (HEVs) and their underlying mechanisms. To achieve fuel-optimal operations, a discontinuous nonlinear optimal control problem is formulated and solved using the Legendre pseudospectral method and the knotting technique. Three optimal cruising strategies in free/fixed-speed cruising scenarios are proposed: vehicle speed pulse-and-glide strategy, state-of-charge (SoC) pulse-and-glide (PnG) strategy, and constant-speed strategy. The performance and optimal behavior of the engine and the motor are presented, and their fuel-saving mechanisms are explained. Finally, two principles to compromise between fuel economy and ride comfort are proposed and studied.

In this paper, we consider nonlinear optimal control problems involving piecewise constant delay functions. A discretization of the Bolza problem by an adaptive Legendre pseudospectral method is considered. The method is based on composite pseudospectral method using the well-known Legendre---Gauss---Lobatto collocation points. In this approach, the main problem is converted into a mathematical optimization problem, whose solution is much more easier than that of the original one. It is shown that, under a simple set of conditions, the operations of discretization and dualization are commutative. The method is used to solve an optimal control problem with continuous variable time lag.

In this article, a direct pseudospectral method based on Lagrange interpolating functions with fractional power terms is used to solve the fractional optimal control problem. As most applied fractional problems have solutions in terms of the fractional power, using appropriate characteristic nodal-based functions with suitable power leads to a more accurate pseudospectral approximation of the solution. The Lagrange interpolating functions and their fractional derivatives belong to the Muntz space; such functions are employed to show that a relationship exists between the Karush---Kukn---Tucker conditions associated with nonlinear programming and the first optimal necessary conditions. Furthermore, the convergence of the method is investigated. The obtained numerical results are an indication of the behavior of the algorithm.

The main purpose of this work is to provide multiple-interval integral Gegenbauer pseudospectral methods for solving optimal control problems. The latest developed single-interval integral Gauss/(flipped Radau) pseudospectral methods can be viewed as special cases of the proposed methods. We present an exact and efficient approach to compute the mesh pseudospectral integration matrices for the Gegenbauer–Gauss and flipped Gegenbauer–Gauss–Radau points. Numerical results on benchmark optimal control problems confirm the ability of the proposed methods to obtain highly accurate solutions.

This paper focuses on a method of changing the orbit of an asteroid by attaching a solar sail to the asteroid. First, the dynamic model of the tethered system is derived. Legendre pseudospectral method is then used to discretize the system, and the sequence of two quadratic programming is utilized to obtain the optimal control law. Simulation results show that the tethered solar sail can efficiently change the asteroid’s orbit. Moreover, the problem of the tether twining around the asteroid caused by the relative orbit motion between the solar sail and the asteroid can be avoided. Finally, the effectiveness of altering an asteroid’s orbit by different solar sails is analyzed. Simulation results show that when the area of the solar sail is 10 6 m 2 , the asteroid can be deflected at 1.227 × 10 8 m by the solar sail after about 20 years, which is better than the effect of a gravitational tractor.

This paper presents an optimal integral sliding mode control method based on a pseudospectral method for a class of affine systems with state and control constraints. First, a general form of an integral sliding mode is presented. Integral sliding mode control cannot deal with the problem of states and control constraints, nor can it satisfy the minimization of the cost function. The pseudospectral method has a high convergence speed and performs well in solving optimal control problems with general performance index, endpoint conditions and path constraints. In consideration of these advantages, an optimal integral sliding mode controller is determined by the pseudospectral method. Then, the stability analysis of optimal pseudospectral sliding mode method is discussed. Finally, numerical simulations show the effectiveness of the proposed method. An application example, consisting of an overhead crane system, is investigated to demonstrate the effectiveness and robustness of the proposed technique.

In the manuscript, a pseudospectral method is developed for approximate and efficient solution of nonlinear singular Lane–Emden–Fowler initial and boundary value problems arising in astrophysics. In the proposed method, the Gauss pseudospectral method is utilized to reduce the problem to the solution of a system of algebraic equations. Furthermore, the Gauss pseudospectral method is developed for finding the first zero of the solution of this equation that gives the radius of the star, in which the numerous properties of the star such as mass, central pressure, and binding energy can be computed through their relations to this solution. The main advantage of the proposed method is that good results are obtained even by using a small number of discretization points and the rate of convergence is high. The accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2015 John Wiley & Sons, Ltd.