Search Results (11)

This paper introduces a novel closed-loop optimal controller that integrates the Extreme Theory of Functional Connections (X-TFC) with receding horizon control (RHC), referred to as X-TFC-RHC. The controller reformulates a sequence of linearized or quasi-linearized optimal control problems into two-point boundary value problems (TPBVPs) using the indirect method of optimal control. X-TFC then solves each TPBVP by approximating the solution with constrained expressions. These expressions consist of radial basis function neural networks (RBFNNs) and terms that satisfy the TPBVP constraints analytically. The RBFNNs are initialized offline using a particle swarm optimizer, which enables X-TFC to solve the TPBVPs efficiently online during each RHC iteration. The effectiveness of X-TFC-RHC is demonstrated through several aerospace guidance applications, which highlight its accuracy and computational efficiency in executing the RHC process. The proposed approach is also compared with state-of-the-art indirect pseudospectral methods and the traditional backward sweep method. Full article

This paper introduces a novel Shifted Gegenbauer Pseudospectral (SGPS) method for approximating Caputo fractional derivatives (FDs) of an arbitrary positive order. The method employs a strategic variable transformation to express the Caputo FD as a scaled integral of the mth-derivative of the Lagrange interpolating polynomial, thereby mitigating singularities and improving numerical stability. Key innovations include the use of shifted Gegenbauer (SG) polynomials to link mth-derivatives with lower-degree polynomials for precise integration via SG quadratures. The developed fractional SG integration matrix (FSGIM) enables efficient, pre-computable Caputo FD computations through matrix–vector multiplications. Unlike Chebyshev or wavelet-based approaches, the SGPS method offers tunable clustering and employs SG quadratures in barycentric forms for optimal accuracy. It also demonstrates exponential convergence, achieving superior accuracy in solving Caputo fractional two-point boundary value problems (TPBVPs) of the Bagley–Torvik type. The method unifies interpolation and integration within a single SG polynomial framework and is extensible to multidimensional fractional problems. Full article

The achievement of full autonomy in Unmanned Aerial Vehicles (UAVs) is significantly dependent on effective motion planning. Specifically, it is crucial to plan collision-free trajectories for smooth transitions from initial to final configurations. However, finding a solution executable by the actual system adds complexity: the planned motion must be dynamically feasible. This involves meeting rigorous criteria, including vehicle dynamics, input constraints, and state constraints. This work addresses optimal kinodynamic motion planning for UAVs in the presence of obstacles by employing a hybrid technique instead of conventional search-based or direct trajectory optimization approaches. This technique involves precomputing a library of motion primitives by solving several Two-Point-Boundary-Value Problems (TPBVP) offline. This library is then repeatedly used online within a graph-search framework. Moreover, to make the method computationally tractable, continuity between consecutive motion primitives is enforced only on a subset of the state variables. This approach is compared with a state-of-the-art quadrotor-tailored search-based approach, which generates motion primitives online through control input discretization and forward propagation of the dynamic equations of a simplified model. The effectiveness of both methods is assessed through simulations and real-world experiments, demonstrating their ability to generate resolution-complete, resolution-optimal, collision-free, and dynamically feasible trajectories. Finally, a comparative analysis highlights the advantages, disadvantages, and optimal usage scenarios for each approach. Full article

This manuscript introduces the first hp-adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. The constrained expression utilized in this work is composed of two parts. The first part consists of Chebyshev orthogonal polynomials, which conform to the solution of differentiation variables. The second part is a summation of products between switching and projection functionals, which satisfy the boundary constraints. The mesh refinement algorithm relies on the truncation error of the constrained expressions to determine the ideal number of basis functions within a segment’s polynomials. Whether to increase the number of basis functions in a segment or divide it is determined by the decay rate of the truncation error. The results show that the proposed algorithm is capable of solving hypersensitive TPBVPs more accurately than MATLAB R2021b’s bvp4c routine and is much better than the standard TFC method that uses global constrained expressions. The proposed algorithm’s main flaw is its long runtime due to the numerical approximation of the Jacobians. Full article

The transfer between two circular, coplanar Keplerian orbits of a spacecraft equipped with a continuous thrust propulsion system is usually studied in an optimal framework by maximizing a given performance index. Using an indirect approach, the optimal trajectory and the maximum value of the performance index are obtained by numerically solving a two-point boundary value problem (TPBVP). In this context, the computation time required by the numerical solution of the TPBVP depends on the guess of unknown initial costates. The aim of this paper is to describe an analytical procedure to accurately approximate the initial costate variables in a coplanar, circle-to-circle, minimum-time transfer. In particular, this method considers a freely steerable propulsive acceleration vector, whose magnitude varies over a finite range with a sufficiently low maximum value. The effectiveness of the analytical method is tested in a set of both geocentric and heliocentric (simplified) mission scenarios, which model the classical LEO-GEO or interplanetary transfers toward Venus, Mars, Jupiter, and comet 29P/Schwassmann–Wachmann 1. Full article

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower temperature leaves the tank through another pipe. We propose a one-dimensional mathematical model that consists of a nonlinear PDE for the temperature along the solid body, coupled to a linear ODE for the temperature in the tank, the boundary and the initial conditions. All equations are converted into a dimensionless form reducing the input parameters to three dimensionless numbers and a dimensionless function. A steady-state analysis is performed. To solve the transient problem, a nontrivial numerical approach is proposed whereby the differential equations are first discretized in time. This reduces the problem to a sequence of nonlinear two-point boundary value problems (TPBVP) and a sequence of linear algebraic equations coupled to it. We show that knowing the temperature in the system at time level n − 1 allows us to decouple the TPBVP and the corresponding algebraic equation at time level n. Thus, starting from the initial conditions, the equations are decoupled and solved sequentially. The TPBVPs are solved by FDM with the Newtonian method. Full article

This article is an independently written continuation of an earlier study with the same title [Mathematics, 2022, 10, 1225] on the Newton Process (NP). This process is applied to solve nonlinear equations. The complementing features are: the smallness of the initial approximation is expressed explicitly in turns of the Lipschitz or Hölder constants and the convergence order $1+p$ is shown for $p\in (0,1].$ The first feature becomes attainable by further simplifying proofs of convergence criteria. The second feature is possible by choosing a bit larger upper bound on the smallness of the initial approximation. This way, the completed convergence analysis is finer and can replace the classical one by Kantorovich and others. A two-point boundary value problem (TPBVP) is solved to complement this article. Full article

In this study, we analyzed the imaging maneuver time, retargeting maneuver time, and attitude maneuvering characteristics in the imaging section (Phase 1) and retargeting maneuver section (Phase 2) when taking multiple-target images in squint spotlight mode in a single pass of a passive SAR satellite. In particular, the synthetic aperture time and attitude maneuvering characteristics in the staring and sliding spotlight modes that can image the wider swath width while maintaining high resolution were compared and analyzed. In the sliding spotlight mode, the rotation center was located below the ground surface when the satellite was maneuvering towards the target. Steering and sliding maneuvers were performed when targeting, and the synthetic aperture time of the sliding spotlight was longer than that of the staring spotlight because overlapping imaging was performed on the point target. The satellite maneuvering during imaging can be considered as a time-fixed problem, because it was performed within synthetic aperture time according to resolution, incidence angle, swath width, etc., by minimizing the Doppler centroid variation. In order to optimize the retargeting maneuver time, an optimal analysis of the attitude maneuvering was carried out and the validity of the optimal analysis algorithm was confirmed. Finally, the scenario was analyzed by assuming a problem of imaging four targets with 5 × 5 km swath width in a 20 km × 20 km densely populated area. It was confirmed that if a squint angle of ±12 degrees is provided in a single pass, four high resolution images of 5 km × 5 km can be imaged in the sliding spotlight mode. Full article

In this work, we introduce Pontryagin Neural Networks (PoNNs) and employ them to learn the optimal control actions for unconstrained and constrained optimal intercept problems. PoNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed for tackling optimal control problems via the Pontryagin Minimum Principle (PMP) application (e.g., indirect method). The PMP provides first-order necessary optimality conditions, which result in a Two-Point Boundary Value Problem (TPBVP). More precisely, PoNNs learn the optimal control actions from the unknown solutions of the arising TPBVP, modeling them with Neural Networks (NNs). The characteristic feature of PoNNs is the use of PINNs combined with a functional interpolation technique, named the Theory of Functional Connections (TFC), which forms the so-called PINN-TFC based frameworks. According to these frameworks, the unknown solutions are modeled via the TFC’s constrained expressions using NNs as free functions. The results show that PoNNs can be successfully applied to learn optimal controls for the class of optimal intercept problems considered in this paper. Full article

The two dimensional gravity turn problem is addressed allowing for the effects of variable rocket mass due to propellant consumption, thrust and thrust vector angle, lift and drag forces at an angle-of-attack and atmospheric mass density varying with altitude; Coriolis and centrifugal forces are neglected. Three distinct analytical solutions are obtained for constant: propellant flow rate, thrust, thrust vector angle, angle-of-attack and acceleration of gravity; the lift and drag are assumed to be proportional to the square of velocity, and the mass density is assumed to decrease exponentially with altitude. The method III uses power series of time for the horizontal (downrange) and vertical (altitude) coordinates; the method II replaces the altitude as variable by the atmospheric mass density and method I by its inverse. Thus the three solutions have distinct properties, e.g., I and III converge best close to lift-off and II close to burn-out. The three solutions: I, II, III, can be applied in isolation (or matched in combination) to the single-point boundary-value problem (SPBVP) of finding the trajectory with given initial conditions at launch. They can also be used as pairs in six distinct ways (I + II, I + III, II + III or reverse orders) to solve the two-point boundary-value problem (TPBVP), viz.: from given conditions at launch achieve one (not more) specified condition at burn-out, e.g., ã desired horizontal velocity for payload release. Each of the six distinct combinations of methods of addressing the TPBVP shares three features: (i) it can determine if there is a solution, viz. if the rocket has enough performance to reach the desired burn-out condition; (ii) if the desired burn-out condition is achievable it can calculate the complete trajectory from launch to burn-out; (iii) it can determine the range of achievable burn-out conditions, e.g., the minimum and maximum possible horizontal velocity at burn-out for given initial conditions at launch. Full article

Pontryagin’s Minimum Principle (PMP) has a significant computational advantage over dynamic programming for energy management issues of hybrid electric vehicles. However, minimizing the total energy consumption for a plug-in hybrid electric vehicle based on PMP is not always a two-point boundary value problem (TPBVP), as the optimal solution of a powering mode will be either a pure-electric driving mode or a hybrid discharging mode, depending on the trip distance. In this paper, based on a plug-in hybrid electric truck (PHET) equipped with an automatic mechanical transmission (AMT), we propose an integrated control strategy to flexibly identify the optimal powering mode in accordance with different trip lengths, where an electric-only-mode decision module is incorporated into the TPBVP by judging the auxiliary power unit state and the final battery state-of-charge (SOC) level. For the hybrid mode, the PMP-based energy management problem is converted to a normal TPBVP and solved by using a shooting method. Moreover, the energy management for the plug-in hybrid electric truck with an AMT involves simultaneously optimizing the power distribution between the auxiliary power unit (APU) and the battery, as well as the gear-shifting choice. The simulation results with long- and short-distance scenarios indicate the flexibility of the PMP-based strategy. Furthermore, the proposed control strategy is compared with dynamic programming (DP) and a rule-based charge-depleting and charge-sustaining (CD-CS) strategy to evaluate its performance in terms of computational accuracy and time efficiency. Full article