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Demonstration of the applicability of methods of centers and of methods of feasible directions to optimal control problems. Presented experimental results show that extensions of Frank-Wolfe (1956), Zoutendijk (1960), and Pironneau-Polak (1971) algorithms for nonlinear programming problems can be quite efficient in solving optimal control problems.
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This study deals with ways of curtailing the rapid growth of cancer cell populations. The performance functional that measures the size of the population at the terminal time as well as the control effort is devised. With use of the discrete maximum principle, the Hamiltonian for this problem is determined and the condition for optimal solutions are developed. The optimal strategy is shown to be a bang-bang control. It is shown that the optimal control for this problem must be on the vertices of an N-dimensional cube contained in the N-dimensional Euclidean space. An algorithm for obtaining a local minimum of the performance function in an orderly fashion is developed. Application of the algorithm to the design of antitumor drug and X-irradiation schedule is discussed.
Parallel window search is applied to single-agent problems by having different processes simultaneously perform iterations of Iterative-Deepening-A(asterisk) (IDA-asterisk) on the same problem but with different cost thresholds. This approach is limited by the time to perform the goal iteration. To overcome this disadvantage, the authors consider node ordering. They discuss how global node ordering by minimum h among nodes with equal f = g + h values can reduce the time complexity of serial IDA-asterisk by reducing the time to perform the iterations prior to the goal iteration. Finally, the two ideas of parallel window search and node ordering are combined to eliminate the weaknesses of each approach while retaining the strengths. The resulting approach, called simply parallel window search, can be used to find a near-optimal solution quickly, improve the solution until it is optimal, and then finally guarantee optimality, depending on the amount of time available.
A method for the parallel solution of triangular sets of equations is described that is appropriate when there are many right-handed sides. By preprocessing, the method can reduce the number of parallel steps required to solve Lx = b compared to parallel forward or backsolve. Applications are to iterative solvers with triangular preconditioners, to structural analysis, or to power systems applications, where there may be many right-handed sides (not all available a priori). The inverse of L is represented as a product of sparse triangular factors. The problem is to find a factored representation of this inverse of L with the smallest number of factors (or partitions), subject to the requirement that no new nonzero elements be created in the formation of these inverse factors. A method from an earlier reference is shown to solve this problem. This method is improved upon by constructing a permutation of the rows and columns of L that preserves triangularity and allow for the best possible such partition. A number of practical examples and algorithmic details are presented. The parallelism attainable is illustrated by means of elimination trees and clique trees.
In this paper, an algorithm is introduced to find a minimum phase transfer function of specified order whose magnitude "tightly" overbounds a specified real-valued nonparametric function of frequency. This method has direct application to transforming nonparametric uncertainty bounds (available from system identification experiments and/or plant modeling) into parametric representations required for modern robust control design software (i.e., a minimum-phase transfer function multiplied by a norm-bounded perturbation).
Depressed Collector Design The microwave traveling wave tube (TWT) is used widely for space communications and high-power airborne transmitting sources. One of the most important features in designing a TWT is overall efficiency. Yet, overall TWT efficiency is strongly dependent on the efficiency of the electron beam collector, particularly for high values of collector efficiency. For these reasons, the NASA Glenn Research Center developed an optimization algorithm based on simulated annealing to quickly design highly efficient multistage depressed collectors (MDC's). Simulated annealing is a strategy for solving highly nonlinear combinatorial optimization problems. Its major advantage over other methods is its ability to avoid becoming trapped in local minima. Simulated annealing is based on an analogy to statistical thermodynamics, specifically the physical process of annealing: heating a material to a temperature that permits many atomic rearrangements and then cooling it carefully and slowly, until it freezes into a strong, minimum-energy crystalline structure. This minimum energy crystal corresponds to the optimal solution of a mathematical optimization problem. The TWT used as a baseline for optimization was the 32-GHz, 10-W, helical TWT developed for the Cassini mission to Saturn. The method of collector analysis and design used was a 2-1/2-dimensional computational procedure that employs two types of codes, a large signal analysis code and an electron trajectory code. The large signal analysis code produces the spatial, energetic, and temporal distributions of the spent beam entering the MDC. An electron trajectory code uses the resultant data to perform the actual collector analysis. The MDC was optimized for maximum MDC efficiency and minimum final kinetic energy of all collected electrons (to reduce heat transfer). The preceding figure shows the geometric and electrical configuration of an optimized collector with an efficiency of 93.8 percent. The results show the improvement in collector efficiency from 89.7 to 93.8 percent, resulting in an increase of three overall efficiency points. In addition, the time to design a highly efficient MDC was reduced from a month to a few days. All work was done in-house at Glenn for the High Rate Data Delivery Program. Future plans include optimizing the MDC and TWT interaction circuit in tandem to further improve overall TWT efficiency.
An algorithm is introduced to find a minimum phase trqansfer function of specified order whose magnitude
This analysis pertains to the applicability of optimal sensitivity information to aerospace vehicle design. The present analysis demonstrates that post-optimality information generated through first-order computations can be used to accurately predict file effect of constraint and parameter perturbations on the optimal solution. This assessment is based on the solution of an aircraft design problem in which the post-optimality estimates are shown to be within a few percent of the true solution over the practical range of constraint and parameter variations. Through solution of a reusable, single-stage-to-orbit, launch vehicle design problem, this optimal sensitivity information is also shown to improve the efficiency of the design process. For a hierarchically decomposed problem, this computational efficiency is realizable by estimating the main-problem objective gradient through optimal sensitivity calculations. By reducing the need for finite differentiation of a re-optimized subproblem, a significant decrease in the number of objective function evaluations required to reach the optimal solution is obtained.
The problem of real time guidance and optimal control of Aeroassisted Orbit Transfer Vehicles (AOTV's) was addressed using singular perturbation theory as an underlying method of analysis. Trajectories were optimized with the objective of minimum energy expenditure in the atmospheric phase of the maneuver. Two major problem areas were addressed: optimal reentry, and synergetic plane change with aeroglide. For the reentry problem, several reduced order models were analyzed with the objective of optimal changes in heading with minimum energy loss. It was demonstrated that a further model order reduction to a single state model is possible through the application of singular perturbation theory. The optimal solution for the reduced problem defines an optimal altitude profile dependent on the current energy level of the vehicle. A separate boundary layer analysis is used to account for altitude and flight path angle dynamics, and to obtain lift and bank angle control solutions. By considering alternative approximations to solve the boundary layer problem, three guidance laws were derived, each having an analytic feedback form. The guidance laws were evaluated using a Maneuvering Reentry Research Vehicle model and all three laws were found to be near optimal. For the problem of synergetic plane change with aeroglide, a difficult terminal boundary layer control problem arises which to date is found to be analytically intractable. Thus a predictive/corrective solution was developed to satisfy the terminal constraints on altitude and flight path angle. A composite guidance solution was obtained by combining the optimal reentry solution with the predictive/corrective guidance method. Numerical comparisons with the corresponding optimal trajectory solutions show that the resulting performance is very close to optimal. An attempt was made to obtain numerically optimized trajectories for the case where heating rate is constrained. A first order state variable inequality constraint was imposed on the full order AOTV point mass equations of motion, using a simple aerodynamic heating rate model.
Westinghouse is pursuing an advanced Nuclear Power Plant design based on Lead Fast Reactor (LFR) technology for global commercialization. To achieve an optimal combination of key attributes, such as safety, sustainability, and economic competitiveness, Westinghouse and ANL partnered in developing and applying a formalized core design optimization strategy. An LFR analysis workflow was developed to automate a suite of reactor physics, fuels performance, safety, and economics simulations on a selected LFR concept. The workflow streamlines analysis of a wide range of LFR designs with different dimensions and fuel types to assess their viability and economic performance, significantly reducing human processing time and risks of processing errors. The LFR optimization exercise was defined, resulting in selection of the design constraints (geometric, neutronics, thermo-mechanical, safety, thermal-hydraulics, and economics) and performance metrics researched (minimization of both the fuels LCOE and the first core inventory cost). A total of 14 varied design parameters were considered, including assembly dimensions, coolant temperature, and enrichment distribution throughout the core. The LFR analysis workflow was connected to DAKOTA for sensitivity and optimization analyses. Due to the extremely large size of the potential LFR optimization solution space relative to the computing time required to characterize one LFR solution, a multi-stage optimization approach was proposed to breakdown the problem into several stages with more reasonable sizes. This optimization approach enabled finding various viable core solutions with different cost tradeoffs that were considered by Westinghouse and justify selection of a smaller core with multi-batch 2-year cycle length.
Singular solutions in optimal control problems, treating singular control surface, flooding technique and allowable switching directions
This paper proposes a novel prediction method for online optimal control allocation that extends the volume of moments achievable with the Moore-Penrose generalized inverse to the entire Attainable Moment Set. This method formulates the control allocation problem using selected basis vectors and associated gains which reduces the optimization problem dimensions and provides physical insight into the resulting optimal solutions. The proposed algorithm finds the entire family of unique optimal control solutions along the desired moment vector from the origin to the boundary of the Attainable Moment Set. Numerical results for the Moore-Penrose prediction method show that the unique minimal controls obtained yield the desired moment with near machine precision accuracy while maintaining control effectors within specified position limits. This method has been fully validated against the unique solution obtained on the boundary of the Attainable Moment Set using the Durham Direct Allocation method. Minimal control solutions obtained for moments in the interior of the Attainable Moment Set, similarly yield the desired moment to near machine precision while providing control solutions that are smaller (i.e. 2-norm) than solutions found with traditional control allocation algorithms (e.g. interior point methods) applied to the minimal control problem. Numerical simulations using a Matlab® autocoded executable (MEX) for the representative real world problem of 3-moments with 20 individual control effectors and prescribed control position limits show a mean computation speed of approximately 125 Hz which is sufficient to enable real-time flight allocation.
An algorithm for optimal design is given which incorporates several of the desirable features of both mathematical programming and optimality criteria, while avoiding some of the undesirable features. The algorithm proceeds by approaching the optimal solution through the solutions of an associated set of constrained optimal design problems. The solutions of the constrained problems are recognized at each stage through the application of optimality criteria based on energy concepts. Two examples are described in which the optimal member size and layout of a truss is predicted, given the joint locations and loads.
A new experimental software package called NETS/PROSSS aimed at reducing the computing time required to solve a complex design problem is described. The software combines a neural network for simulating the analysis program with an optimization program. The neural network is applied to approximate results of a finite element analysis program to quickly obtain a near-optimal solution. Results of the NETS/PROSSS optimization process can also be used as an initial design in a normal optimization process and make it possible to converge to an optimum solution with significantly fewer iterations.
Fuel optimal solutions for the reorientation of an inertially symmetric rigid spacecraft with independent 3-axes controls are investigated. All possible optimal control strategies are identified. These include bangbang solutions, finite order singular arcs and infinite order singular arcs. Higher order necessary conditions for optimality of finite order singular arcs are presented. Numerical examples of fuel optimal solutions with fixed maneuver time are presented involving all of the theoretically possible control logics.
Mission proposals that land on asteroids are becoming popular. However, in order to have a successful mission the spacecraft must reliably and softly land at the intended landing site. The problem under investigation is how to design a fuel-optimal powered descent trajectory that can be quickly computed on- board the spacecraft, without interaction from ground control. An optimal trajectory designed immediately prior to the descent burn has many advantages. These advantages include the ability to use the actual vehicle starting state as the initial condition in the trajectory design and the ease of updating the landing target site if the original landing site is no longer viable. For long trajectories, the trajectory can be updated periodically by a redesign of the optimal trajectory based on current vehicle conditions to improve the guidance performance. One of the key drivers for being completely autonomous is the infrequent and delayed communication between ground control and the vehicle. Challenges that arise from designing an asteroid powered descent trajectory include complicated nonlinear gravity fields, small rotating bodies and low thrust vehicles. There are two previous studies that form the background to the current investigation. The first set looked in-depth at applying convex optimization to a powered descent trajectory on Mars with promising results.1, 2 This showed that the powered descent equations of motion can be relaxed and formed into a convex optimization problem and that the optimal solution of the relaxed problem is indeed a feasible solution to the original problem. This analysis used a constant gravity field. The second area applied a successive solution process to formulate a second order cone program that designs rendezvous and proximity operations trajectories.3, 4 These trajectories included a Newtonian gravity model. The equivalence of the solutions between the relaxed and the original problem is theoretically established. The proposed solution for designing the asteroid powered descent trajectory is to use convex optimization, a gravity model with higher fidelity than Newtonian, and an iterative solution process to design the fuel optimal trajectory. The solution to the convex optimization problem is the thrust profile, magnitude and direction, that will yield the minimum fuel trajectory for a soft landing at the target site, subject to various mission and operational constraints. The equations of motion are formulated in a rotating coordinate system and includes a high fidelity gravity model. The vehicle's thrust magnitude can vary between maximum and minimum bounds during the burn. Also, constraints are included to ensure that the vehicle does not run out of propellant, or go below the asteroid's surface, and any vehicle pointing requirements. The equations of motion are discretized and propagated with the trapezoidal rule in order to produce equality constraints for the optimization problem. These equality constraints allow the optimization algorithm to solve the entire problem, without including a propagator inside the optimization algorithm.