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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Optimizing for minimum weight when two different finite element models and analyses are required

The Finite Element Structural Optimization Program's (FESOP) ability to perform minimum weight optimization using two different finite element analyses and models is discussed. FESOP uses the ADS optimizer developed by Dr. Garret Vanderplaats to solve the nonlinear constrained optimization problem. The design optimization problem requires a response spectrum analysis and model to evaluate the stress and displacement constraints. However, the problem needs a frequency analysis and model to calculate the natural frequencies used to evaluate the frequency range constraints. The results of both the successful and unsuccessful approaches used to solve this difficult weight minimization problem are summarized. The results show that no one ADS optimization algorithm worked in all cases. However, the Sequential Convex Programming and Modified Method of Feasible Directions algorithms were the most successful.

Hall, Jeffrey C.↗
NASA NTRS2002

Bi-Level Integrated System Synthesis (BLISS) for Concurrent and Distributed Processing

The paper introduces a new version of the Bi-Level Integrated System Synthesis (BLISS) methods intended for optimization of engineering systems conducted by distributed specialty groups working concurrently and using a multiprocessor computing environment. The method decomposes the overall optimization task into subtasks associated with disciplines or subsystems where the local design variables are numerous and a single, system-level optimization whose design variables are relatively few. The subtasks are fully autonomous as to their inner operations and decision making. Their purpose is to eliminate the local design variables and generate a wide spectrum of feasible designs whose behavior is represented by Response Surfaces to be accessed by a system-level optimization. It is shown that, if the problem is convex, the solution of the decomposed problem is the same as that obtained without decomposition. A simplified example of an aircraft design shows the method working as intended. The paper includes a discussion of the method merits and demerits and recommendations for further research.

Sobieszczanski-Sobieski, Jaroslaw↗
NASA NTRS2015

Random Predictor Models for Rigorous Uncertainty Quantification: Part 2

This and a companion paper propose techniques for constructing parametric mathematical models describing key features of the distribution of an output variable given input-output data. By contrast to standard models, which yield a single output value at each value of the input, Random Predictors Models (RPMs) yield a random variable at each value of the input. Optimization-based strategies for calculating RPMs having a polynomial dependency on the input and a linear dependency on the parameters are proposed. These formulations yield RPMs having various levels of fidelity in which the mean, the variance, and the range of the model's parameter, thus of the output, are prescribed. As such they encompass all RPMs conforming to these prescriptions. The RPMs are optimal in the sense that they yield the tightest predictions for which all (or, depending on the formulation, most) of the observations are less than a fixed number of standard deviations from the mean prediction. When the data satisfies mild stochastic assumptions, and the optimization problem(s) used to calculate the RPM is convex (or, when its solution coincides with the solution to an auxiliary convex problem), the model's reliability, which is the probability that a future observation would be within the predicted ranges, is bounded rigorously.

Crespo, Luis G.↗
NASA NTRS2015

Random Predictor Models for Rigorous Uncertainty Quantification: Part 1

This and a companion paper propose techniques for constructing parametric mathematical models describing key features of the distribution of an output variable given input-output data. By contrast to standard models, which yield a single output value at each value of the input, Random Predictors Models (RPMs) yield a random variable at each value of the input. Optimization-based strategies for calculating RPMs having a polynomial dependency on the input and a linear dependency on the parameters are proposed. These formulations yield RPMs having various levels of fidelity in which the mean and the variance of the model's parameters, thus of the predicted output, are prescribed. As such they encompass all RPMs conforming to these prescriptions. The RPMs are optimal in the sense that they yield the tightest predictions for which all (or, depending on the formulation, most) of the observations are less than a fixed number of standard deviations from the mean prediction. When the data satisfies mild stochastic assumptions, and the optimization problem(s) used to calculate the RPM is convex (or, when its solution coincides with the solution to an auxiliary convex problem), the model's reliability, which is the probability that a future observation would be within the predicted ranges, can be bounded tightly and rigorously.

Crespo, Luis G.↗
NASA NTRS2022

Model Predictive Control in the Three-body Problem Using Invariant Funnels As Terminal Sets

This paper describes a method for augmenting Model Predictive Control techniques using invariant funnels computed in the Circular Restricted Three-Body Problem. We use ellipsoids that roughly approximate the boundary of the invariant funnel as convex terminal sets for a short look-ahead optimization problem at each time step. We apply this method to a hypothetical low-thrust mission to land on Jupiter’s mooon Europa and show that including the invariant funnels as terminal sets reduces the amount of control effort required by almost an order of magnitude.

Close, Sigrid↗
NASA NTRS2005

Robust Control Design for Systems With Probabilistic Uncertainty

This paper presents a reliability- and robustness-based formulation for robust control synthesis for systems with probabilistic uncertainty. In a reliability-based formulation, the probability of violating design requirements prescribed by inequality constraints is minimized. In a robustness-based formulation, a metric which measures the tendency of a random variable/process to cluster close to a target scalar/function is minimized. A multi-objective optimization procedure, which combines stability and performance requirements in time and frequency domains, is used to search for robustly optimal compensators. Some of the fundamental differences between the proposed strategy and conventional robust control methods are: (i) unnecessary conservatism is eliminated since there is not need for convex supports, (ii) the most likely plants are favored during synthesis allowing for probabilistic robust optimality, (iii) the tradeoff between robust stability and robust performance can be explored numerically, (iv) the uncertainty set is closely related to parameters with clear physical meaning, and (v) compensators with improved robust characteristics for a given control structure can be synthesized.

Crespo, Luis G.↗
NASA NTRS1991

Lanczos eigensolution method for high-performance computers

The theory, computational analysis, and applications are presented of a Lanczos algorithm on high performance computers. The computationally intensive steps of the algorithm are identified as: the matrix factorization, the forward/backward equation solution, and the matrix vector multiples. These computational steps are optimized to exploit the vector and parallel capabilities of high performance computers. The savings in computational time from applying optimization techniques such as: variable band and sparse data storage and access, loop unrolling, use of local memory, and compiler directives are presented. Two large scale structural analysis applications are described: the buckling of a composite blade stiffened panel with a cutout, and the vibration analysis of a high speed civil transport. The sequential computational time for the panel problem executed on a CONVEX computer of 181.6 seconds was decreased to 14.1 seconds with the optimized vector algorithm. The best computational time of 23 seconds for the transport problem with 17,000 degs of freedom was on the the Cray-YMP using an average of 3.63 processors.

Bostic, Susan W.↗
NASA NTRS1988

A methodology for the synthesis of robust feedback systems

A new methodology is developed for the synthesis of linear, time-variant (LTI) controllers for multivariable LTI systems. The resulting closed-loop system is nominally stable and exhibits a known level of performance. In addition, robustness of the feedback system is guaranteed, i.e., stability and performance are retained in the presence of multiple unstructured uncertainty blocks located at various points in the feedback loop. The design technique is referred to as the Causality Recovery Methodology (CRM). The CRM relies on the Youla parameterization of all stabilizing compensators to ensure nominal stability of the feedback system. A frequency-domain inequality in terms of the structured singular value mu defines the robustness specification. The optimal compensator, with respect to the mu condition, is shown to be noncausal in general. The aim of the CRM is to find a stable, causal transfer function matrix that approximates the robustness characteristics of the optimal solution. The CRM, via a series of infinite-dimensional convex programs, produces a closed-loop system whose performance robustness is at least as good as that of any initial design. The algorithm is approximated by a finite dimensional process for the purposes of implementation. Two numerical examples confirm the potential viability of the CRM concept; however, the robustness improvement comes at the expense of increased computational burden and compensator complexity.

Milich, David Albert↗
NASA NTRS1992

Recursive optimal pruning with applications to tree structured vector quantizers

A pruning algorithm of Chou et al. (1989) for designing optimal tree structures identifies only those codebooks which lie on the convex hull of the original codebook's operational distortion rate function. The authors introduce a modified version of the original algorithm, which identifies a large number of codebooks having minimum average distortion, under the constraint that, in each step, only modes having no descendents are removed from the tree. All codebooks generated by the original algorithm are also generated by this algorithm. The new algorithm generates a much larger number of codebooks in the middle- and low-rate regions. The additional codebooks permit operation near the codebook's operational distortion rate function without time sharing by choosing from the increased number of available bit rates. Despite the statistical mismatch which occurs when coding data outside the training sequence, these pruned codebooks retain their performance advantage over full search vector quantizers (VQs) for a large range of rates.

Kiang, Shei-Zein↗
NASA NTRS1994

Algorithms for bilevel optimization

General multilevel nonlinear optimization problems arise in design of complex systems and can be used as a means of regularization for multi-criteria optimization problems. Here, for clarity in displaying our ideas, we restrict ourselves to general bi-level optimization problems, and we present two solution approaches. Both approaches use a trust-region globalization strategy, and they can be easily extended to handle the general multilevel problem. We make no convexity assumptions, but we do assume that the problem has a nondegenerate feasible set. We consider necessary optimality conditions for the bi-level problem formulations and discuss results that can be extended to obtain multilevel optimization formulations with constraints at each level.

Alexandrov, Natalia↗
NASA NTRS1988

Efficient approximation concepts using second order information

The application of second derivative information for solving structural optimization problems is considered. In the present method, rather than building approximate nonlinear forms for the objective function and constraints, only linear approximations are used. A separable quadratic approximation of the Lagrangian function is included in the subproblem statement. The method has been successfully used for simple problems that can be solved in closed form, in addition to the sizing optimization of trusses, and it is shown to converge faster than the convex linearization method or the method of moving asymptotes.

Fleury, Claude↗
NASA NTRS2025

Robust Trajectory Optimization for NRHO Rendezvous Using SPICE Kernel Relative Motion

In this paper, robust optimization is performed on trajectory correction maneuvers during the lunar lander return phase of an Artemis mission, treating the trajectory from one hour after low lunar orbit departure to arrival in the vicinity of the lunar Gateway as a relative motion problem. To enable rapid stochastic optimization techniques requiring many candidate trajectories, SPICE kernel relative motion as implemented by the Quadratic Interpolated State Transition (QIST) system is used as the underlying dynamics propagation. The optimization is performed with a genetic optimizer using linear covariance (LinCov) software in a simplified operational context, taking into account the availability of navigation sensors with varying measurement models, ranges, and accuracies. No numerical integration is used, since the relative motion around Gateway is fully characterized with the a priori computation of the QIST coefficients. Maneuver placements are computed to optimize the minimum 3σ delta-v of the trajectory, the position dispersion at a target point, and a convex combination of these two metrics. An order of magnitude runtime improvement is provided over legacy methods with less than 10% error introduced. All QIST results are shown to be in-family with legacy methods. The tradespace for optimal delta-v design is found to range from 77.0 to 93.9 m/s, while the range of optimal dispersion is between 1.4 and 11.7 km.

Relative Motion↗
NASA NTRS1993

Hodograph analysis in aircraft trajectory optimization

An account is given of key geometrical concepts involved in the use of a hodograph as an optimal control theory resource which furnishes a framework for geometrical interpretation of the minimum principle. Attention is given to the effects of different convexity properties on the hodograph, which bear on the existence of solutions and such types of controls as chattering controls, 'bang-bang' control, and/or singular control. Illustrative aircraft trajectory optimization problems are examined in view of this use of the hodograph.

Cliff, Eugene M.↗
NASA NTRS2006

A Convex Guidance Algorithm for Formation Reconfiguration

In this paper, a reconfiguration guidance algorithm for formation flying spacecraft is presented. The formation reconfiguration guidance problem is first formulated as a continuous-time minimum-fuel or minimum-energy optimal control problem with collision avoidance and control constraints. The optimal control problem is then discretized to obtain a finite dimensional parameter optimization problem. In this formulation, the collision avoidance constraints are imposed via separating planes between each pair of spacecraft. A heuristic is introduced to choose these separating planes that leads to the convexification of the collision avoidance constraints. Additionally, convex constraints are imposed to guarantee that no collisions occur between discrete time samples. The resulting finite dimensional optimization problem is a second order cone program, for which standard algorithms can compute the global optimum with deterministic convergence and a prescribed level of accuracy. Consequently, the formation reconfiguration algorithm can be implemented onboard a spacecraft for real-time operations.

formation reconfiguration↗
NASA NTRS1991

New displacement-based methods for optimal truss topology design

Two alternate methods for maximum stiffness truss topology design are presented. The ground structure approach is used, and the problem is formulated in terms of displacements and bar areas. This large, nonconvex optimization problem can be solved by a simultaneous analysis and design approach. Alternatively, an equivalent, unconstrained, and convex problem in the displacements only can be formulated, and this problem can be solved by a nonsmooth, steepest descent algorithm. In both methods, the explicit solving of the equilibrium equations and the assembly of the global stiffness matrix are circumvented. A large number of examples have been studied, showing the attractive features of topology design as well as exposing interesting features of optimal topologies.

Bendsoe, Martin P.↗
NASA NTRS1993

On convexity of H-infinity Riccati solutions and its applications

The celebrated two-Riccati-equation solution to a standard H-infinity control problem can be used to characterize all possible stabilizing optimal or suboptimal H-infinity controllers if the optimal H-infinity norm or gamma, an upper bound of a suboptimal H-infinity norm, is given. In this note, some properties of these H-infinity Riccati solutions are revealed. Among them, the most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of gamma on the domain of interest. Based on these properties, a quadratically convergent algorithm is developed to compute the optimal H-infinity norm.

Li, X. P.↗
NASA NTRS1991

Robust control of systems with real parameter uncertainty and unmodelled dynamics

During this research period we have made significant progress in the four proposed areas: (1) design of robust controllers via H infinity optimization; (2) design of robust controllers via mixed H2/H infinity optimization; (3) M-delta structure and robust stability analysis for structured uncertainties; and (4) a study on controllability and observability of perturbed plant. It is well known now that the two-Riccati-equation solution to the H infinity control problem can be used to characterize all possible stabilizing optimal or suboptimal H infinity controllers if the optimal H infinity norm or gamma, an upper bound of a suboptimal H infinity norm, is given. In this research, we discovered some useful properties of these H infinity Riccati solutions. Among them, the most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of gamma in the domain of interest. Based on these properties, quadratically convergent algorithms are developed to compute the optimal H infinity norm. We also set up a detailed procedure for applying the H infinity theory to robust control systems design. The desire to design controllers with H infinity robustness but H(exp 2) performance has recently resulted in mixed H(exp 2) and H infinity control problem formulation. The mixed H(exp 2)/H infinity problem have drawn the attention of many investigators. However, solution is only available for special cases of this problem. We formulated a relatively realistic control problem with H(exp 2) performance index and H infinity robustness constraint into a more general mixed H(exp 2)/H infinity problem. No optimal solution yet is available for this more general mixed H(exp 2)/H infinity problem. Although the optimal solution for this mixed H(exp 2)/H infinity control has not yet been found, we proposed a design approach which can be used through proper choice of the available design parameters to influence both robustness and performance. For a large class of linear time-invariant systems with real parametric perturbations, the coefficient vector of the characteristic polynomial is a multilinear function of the real parameter vector. Based on this multilinear mapping relationship together with the recent developments for polytopic polynomials and parameter domain partition technique, we proposed an iterative algorithm for coupling the real structured singular value.

Chang, Bor-Chin↗