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All sourcesNASA NTRSDOE OSTI
NASA NTRS2008

Hard Constraints in Optimization Under Uncertainty

This paper proposes a methodology for the analysis and design of systems subject to parametric uncertainty where design requirements are specified via hard inequality constraints. Hard constraints are those that must be satisfied for all parameter realizations within a given uncertainty model. Uncertainty models given by norm-bounded perturbations from a nominal parameter value, i.e., hyper-spheres, and by sets of independently bounded uncertain variables, i.e., hyper-rectangles, are the focus of this paper. These models, which are also quite practical, allow for a rigorous mathematical treatment within the proposed framework. Hard constraint feasibility is determined by sizing the largest uncertainty set for which the design requirements are satisfied. Analytically verifiable assessments of robustness are attained by comparing this set with the actual uncertainty model. Strategies that enable the comparison of the robustness characteristics of competing design alternatives, the description and approximation of the robust design space, and the systematic search for designs with improved robustness are also proposed. Since the problem formulation is generic and the tools derived only require standard optimization algorithms for their implementation, this methodology is applicable to a broad range of engineering problems.

Crespo, Luis G.↗
NASA NTRS1989

Extensions of output variance constrained controllers to hard constraints

Covariance Controllers assign specified matrix values to the state covariance. A number of robustness results are directly related to the covariance matrix. The conservatism in known upperbounds on the H infinity, L infinity, and L (sub 2) norms for stability and disturbance robustness of linear uncertain systems using covariance controllers is illustrated with examples. These results are illustrated for continuous and discrete time systems. **** ONLY 2 BLOCK MARKERS FOUND -- RETRY *****

Skelton, R.↗
NASA NTRS1990

Bounding the states of systems with unknown-but-bounded disturbances

Control systems with hard constraints on certain variables are characterized, in an analytical review of recent investigations based on an unknown-but-bounded description of magnitude uncertainties. Hard-constraint problems typically arise in the design of controllers for potentially hazardous systems such as nuclear power plants. Consideration is given to norm bounds based on matrix measures, extensions of Schweppe's (1968) ellipsoid bounds, and set-theoretic regulator design. The performance of these approaches is evaluated by means of numerical simulations involving a third-order nonlinear steam-boiler model; the results are presented in graphs, and it is found that ellipsoid bounds are tightest in the general case, but that box bounds are even tighter for linear systems with Metzler system matrices.

Tsai, Wei K.↗
NASA NTRS2006

Hard and Soft Constraints in Reliability-Based Design Optimization

This paper proposes a framework for the analysis and design optimization of models subject to parametric uncertainty where design requirements in the form of inequality constraints are present. Emphasis is given to uncertainty models prescribed by norm bounded perturbations from a nominal parameter value and by sets of componentwise bounded uncertain variables. These models, which often arise in engineering problems, allow for a sharp mathematical manipulation. Constraints can be implemented in the hard sense, i.e., constraints must be satisfied for all parameter realizations in the uncertainty model, and in the soft sense, i.e., constraints can be violated by some realizations of the uncertain parameter. In regard to hard constraints, this methodology allows (i) to determine if a hard constraint can be satisfied for a given uncertainty model and constraint structure, (ii) to generate conclusive, formally verifiable reliability assessments that allow for unprejudiced comparisons of competing design alternatives and (iii) to identify the critical combination of uncertain parameters leading to constraint violations. In regard to soft constraints, the methodology allows the designer (i) to use probabilistic uncertainty models, (ii) to calculate upper bounds to the probability of constraint violation, and (iii) to efficiently estimate failure probabilities via a hybrid method. This method integrates the upper bounds, for which closed form expressions are derived, along with conditional sampling. In addition, an l(sub infinity) formulation for the efficient manipulation of hyper-rectangular sets is also proposed.

Crespo, L.uis G.↗
NASA NTRS2018

LPV Modeling and Control for Active Flutter Suppression of a Smart Airfoil

In this paper, a novel technique of linear parameter varying (LPV) modeling and control of a smart airfoil for active flutter suppression is proposed, where the smart airfoil has a groove along its chord and contains a moving mass that is used to control the airfoil pitching and plunging motions. The new LPV modeling technique is proposed that uses mass position as a scheduling parameter to describe the physical constraint of the moving mass, in addition the hard constraint at the boundaries is realized by proper selection of the parameter varying function. Therefore, the position of the moving mass and the free stream airspeed are considered the scheduling parameters in the study. A state-feedback based LPV gain-scheduling controller with guaranteed H infinity performance is presented by utilizing the dynamics of the moving mass as scheduling parameter at a given airspeed. The numerical simulations demonstrate the effectiveness of the proposed LPV control architecture by significantly improving the performance while reducing the control effort.

Smart Airfoil↗
NASA NTRS2017

Quantum Approximate Optimization with Hard and Soft Constraints

Challenging computational problems arising in the practical world are frequently tackled by heuristic algorithms. Small universal quantum computers will emerge in the next year or two, enabling a substantial broadening of the types of quantum heuristics that can be investigated beyond quantum annealing. The immediate question is What experiments should we prioritize that will give us insight into quantum heuristics? One leading candidate is the quantum approximate optimization algorithm (QAOA) metaheuristic. Here, we provide a framework for designing QAOA circuits for a variety of combinatorial optimization problems with both hard constraints that must be met and soft constraints whose violation we wish to minimize. We work through a number of examples, and discuss design principles and implementation considerations.

Hadfield, Stuart↗
NASA NTRS2019

Exploring Network-Related Optimization Problems Using Quantum Heuristics

Network-related connectivity optimization problems are underlying a wide range of applications and are also of high computational complexity. We consider studying network optimization problems using two types of quantum heuristics.One is quantum annealing, and the other Quantum Alternating Operator Ansatz, an extension of the Quantum Approximate Optimization Algorithms for gate-model quantum computation, in which a cost-function based unitary and a non-commuting mixing unitary are applied alternately. We present problem mappings for problems of finding the spanning-tree or spanning-graph of a graph that optimizes certain costs, and a variant that further requires the spanning-tree be degree-bounded. With quantum annealing, all constraints are cast into penalty terms in the cost Hamiltonian, and the solution is encoded as the ground state of the Hamiltonian. We provide three mappings to the quadratic unconstrained binary optimization (QUBO) form, compare the resource requirements, and analyze the tradeoffs. For QAOA, we give special focus on the design of mixers based on the constraints presented in the problem, such that the system evolution remains in a subspace of the full Hilbert space where all constraints are satisfied. In the spanning-tree problem, one such hard constraint is that a mixer applied to a spanning-tree needs also be a spanning tree. This involves checking the connectivity of a subgraph, which is a global condition common for most network-related problems. We show how this feature can be efficiently represented in the mixer in a quantum coherent way, based on manipulation of a descendant-matrix and an adjacent matrix. We further develop a mixer for the spanning-graphs based on the spanning-tree mixer.

Wang, Zhihui↗
NASA NTRS2019

Study network-related optimization problems using quantum alternating optimization ansatz

Network-related connectivity optimization problems are underlying a wide range of applications and are also of high computational complexity. We consider studying network optimization problems using two types of quantum heuristics. One is quantum annealing, and the other Quantum Alternating Operator Ansatz, an extension of the Quantum Approximate Optimization Algorithms for gate-model quantum computation, in which a cost-function based unitary and a non-commuting mixing unitary are applied alternately. We present problem mappings for problems of finding the spanning-tree or spanning-graph of a graph that optimizes certain costs, and a variant that further requires the spanning-tree be degree-bounded. With quantum annealing, all constraints are cast into penalty terms in the cost Hamiltonian, and the solution is encoded as the ground state of the Hamiltonian. We provide three mappings to the quadratic unconstrained binary optimization (QUBO) form, compare the resource requirements, and analyze the tradeoffs. For QAOA, we give special focus on the design of mixers based on the constraints presented in the problem, such that the system evolution remains in a subspace of the full Hilbert space where all constraints are satisfied. In the spanning-tree problem, one such hard constraint is that a mixer applied to a spanning-tree needs also be a spanning tree. This involves checking the connectivity of a subgraph, which is a global condition common for most network-related problems. We show how this feature can be efficiently represented in the mixer in a quantum coherent way, based on manipulation of a descendant-matrix and an adjacent matrix. We further develop a mixer for the spanning-graphs based on the spanning-tree mixer.

Zhihui Wang↗
NASA NTRS2003

Kalman Filtering with Inequality Constraints for Turbofan Engine Health Estimation

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This paper develops two analytic methods of incorporating state variable inequality constraints in the Kalman filter. The first method is a general technique of using hard constraints to enforce inequalities on the state variable estimates. The resultant filter is a combination of a standard Kalman filter and a quadratic programming problem. The second method uses soft constraints to estimate state variables that are known to vary slowly with time. (Soft constraints are constraints that are required to be approximately satisfied rather than exactly satisfied.) The incorporation of state variable constraints increases the computational effort of the filter but significantly improves its estimation accuracy. The improvement is proven theoretically and shown via simulation results. The use of the algorithm is demonstrated on a linearized simulation of a turbofan engine to estimate health parameters. The turbofan engine model contains 16 state variables, 12 measurements, and 8 component health parameters. It is shown that the new algorithms provide improved performance in this example over unconstrained Kalman filtering.

Simon, Dan↗
NASA NTRS1996

A System for Automatically Generating Scheduling Heuristics

The goal of this research is to improve the performance of automated schedulers by designing and implementing an algorithm by automatically generating heuristics by selecting a schedule. The particular application selected by applying this method solves the problem of scheduling telescope observations, and is called the Associate Principal Astronomer. The input to the APA scheduler is a set of observation requests submitted by one or more astronomers. Each observation request specifies an observation program as well as scheduling constraints and preferences associated with the program. The scheduler employs greedy heuristic search to synthesize a schedule that satisfies all hard constraints of the domain and achieves a good score with respect to soft constraints expressed as an objective function established by an astronomer-user.

Morris, Robert↗
NASA NTRS1973

Real-time optimal guidance for orbital maneuvering.

A new formulation for soft-constraint trajectory optimization is presented as a real-time optimal feedback guidance method for multiburn orbital maneuvers. Control is always chosen to minimize burn time plus a quadratic penalty for end condition errors, weighted so that early in the mission (when controllability is greatest) terminal errors are held negligible. Eventually, as controllability diminishes, the method partially relaxes but effectively still compensates perturbations in whatever subspace remains controllable. Although the soft-constraint concept is well-known in optimal control, the present formulation is novel in addressing the loss of controllability inherent in multiple burn orbital maneuvers. Moreover the necessary conditions usually obtained from a Bolza formulation are modified in this case so that the fully hard constraint formulation is a numerically well behaved subcase. As a result convergence properties have been greatly improved.

Cohen, A. O.↗
NASA NTRS1980

Pareto-optimal multi-objective design of airplane control systems

A constrained minimization algorithm for the computer aided design of airplane control systems to meet many requirements over a set of flight conditions is generalized using the concept of Pareto-optimization. The new algorithm yields solutions on the boundary of the achievable domain in objective space in a single run, whereas the older method required a sequence of runs to approximate such a limiting solution. However, Pareto-optimality does not guarantee a satisfactory design, since such solutions may emphasize some objectives at the expense of others. The designer must still interact with the program to obtain a well-balanced set of objectives. Using the example of a fighter lateral stability augmentation system (SAS) design over five flight conditions, several effective techniques are developed for obtaining well-balanced Pareto-optimal solutions. For comparison, one of these techniques is also used in a recently developed algorithm of Kreisselmeier and Steinhauser, which replaces the hard constraints with soft constraints, using a special penalty function. It is shown that comparable results can be obtained.

Schy, A. A.↗
NASA NTRS1993

Primary differentiation in the early Earth: Nd and Sr isotopic evidence from diamondiferous eclogites for both old depleted and old enriched mantle, Yakutia, Siberia

Ancient, stable, continental cratons possess thick, subcontinental-lithospheric mantle 'keels' which favor particularly the emplacement of diamondiferous kimberlites and included peridotites and eclogites. These refractory mantle samples of the roots provide hard constraints on the theories of formation, growth, and evolution of these cratons. Xenoliths containing only primary garnet and clinopyroxene (eclogites), although rare in most kimberlites, can retain the geochemical signatures of their parent protoliths (e.g., subducted oceanic crust, ancient mantle) thus offering the opportunity to address mantle processes which may have taken place at earlier times in the Earth's history. In fact, it has been postulated that some eclogites are residues from the accretion of the early Earth. Nd and Sr isotopic data are presented which may be interpreted as evidence of an early (greater than 4 Ga) mantle differentiation event. The kimberlites of Yakutia are located both marginal and central to the Siberian craton, and a wide variety of xenoliths are present within them. The Siberian mantle samples have received little attention in the western world, largely because suitable suites of Yakutian samples have not been readily available. Importantly, there is evidence that metasomatism of the Siberian lithosphere has been considerably less intense or extensive than for the Kaapvaal craton. Therefore, it should be considerably easier to elicit the igneous/metamorphic histories of Siberian kimberlitic xenoliths. One of the notable features of the Siberian eclogites is the common appearance of diamonds, especially in the Mir and Udachnaya pipes. In all, eight eclogite samples (eight garnet separates and eight clinopyroxene separates) have been analyzed to date on the Udachnaya pipe, seven from our group.

Snyder, Gregory A.↗
NASA NTRS2005

Precision Manipulation with Cooperative Robots

This work addresses several challenges of cooperative transportThis work addresses several challenges of cooperative transport and precision manipulation. Precision manipulation requires a rigid grasp, which places a hard constraint on the relative rover formation that must be accommodated, even though the rovers cannot directly observe their relative poses. Additionally, rovers must jointly select appropriate actions based on all available sensor information. Lastly, rovers cannot act on independent sensor information, but must fuse information to move jointly; the methods for fusing information must be determined.

multi-robot teams↗
NASA NTRS2008

Optimization of Angular-Momentum Biases of Reaction Wheels

RBOT [RWA Bias Optimization Tool (wherein RWA signifies Reaction Wheel Assembly )] is a computer program designed for computing angular momentum biases for reaction wheels used for providing spacecraft pointing in various directions as required for scientific observations. RBOT is currently deployed to support the Cassini mission to prevent operation of reaction wheels at unsafely high speeds while minimizing time in undesirable low-speed range, where elasto-hydrodynamic lubrication films in bearings become ineffective, leading to premature bearing failure. The problem is formulated as a constrained optimization problem in which maximum wheel speed limit is a hard constraint and a cost functional that increases as speed decreases below a low-speed threshold. The optimization problem is solved using a parametric search routine known as the Nelder-Mead simplex algorithm. To increase computational efficiency for extended operation involving large quantity of data, the algorithm is designed to (1) use large time increments during intervals when spacecraft attitudes or rates of rotation are nearly stationary, (2) use sinusoidal-approximation sampling to model repeated long periods of Earth-point rolling maneuvers to reduce computational loads, and (3) utilize an efficient equation to obtain wheel-rate profiles as functions of initial wheel biases based on conservation of angular momentum (in an inertial frame) using pre-computed terms.

Lee, Clifford↗
NASA NTRS2006

Strict Constraint Feasibility in Analysis and Design of Uncertain Systems

This paper proposes a methodology for the analysis and design optimization of models subject to parametric uncertainty, where hard inequality constraints are present. Hard constraints are those that must be satisfied for all parameter realizations prescribed by the uncertainty model. Emphasis is given to uncertainty models prescribed by norm-bounded perturbations from a nominal parameter value, i.e., hyper-spheres, and by sets of independently bounded uncertain variables, i.e., hyper-rectangles. These models make it possible to consider sets of parameters having comparable as well as dissimilar levels of uncertainty. Two alternative formulations for hyper-rectangular sets are proposed, one based on a transformation of variables and another based on an infinity norm approach. The suite of tools developed enable us to determine if the satisfaction of hard constraints is feasible by identifying critical combinations of uncertain parameters. Since this practice is performed without sampling or partitioning the parameter space, the resulting assessments of robustness are analytically verifiable. Strategies that enable the comparison of the robustness of competing design alternatives, the approximation of the robust design space, and the systematic search for designs with improved robustness characteristics are also proposed. Since the problem formulation is generic and the solution methods only require standard optimization algorithms for their implementation, the tools developed are applicable to a broad range of problems in several disciplines.

Crespo, Luis G.↗
NASA NTRS1994

Empirical results on scheduling and dynamic backtracking

At the Honeywell Technology Center (HTC), we have been working on a scheduling problem related to commercial avionics. This application is large, complex, and hard to solve. To be a little more concrete: 'large' means almost 20,000 activities, 'complex' means several activity types, periodic behavior, and assorted types of temporal constraints, and 'hard to solve' means that we have been unable to eliminate backtracking through the use of search heuristics. At this point, we can generate solutions, where solutions exist, or report failure and sometimes why the system failed. To the best of our knowledge, this is among the largest and most complex scheduling problems to have been solved as a constraint satisfaction problem, at least that has appeared in the published literature. This abstract is a preliminary report on what we have done and how. In the next section, we present our approach to treating scheduling as a constraint satisfaction problem. The following sections present the application in more detail and describe how we solve scheduling problems in the application domain. The implemented system makes use of Ginsberg's Dynamic Backtracking algorithm, with some minor extensions to improve its utility for scheduling. We describe those extensions and the performance of the resulting system. The paper concludes with some general remarks, open questions and plans for future work.

Boddy, Mark S.↗
NASA NTRS1988

Analytic limits on the forms of spectra possible from optically thin collisional bremsstrahlung source models

The constraints on hard X-ray bremsstrahlung spectral forms required in order for them to correspond to physically acceptable (nonnegative) electron distributions in thin-target, thick-target, and thermal source models are discussed. The extent to which various spectra can be attributed to the different models is examined, showing that many possible spectra cannot be described by all, or in some cases, any of the models. It is shown that for any bremsstrahlung cross section, the thick-target and thermal models require that successively higher derivatives of the thin-target constraint have the appropriate sign. It is found that thermal models are the most restrictive, and that thin-target models are the least restrictive. Explicit analytic constraint expressions are derived for all three cases for the Kramers cross section and examples of acceptable and unacceptable spectra are given. Application of these criteria to the testing and exclusion of models is discussed.

Brown, John C.↗