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NASA NTRS1988

Digital robust active control law synthesis for large order flexible structure using parameter optimization

A generic procedure for the parameter optimization of a digital control law for a large-order flexible flight vehicle or large space structure modeled as a sampled data system is presented. A linear quadratic Guassian type cost function was minimized, while satisfying a set of constraints on the steady-state rms values of selected design responses, using a constrained optimization technique to meet multiple design requirements. Analytical expressions for the gradients of the cost function and the design constraints on mean square responses with respect to the control law design variables are presented.

Mukhopadhyay, V.↗

NASA NTRS1984

Efficient calculation of optimum design sensitivity

Two methods for obtaining the sensitivity of an optimized design to changes in problem parameters are presented. The first-order method uses gradient information to estimate the required sensitivity. The second-order method uses second derivatives of the design objective and constraint functions to provide a quadratic approximation to the new design which results from changing the specified parameter. It is shown that the sensitivity of the optimized design may be discontinuous with respect to the new parameter. This discontinuity is accounted for in the present methods. These methods are compared to a previous approach based on the Kuhn-Tucker necessary conditions for optimality. It is shown that the present methods provide the appropriate sensitivity information in an efficient manner. The methods are demonstrated by a structural synthesis example.

Vanderplaats, G. N.↗

NASA NTRS1991

Design of 3-dimensional complex airplane configurations with specified pressure distribution via optimization

A subcritical panel method applied to flow analysis and aerodynamic design of complex aircraft configurations is presented. The analysis method is based on linearized, compressible, subsonic flow equations and indirect Dirichlet boundary conditions. Quadratic dipol and linear source distribution on flat panels are applied. In the case of aerodynamic design, the geometry which minimizes differences between design and actual pressure distribution is found iteratively, using numerical optimization technique. Geometry modifications are modeled by surface transpiration concept. Constraints in respect to resulting geometry can be specified. A number of complex 3-dimensional design examples are presented. The software is adopted to personal computers, and as result an unexpected low cost of computations is obtained.

Kubrynski, Krzysztof↗

DOE OSTI2019

A unified form of low-energy nodal electronic interactions in hole-doped cuprate superconductors

Using angle resolved photoemission spectroscopy measurements of Bi 2 Sr 2 CaCu 2 O 8+δ over a wide range of doping levels, we present a universal form for the non-Fermi liquid electronic interactions in the nodal direction in the exotic normal state phase. It is described by a continuously varying power law exponent versus energy and temperature (hence named a Power Law Liquid or PLL), which with doping varies smoothly from a quadratic Fermi Liquid in the overdoped regime, to a linear Marginal Fermi Liquid at optimal doping, to a non-quasiparticle non-Fermi Liquid in the underdoped regime. The coupling strength is essentially constant across all regimes and is consistent with Planckian dissipation. Using the extracted PLL parameters we reproduce the experimental optics and resistivity over a wide range of doping and normal-state temperature values, including the T* pseudogap temperature scale observed in the resistivity curves. This breaks the direct link to the pseudogapping of antinodal spectral weight observed at similar temperature scales and gives an alternative direction for searches of the microscopic mechanism.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1973

Equivalence of quadratic performance criteria.

Necessary and sufficient conditions are derived in terms of system parameters and quadratic weighting matrices for two quadratic cost functionals that are defined to be equivalent if they generate the same optimal control law. The derived conditions lie between the conditions of Tanaka and Asai (1971) and those of Kreindler and Hedrick (1970). Sufficient conditions for a vector valued function to attain an infimum are stated.

Martin, C.↗

NASA NTRS1980

Alternatives for jet engine control

Alternatives to linear quadratic regulator theory in the linear case are examined along with nonlinear modelling and optimization approaches for global control. Context for the studies has been set by the DYNGEN digital simulator and by models generated for various phases of the F100 Multivariable Control Synthesis Program. With respect to the linear alternatives, the multivariable frequency domain is stressed. Progress is reported in both the direct algebraic approach to exact model matching, by means of stimulating work on the basic computational issues, and in the indirect generalized Nyquist approach. With respect to nonlinear modelling and optimization, the emphasis is twofold: the development of analytical nonlinear models of the jet engine and the use of these models in conjunction with techniques of mathematical programming in order to study global control over nonincremental portions of the flight envelope. The possibility of using tensor methods is explored.

Sain, M. K.↗

NASA NTRS1980

Model error estimation for large space systems

In-flight estimation of large structure model errors may have to be carried out in order to detect inevitable deficiencies in large structure controller/estimator models. These error estimates can most efficiently be obtained by the minimization of a quadratic functional of the model errors and on the subsequent analysis of the resulting optimal model error estimates. An integral operator approach to estimation leads to a geometrical interpretation of the model error estimation process. One of the significant insights gained with this interpretation is that the actual but unknown model errors can be decomposed as the sum of two distinct components that are orthogonal in some sense. One of these components is a so-called minimal error vector that retains many of the significant dynamics of the actual errors. The basic ideas in the model error estimation approach are first set forth with a two-dimensional analogy that has most of the essential features of the general estimation problem. The generalized results are then established and their application to a reference large structure model illustrated.

Rodriguez, G.↗

NASA NTRS1983

Structural modelling and control design under incomplete parameter information: The maximum-entropy approach

A stochastic structural control model is described. In contrast to the customary deterministic model, the stochastic minimum data/maximum entropy model directly incorporates the least possible a priori parameter information. The approach is to adopt this model as the basic design model, thus incorporating the effects of parameter uncertainty at a fundamental level, and design mean-square optimal controls (that is, choose the control law to minimize the average of a quadratic performance index over the parameter ensemble).

Hyland, D. C.↗

NASA NTRS1991

Coupled Riccati equations for complex plane constraint

A new Linear Quadratic Gaussian design method is presented which provides prescribed imaginary axis pole placement for optimal control and estimation systems. This procedure contributes another degree of design freedom to flexible spacecraft control. Current design methods which interject modal damping into the system tend to have little affect on modal frequencies, i.e., they predictably shift open plant poles horizontally in the complex plane to form the closed loop controller or estimator pole constellation, but make little provision for vertical (imaginary axis) pole shifts. Imaginary axis shifts which reduce the closed loop model frequencies (the bandwidths) are desirable since they reduce the sensitivity of the system to noise disturbances. The new method drives the closed loop modal frequencies to predictable (specified) levels, frequencies as low as zero rad/sec (real axis pole placement) can be achieved. The design procedure works through rotational and translational destabilizations of the plant, and a coupling of two independently solved algebraic Riccati equations through a structured state weighting matrix. Two new concepts, gain transference and Q equivalency, are introduced and their use shown.

Strong, Kristin M.↗

NASA NTRS1993

Robust neighboring extremal guidance for the advanced launch system

With the availability of modern flight computers, realtime neighboring extremal guidance seems feasible. To overcome sensitivity to unknown system parameters and environmental uncertainties, a robust neighboring extremal guidance scheme is proposed. About the optimal trajectory, the accessory problem in the calculus of variations is formed, generating a quadratic cost criterion in the perturbed states and controls. By formulating a disturbance attenuation problem based upon the second variation cost criterion, a differential game is formulated. The game theoretic cost criterion is minimized with respect to the perturbed control but maximized with respect to the unknown parameters in the linearized dynamics. The resulting differential game problem gives rise to a two-point boundary-value problem solved using the sweep method. The sweep method solution provides a linear robust neighboring extremal guidance scheme that is applied to the Advanced Launch System.

Bain, John↗

DOE OSTI2017

Maximum a posteriori CMB lensing reconstruction

Gravitational lensing of the CMB is a valuable cosmological signal that correlates to tracers of large-scale structure and acts as a important source of confusion for primordial B-mode polarization. State-of-the-art lensing reconstruction analyses use quadratic estimators, which are easily applicable to data. Yet, these estimators are known to be suboptimal, in particular for polarization, and large improvements are expected to be possible for high signal-to-noise polarization experiments. We develop a method and numerical code, LensIt, that is able to find efficiently the most probable lensing map, introducing no significant approximations to the lensed CMB likelihood, and applicable to beamed and masked data with inhomogeneous noise. It works by iteratively reconstructing the primordial unlensed CMB using a deflection estimate and its inverse, and removing residual lensing from these maps with quadratic estimator techniques. Roughly linear computational cost is maintained due to fast convergence of iterative searches, combined with the local nature of lensing. The technique achieves the maximal improvement in signal to noise expected from analytical considerations on the unmasked parts of the sky. Delensing with this optimal map leads to forecast tensor-to-scalar ratio parameter errors improved by a factor ≃2 compared to the quadratic estimator in a CMB stage IV configuration.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

Online Optimization as a Feedback Controller: Stability and Tracking

This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time-invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time-varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that, under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. The paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power re-balancing and tracking of DC optimal power flow points.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1988

Optimal discrete-time LQR problems for parabolic systems with unbounded input: Approximation and convergence

An abstract approximation and convergence theory for the closed-loop solution of discrete-time linear-quadratic regulator problems for parabolic systems with unbounded input is developed. Under relatively mild stabilizability and detectability assumptions, functional analytic, operator techniques are used to demonstrate the norm convergence of Galerkin-based approximations to the optimal feedback control gains. The application of the general theory to a class of abstract boundary control systems is considered. Two examples, one involving the Neumann boundary control of a one-dimensional heat equation, and the other, the vibration control of a cantilevered viscoelastic beam via shear input at the free end, are discussed.

Rosen, I. G.↗

NASA NTRS2004

A Study of Penalty Function Methods for Constraint Handling with Genetic Algorithm

COMETBOARDS (Comparative Evaluation Testbed of Optimization and Analysis Routines for Design of Structures) is a design optimization test bed that can evaluate the performance of several different optimization algorithms. A few of these optimization algorithms are the sequence of unconstrained minimization techniques (SUMT), sequential linear programming (SLP) and the sequential quadratic programming techniques (SQP). A genetic algorithm (GA) is a search technique that is based on the principles of natural selection or "survival of the fittest". Instead of using gradient information, the GA uses the objective function directly in the search. The GA searches the solution space by maintaining a population of potential solutions. Then, using evolving operations such as recombination, mutation and selection, the GA creates successive generations of solutions that will evolve and take on the positive characteristics of their parents and thus gradually approach optimal or near-optimal solutions. By using the objective function directly in the search, genetic algorithms can be effectively applied in non-convex, highly nonlinear, complex problems. The genetic algorithm is not guaranteed to find the global optimum, but it is less likely to get trapped at a local optimum than traditional gradient-based search methods when the objective function is not smooth and generally well behaved. The purpose of this research is to assist in the integration of genetic algorithm (GA) into COMETBOARDS. COMETBOARDS cast the design of structures as a constrained nonlinear optimization problem. One method used to solve constrained optimization problem with a GA to convert the constrained optimization problem into an unconstrained optimization problem by developing a penalty function that penalizes infeasible solutions. There have been several suggested penalty function in the literature each with there own strengths and weaknesses. A statistical analysis of some suggested penalty functions is performed in this study. Also, a response surface approach to robust design is used to develop a new penalty function approach. This new penalty function approach is then compared with the other existing penalty functions.

Ortiz, Francisco↗

NASA NTRS1981

Antenna optimization and cost consideration for the Solar Power Satellite microwave system

The sizing, criteria, cost analysis, and optimized taper of the Solar Power Satellite (SPS) transmitting antenna are discussed. The sizing parameters considered were a thermal limit of 23 kW/sq m in the antenna, a peak power density of 23 mW/sq cm in the ionosphere, and cost effectiveness. Cost schedules and equations are presented for the SPS, and four antenna tapers are analyzed and compared, including the reference 10 dB Gaussian taper. An even powered quadratic series is formulated to minimize electricity cost and stay within thermal and ionospheric power level limits. The optimized Johnson Space Center taper is found to display the lowest energy costs, 4% below the 10 dB Gaussian, and can deliver 5.69 GW at 45.4 mills/kWh to the grid. Further studies are indicated for the optimal antenna and rectenna sizes

Kerwin, E. M.↗

NASA NTRS1989

Locally optimal-digital redesign of continuous-time systems

This paper presents a new optimal digital redesign technique for finding a dynamic digital control law from the given continuous-time counterpart by minimizing a local quadratic performance index. The quadratic performance index is chosen as the integral of the weighted squared difference between the states of the original closed-loop system and those of the digitally controlled closed-loop system at any instant between each sampling period. The developed optimal digital redesign control law enables the states of the digitally controlled closed-loop system 10 closely match those of the original closed-loop system at any instant between each sampling period, and it can easily be implemented using microcomputers with a relatively large sampling period.

Shieh, Leang-San↗

NASA NTRSDate not supplied

Path Planning: Differential Dynamic Programming and Model Predictive Path Integral Control on VTOL Aircraft

This paper explores two optimal control approaches, widely used in robotics, to establish their viability as real-time trajectory planners for vehicle configurations envisioned for the emerging aviation sector of Urban Air Mobility (UAM). Differential Dynamic Programming (DDP) enables planning over highly nonlinear dynamics using second-order approximations along a nominal trajectory, and displays quadratic convergence to a local solution. Model Predictive Path Integral (MPPI) is a stochastic sampling-based algorithm that can optimize for general cost criteria, including potentially highly nonlinear formulations, and supports parallel computation through the use of modern GPU hardware. In this work, DDP and MPPI were implemented using model predictive control (MPC), and the results indicate they are able to successfully transition the aircraft over different flight envelopes and generate trajectories unique to UAM vehicles.

Differential Dynamic Programming↗

DOE OSTI2016

TH-CD-207B-09: Task-Driven Fluence Field Modulation Design for Model-Based Iterative Reconstruction in CT

Purpose: There has been increasing interest in integrating fluence field modulation (FFM) devices with diagnostic CT scanners for dose reduction purposes. Conventional FFM strategies, however, are often either based on heuristics or the analysis of filtered-backprojection (FBP) performance. This work investigates a prospective task-driven optimization of FFM for model-based iterative reconstruction (MBIR) in order to improve imaging performance at the same total dose as conventional strategies. Methods: The task-driven optimization framework utilizes an ultra-low dose 3D scout as a patient-specific anatomical model and a mathematical formation of the imaging task. The MBIR method investigated is quadratically penalized-likelihood reconstruction. The FFM objective function uses detectability index, d’, computed as a function of the predicted spatial resolution and noise in the image. To optimize performance throughout the object, a maxi-min objective was adopted where the minimum d’ over multiple locations is maximized. To reduce the dimensionality of the problem, FFM is parameterized as a linear combination of 2D Gaussian basis functions over horizontal detector pixels and projection angles. The coefficients of these bases are found using the covariance matrix adaptation evolution strategy (CMA-ES) algorithm. The task-driven design was compared with three other strategies proposed for FBP reconstruction for a calcification cluster discrimination task in an abdomen phantom. Results: The task-driven optimization yielded FFM that was significantly different from those designed for FBP. Comparing all four strategies, the task-based design achieved the highest minimum d’ with an 8–48% improvement, consistent with the maxi-min objective. In addition, d’ was improved to a greater extent over a larger area within the entire phantom. Conclusion: Results from this investigation suggests the need to re-evaluate conventional FFM strategies for MBIR. The task-based optimization framework provides a promising approach that maximizes imaging performance under the same total dose constraint.

60 APPLIED LIFE SCIENCES↗