Nonlinear programming and optimal control
Generalization of nonlinear programming problem - optimization problems
MATHEMATICAL MODEL↗
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An introduction to optimal power flow: Theory, formulation, and examples
The set of optimization problems in electric power systems engineering known collectively as Optimal Power Flow (OPF) is one of the most practically important and well-researched subfields of constrained nonlinear optimization. OPF has enjoyed a rich history of research, innovation, and publication since its debut five decades ago. Nevertheless, entry into OPF research is a daunting task for the uninitiated--both due to the sheer volume of literature and because OPF's ubiquity within the electric power systems community has led authors to assume a great deal of prior knowledge that readers unfamiliar with electric power systems may not possess. This article provides an introduction to OPF from an operations research perspective; it describes a complete and concise basis of knowledge for beginning OPF research. The discussion is tailored for the operations researcher who has experience with nonlinear optimization but little knowledge of electrical engineering. Topics covered include power systems modeling, the power flow equations, typical OPF formulations, and common OPF extensions.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Online Modelling and Optimization of Nonlinear Integrable Systems
Nonlinear integrable optics was recently proposed as a design approach to increase the limits on beam brightness and intensity imposed by fast collective instabilities. To study these systems experimentally, a new research electron and proton storage ring, the Integrable Optics Test Accelerator, was constructed and recently commissioned at Fermilab. Beam-based diagnostics and online modelling of nonlinear systems presents unique challenges - in this paper, we report on our efforts to develop optimization methods suited for such lattices. We explore the effectiveness of neural networks as fast online surrogate estimators, and integrate them into a beam-based tuning algorithm. We also develop a method of knob dimensionality reduction and subsequent robust multivariate optimization for maximizing key performance metrics under complicated lattice optics constraints.
43 PARTICLE ACCELERATORS↗
Spacecraft protective structures design optimization
The optimization of spacecraft protective structures design to defeat hypervelocity impacts of meteoroids and space debris is presented. A space debris environment model is incorporated into an overall optimization methodology employing engineering models developed to predict protective structures design requirements for hypervelocity impact loads. Several nonlinear optimization techniques are used to generate design parametrics based on environment, mission, and configuration variables for the Space Station Core Module Configuration. Results indicate that careful consideration of the spacecraft structural configuration and materials can partially offset the design consequences of dramatic increases in the orbital space debris environment. Furthermore, the use of nonlinear optimization techniques coupled with hypervelocity impact engineering models can provide significant design tradeoff insight through the use of parametric analyses.
Mog, Robert A.↗
Nonlinear programming extensions to rational function approximations of unsteady aerodynamics
This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.
Tiffany, Sherwood H.↗
Design of High-Accuracy Multiple Flyby Trajectories Using Constrained Optimization
The trajectory optimization technique described in this paper provides several distinct advantages over previous formulations. First, fully numerically integrated trajectory modeling is used. That is, no approximations to the trajectory are made and the inclusion of any level of complicated force models desired is allowed. Second, only trajectory propagation is used so there is no requirement for optimization. This is accomplished by the novel method of splitting the trajectory into independent legs, which are then subjected to constrained optimization. Third, each of the trajectory legs may be specified by any convenient set of parameters particularly useful for that leg. Any of these parameters may then be subject to constraints. Fourth, the nonlinear optimization problem is solved by solving a sequence of linear problems which converges to the optimal nonlinear solution. Fifth, the robustness of this formulation requires little or no user interaction with the optimization once a feasible problem has been posed.
flyby↗
Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces
The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.
Tiffany, Sherwood H.↗
Powered Descent Guidance with General Thrust-Pointing Constraints
The Powered Descent Guidance (PDG) algorithm and software for generating Mars pinpoint or precision landing guidance profiles has been enhanced to incorporate thrust-pointing constraints. Pointing constraints would typically be needed for onboard sensor and navigation systems that have specific field-of-view requirements to generate valid ground proximity and terrain-relative state measurements. The original PDG algorithm was designed to enforce both control and state constraints, including maximum and minimum thrust bounds, avoidance of the ground or descent within a glide slope cone, and maximum speed limits. The thrust-bound and thrust-pointing constraints within PDG are non-convex, which in general requires nonlinear optimization methods to generate solutions. The short duration of Mars powered descent requires guaranteed PDG convergence to a solution within a finite time; however, nonlinear optimization methods have no guarantees of convergence to the global optimal or convergence within finite computation time. A lossless convexification developed for the original PDG algorithm relaxed the non-convex thrust bound constraints. This relaxation was theoretically proven to provide valid and optimal solutions for the original, non-convex problem within a convex framework. As with the thrust bound constraint, a relaxation of the thrust-pointing constraint also provides a lossless convexification that ensures the enhanced relaxed PDG algorithm remains convex and retains validity for the original nonconvex problem. The enhanced PDG algorithm provides guidance profiles for pinpoint and precision landing that minimize fuel usage, minimize landing error to the target, and ensure satisfaction of all position and control constraints, including thrust bounds and now thrust-pointing constraints.
Carson, John M., III↗
Transition Delay in Hypersonic Boundary Layers via Optimal Perturbations
The effect of nonlinear optimal streaks on disturbance growth in a Mach 6 axisymmetric flow over a 7deg half-angle cone is investigated in an e ort to expand the range of available techniques for transition control. Plane-marching parabolized stability equations are used to characterize the boundary layer instability in the presence of azimuthally periodic streaks. The streaks are observed to stabilize nominally planar Mack mode instabilities, although oblique Mack mode disturbances are destabilized. Experimentally measured transition onset in the absence of any streaks correlates with an amplification factor of N = 6 for the planar Mack modes. For high enough streak amplitudes, the transition threshold of N = 6 is not reached by the Mack mode instabilities within the length of the cone, but subharmonic first mode instabilities, which are destabilized by the presence of the streaks, reach N = 6 near the end of the cone. These results suggest a passive flow control strategy of using micro vortex generators to induce streaks that would delay transition in hypersonic boundary layers.
Boundary layer transition↗
Gain optimization with nonlinear controls
An algorithm has been developed for the analysis and design of controls for nonlinear systems. The technical approach is to use statistical linearization to model the nonlinear dynamics of a system. A covariance analysis is performed to determine the behavior of the dynamical system and a quadratic cost function. Expressions for the cost function and its derivatives are determined so that numerical optimization techniques can be applied to determine optimal feedback laws. The primary application for this report is centered about the design of controls for nominally linear systems but where the controls are saturated or limited by fixed constraints. The analysis is general however and numerical computation requires only that the specific nonlinearity be considered in the analysis.
Slater, G. L.↗
Topology Optimization for Nonlinear Transient Applications Using a Minimally Invasive Approach (LDRD Final Report)
The purpose of this project was to devise, implement, and demonstrate a method that can use Sandia's existing analysis codes (e.g., Sierra, Alegra, the CTH hydro code) with minimal modification to generate objective function gradients for optimization-based design in transient, non-linear, coupled-physics applications. The approach uses a Moving Least Squares representation of the geometry to substantially reduce the number of geometric degrees of freedom. A Multiple-Program Multiple-Data computing model is then used to compute objective gradients via finite differencing. Details of the formulation and implementation are provided, and example applications are presented that show effectiveness and scalability of the approach.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Convergence map with action-angle variables based on square matrix for nonlinear lattice optimization
Not Available
36 MATERIALS SCIENCE↗
Research on an augmented Lagrangian penalty function algorithm for nonlinear programming
The augmented Lagrangian (ALAG) Penalty Function Algorithm for optimizing nonlinear mathematical models is discussed. The mathematical models of interest are deterministic in nature and finite dimensional optimization is assumed. A detailed review of penalty function techniques in general and the ALAG technique in particular is presented. Numerical experiments are conducted utilizing a number of nonlinear optimization problems to identify an efficient ALAG Penalty Function Technique for computer implementation.
Frair, L.↗
Recent Developments in Security-Constrained AC Optimal Power Flow: Overview of Challenge 1 in the ARPA-E Grid Optimization Competition
In “Recent Developments in Security-Constrained AC Optimal Power Flow: Overview of Challenge 1 in the ARPA-E Grid Optimization Competition,” we review the state of the art in practical algorithms for scheduling power-systems operations in the short term and the results of the recent competition organized by the U.S. Advanced Research Projects Agency–Energy. We explain the mixed-integer nonlinear formulation used in the competition for nonspecialists in electrical engineering, the context and organization of the competition, and the performance of competitors. We find that the collective approaches and results of competitors provide support for efforts to move nonlinear optimization techniques into industrial applications, as they have proven to be a robust and efficient alternative to current linear approximation techniques.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Cooperative control of two arms in the transport of an inertial load in zero gravity
In designing a robot control system for dual arm configurations, the control engineer is faced with two challenges: to derive the equations of motion for a given situation, and to meet certain desired control requirements (for instance, minimum energy). The former may involve closed kinematic chains, such as the case when the two arms are grasping a common object. The latter usually involves nonlinear optimization. These issues are considered in the context of transporting an inertial load using two planar three-link arms. A generalized 'reduction transformation' is applied to the dynamics to remove the singularity in the system equations. A suboptimal minimum energy method is presented to reduce a difficult 12-state, six-control nonlinear optimization to two independent, nonconflicting suboptimizations. A simulation example is provided to illustrate the degree of energy reduction possible using the optimal arm torque distribution that was developed.
Carignan, Craig R.↗
Electric Field Conjugation in the Presence of Model Uncertainty
The Wide-Field Infrared Survey Telescope (WFIRST) is a 2.4-meter-diameter space telescope NASA program. The payload will include a coronagraph instrument (CGI). The CGI designs under development use deformable mirrors (DM) to create a point spread function (PSF) with a dark region around the obscured star object. Electric field conjugation (EFC) is an iterative nonlinear optimization procedure that uses measurements of the electric field at the image to determine the DM displacements to modify the PSF to create the region of high contrast in the image. EFC requires a numerical model of the coronagraph to calculate the Jacobian of the system, which is used, along with regularization, to solve for the DM displacements for each iteration of the nonlinear optimization. Ideally, the coronagraph is aligned and calibrated, and the calibration data are used in the numerical model for calculating the Jacobian. However, calibration and alignment measurements always contain uncertainty resulting in calibration error. Therefore, the Jacobian calculated from the numerical model is not an exact representation of the physical coronagraph, and the resulting DM solution for an EFC iteration does not have the exact impact on the electric field of the coronagraph as predicted by the EFC. The result is slow convergence, and, as will be shown, drives the use of more restrictive regularization. Using Monte Carlo trials, we investigate the effect of calibration error on EFC convergence and regulatization. Comparison to results from the High Contrast Imaging Testbed Hybrid Lyot Coronagraph are also presented.
Marx, David↗
Bilinear modeling and nonlinear estimation
New methods are illustrated for online nonlinear estimation applied to the lateral deflection of an elastic beam on board measurements of angular rates and angular accelerations. The development of the filter equations, together with practical issues of their numerical solution as developed from global linearization by nonlinear output injection are contrasted with the usual method of the extended Kalman filter (EKF). It is shown how nonlinear estimation due to gyroscopic coupling can be implemented as an adaptive covariance filter using off-the-shelf Kalman filter algorithms. The effect of the global linearization by nonlinear output injection is to introduce a change of coordinates in which only the process noise covariance is to be updated in online implementation. This is in contrast to the computational approach which arises in EKF methods arising by local linearization with respect to the current conditional mean. Processing refinements for nonlinear estimation based on optimal, nonlinear interpolation between observations are also highlighted. In these methods the extrapolation of the process dynamics between measurement updates is obtained by replacing a transition matrix with an operator spline that is optimized off-line from responses to selected test inputs.
Dwyer, Thomas A. W., III↗
Smoothing-based compressed state Kalman filter for joint state-parameter estimation: Applications in reservoir characterization and CO 2 storage monitoring
The operation of most engineered hydrogeological systems relies on simulating physical processes using numerical models with uncertain parameters and initial conditions. Predictions by such uncertain models can be greatly improved by Kalman-filter techniques that sequentially assimilate monitoring data. Each assimilation constitutes a nonlinear optimization, which is solved by linearizing an objective function about the model prediction and applying a linear correction to this prediction. However, if model parameters and initial conditions are uncertain, the optimization problem becomes strongly nonlinear and a linear correction may yield unphysical results. In this paper, we investigate the utility of one-step ahead smoothing, a variant of the traditional filtering process, to eliminate nonphysical results and reduce estimation artifacts caused by nonlinearities. Here, we present the smoothing-based compressed state Kalman filter (sCSKF), an algorithm that combines one step ahead smoothing, in which current observations are used to correct the state and parameters one step back in time, with a nonensemble covariance compression scheme, that reduces the computational cost by efficiently exploring the high-dimensional state and parameter space. Numerical experiments show that when model parameters are uncertain and the states exhibit hyperbolic behavior with sharp fronts, as in CO 2 storage applications, one-step ahead smoothing reduces overshooting errors and, by design, gives physically consistent state and parameter estimates. We compared sCSKF with commonly used data assimilation methods and showed that for the same computational cost, combining one step ahead smoothing and nonensemble compression is advantageous for real-time characterization and monitoring of large-scale hydrogeological systems with sharp moving fronts.
42 ENGINEERING↗