Structure and stability of discrete-time optimal systems
Discrete-time optimal linear systems structure and stability, assuming quadratic costs and perfect observability
Rappaport, D.↗
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Results for “Discrete Optimization”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Solar collector parameter identification from unsteady data by a discrete-gradient optimization algorithm
A discrete-gradient optimization algorithm is used to identify the parameters in a one-node and a two-node capacitance model of a flat-plate collector. Collector parameters are first obtained by a linear-least-squares fit to steady state data. These parameters, together with the collector heat capacitances, are then determined from unsteady data by use of the discrete-gradient optimization algorithm with less than 10 percent deviation from the steady state determination. All data were obtained in the indoor solar simulator at the NASA Lewis Research Center.
Hotchkiss, G. B.↗
Necessary conditions for discrete parameter stochastic optimization problems
Discrete parameter stochastic optimization problems necessary conditions, deriving maximum principle
Kushner, H. J.↗
Three-dimensional aerodynamic shape optimization using discrete sensitivity analysis
An aerodynamic shape optimization procedure based on discrete sensitivity analysis is extended to treat three-dimensional geometries. The function of sensitivity analysis is to directly couple computational fluid dynamics (CFD) with numerical optimization techniques, which facilitates the construction of efficient direct-design methods. The development of a practical three-dimensional design procedures entails many challenges, such as: (1) the demand for significant efficiency improvements over current design methods; (2) a general and flexible three-dimensional surface representation; and (3) the efficient solution of very large systems of linear algebraic equations. It is demonstrated that each of these challenges is overcome by: (1) employing fully implicit (Newton) methods for the CFD analyses; (2) adopting a Bezier-Bernstein polynomial parameterization of two- and three-dimensional surfaces; and (3) using preconditioned conjugate gradient-like linear system solvers. Whereas each of these extensions independently yields an improvement in computational efficiency, the combined effect of implementing all the extensions simultaneously results in a significant factor of 50 decrease in computational time and a factor of eight reduction in memory over the most efficient design strategies in current use. The new aerodynamic shape optimization procedure is demonstrated in the design of both two- and three-dimensional inviscid aerodynamic problems including a two-dimensional supersonic internal/external nozzle, two-dimensional transonic airfoils (resulting in supercritical shapes), three-dimensional transport wings, and three-dimensional supersonic delta wings. Each design application results in realistic and useful optimized shapes.
Burgreen, Gregory W.↗
A Bell-Curved Based Algorithm for Mixed Continuous and Discrete Structural Optimization
An evolutionary based strategy utilizing two normal distributions to generate children is developed to solve mixed integer nonlinear programming problems. This Bell-Curve Based (BCB) evolutionary algorithm is similar in spirit to (mu + mu) evolutionary strategies and evolutionary programs but with fewer parameters to adjust and no mechanism for self adaptation. First, a new version of BCB to solve purely discrete optimization problems is described and its performance tested against a tabu search code for an actuator placement problem. Next, the performance of a combined version of discrete and continuous BCB is tested on 2-dimensional shape problems and on a minimum weight hub design problem. In the latter case the discrete portion is the choice of the underlying beam shape (I, triangular, circular, rectangular, or U).
Kincaid, Rex K.↗
Endogenous Interface Pricing for Consistent Transmission–Distribution Co-Optimization With Discrete Distribution Controls
This paper proposes an endogenous interface pricing model for day-ahead transmission–distribution co-optimization that co-determines the interface locational marginal price (LMP) and the transmission–distribution exchange, ensuring price–dispatch consistency while optimally scheduling discrete distribution controls. The formulation couples a DC optimal power flow (OPF) with a branch-flow AC OPF that schedules distributed energy resources (DERs), tap-changer settings, capacitor banks (CBs), and multi-period energy storage systems (ESSs) under feeder voltage and current limits, and is solved as a mixed-integer second-order cone program (MISOCP). In a T14–D33 system, coordinated device scheduling recovers about 90% of the distribution-to-transmission export achievable in a reference case that ignores distribution network (DN) limits, while satisfying a 1.05 p.u. voltage upper bound. In a T39–D34/D37/D123 system, a sequential decoupled benchmark produces interface LMP distortions up to 12.5% and a 7.28% mismatch in net export energy, whereas the proposed model removes these distortions and the associated settlement mismatches. Second-order cone (SOC) relaxation gaps remain below $10^{-3}$ in all cases.
Noh, Seung-Gil↗
Maneuvering of large flexible space antennas using optimal discrete-time controllers
The problem of maneuvering large space structures is formulated as a discrete-time, finite-duration optimal regulator problem. Two methods are considered for minimizing the elastic motion during the maneuver. The methods are applied to retargetting maneuver of a large space antenna, and the results are compared with those obtained using bang-bang control. The optimal regulator designed using the model error sensitivity supression method gives the best performance.
Joshi, S. M.↗
Optimal discrete-time dynamic output-feedback design - A w-domain approach
An alternative method for optimal digital control design is described in this paper. The method is based on the usage of the w-transform and has many attractive design features. One of these is its immediate connection with frequency loop-shaping techniques that are now popular and effective for multivariable control synthesis in continuous-time domain. Furthermore, any design algorithms originally developed for continuous-time systems can now be immediately extended to the discrete-time domain. The main results presented in this paper are the exact problem formulation and solution of an optimal discrete-time dynamic output-feedback design in the w-domain involving a quadratic performance index to random disturbances. In addition, necessary conditions for optimality are obtained for the numerical solution of the optimal output-feedback compensator design. A numerical example is presented illustrating its application to the design of a low-order dynamic compensator in a stability augmentation system of a commercial transport.
Ha, Cheolkeun↗
Adaptive proximate time optimal servomechanisms: Discrete time case
Following a review of the continuous time proximate time-optimal servomechanism (PTOS), a discrete-time PTOS is developed, along with conditions for its stability. An algorithm is proposed for adapting PTOS when the plant parameters are not known yielding APTOS (adaptive PTOS). Under ideal conditions APTOs is shown to be uniformly asymptotically stable. Simulation results demonstrate the predicted performance, and experimental results validate the practically of APTOS.
Workman, M. L.↗
Recursive multibody dynamics and discrete-time optimal control
A recursive algorithm is developed for the solution of the simulation dynamics problem for a chain of rigid bodies. Arbitrary joint constraints are permitted, that is, joints may allow translational and/or rotational degrees of freedom. The recursive procedure is shown to be identical to that encountered in a discrete-time optimal control problem. For each relevant quantity in the multibody dynamics problem, there exists an analog in the context of optimal control. The performance index that is minimized in the control problem is identified as Gibbs' function for the chain of bodies.
Deleuterio, G. M. T.↗
On the optimization of discrete structures with aeroelastic constraints
The paper deals with the problem of dynamic structural optimization where constraints relating to flutter of a wing (or other dynamic aeroelastic performance) are imposed along with conditions of a more conventional nature such as those relating to stress under load, deflection, minimum dimensions of structural elements, etc. The discussion is limited to a flutter problem for a linear system with a finite number of degrees of freedom and a single constraint involving aeroelastic stability, and the structure motion is assumed to be a simple harmonic time function. Three search schemes are applied to the minimum-weight redesign of a particular wing: the first scheme relies on the method of feasible directions, while the other two are derived from necessary conditions for a local optimum so that they can be referred to as optimality-criteria schemes. The results suggest that a heuristic redesign algorithm involving an optimality criterion may be best suited for treating multiple constraints with large numbers of design variables.
Mcintosh, S. C., Jr.↗
Multi-level adaptive finite element methods. 1: Variation problems
A general numerical strategy for solving partial differential equations and other functional problems by cycling between coarser and finer levels of discretization is described. Optimal discretization schemes are provided together with very fast general solvers. It is described in terms of finite element discretizations of general nonlinear minimization problems. The basic processes (relaxation sweeps, fine-grid-to-coarse-grid transfers of residuals, coarse-to-fine interpolations of corrections) are directly and naturally determined by the objective functional and the sequence of approximation spaces. The natural processes, however, are not always optimal. Concrete examples are given and some new techniques are reviewed. Including the local truncation extrapolation and a multilevel procedure for inexpensively solving chains of many boundary value problems, such as those arising in the solution of time-dependent problems.
Brandt, A.↗
Numerical approximation for the infinite-dimensional discrete-time optimal linear-quadratic regulator problem
An abstract approximation framework is developed for the finite and infinite time horizon discrete-time linear-quadratic regulator problem for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional Hilbert space. The schemes included the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed.
Gibson, J. S.↗
Numerical approximation for the infinite-dimensional discrete-time optimal linear-quadratic regulator problem
An abstract approximation framework is developed for the finite and infinite time horizon discrete-time linear-quadratic regulator problem for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional Hilbert space. The schemes included the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed.
Gibson, J. S.↗
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.↗
Selection of sampling rate for digital control of aircrafts
The considerations in selecting the sample rates for digital control of aircrafts are identified and evaluated using the optimal discrete method. A high performance aircraft model which includes a bending mode and wind gusts was studied. The following factors which influence the selection of the sampling rates were identified: (1) the time and roughness response to control inputs; (2) the response to external disturbances; and (3) the sensitivity to variations of parameters. It was found that the time response to a control input and the response to external disturbances limit the selection of the sampling rate. The optimal discrete regulator, the steady state Kalman filter, and the mean response to external disturbances are calculated.
Katz, P.↗
Optimal stabilization of discrete event systems
An optimal stabilization problem for discrete event systems (DES) is addressed. A class of not necessarily finite state 'logical' DES models is utilized which can also model the costs for events to occur. Let P and A denote two such models. Suppose that P characterizes the valid behavior of a dynamical system and A represents certain design objectives which specify the allowable DES behavior which is 'contained in' the valid behavior. An optimal control problem for P and A is how to choose the sequence of inputs to P so that the DES behavior lies in A (i.e., it is allowable) and so that a performance index defined in terms of the costs of the events is minimized. Two solutions are provided to an optimal stabilization problem, i.e. how to find a sequence of inputs that results in an optimal state trajectory which cycles in a pre-specified set.
Passino, Kevin M.↗
A penalty approach for nonlinear optimization with discrete design variables
Introduced here is a simple approach to minimization problems with discrete design variables by modifying the penaly function approach of converting the constrained problems into sequential unconstrained minimization technique (SUMT) problems. It was discovered, during the course of the present work, that a similar idea was suggested by Marcal and Gellatly. However, no further work has been encountered. A brief description of the SUMT is presented. The form of the penalty function for the discrete-valued design variables and strategy used for the implementation of the procedure is discussed next. Finally, several design examples are used to demonstrate the procedure, and results are compared with the ones available in the literature.
Shin, Dong K.↗