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NASA NTRS1971

Parameter identification using a creeping-random-search algorithm

A creeping-random-search algorithm is applied to different types of problems in the field of parameter identification. The studies are intended to demonstrate that a random-search algorithm can be applied successfully to these various problems, which often cannot be handled by conventional deterministic methods, and, also, to introduce methods that speed convergence to an extremal of the problem under investigation. Six two-parameter identification problems with analytic solutions are solved, and two application problems are discussed in some detail. Results of the study show that a modified version of the basic creeping-random-search algorithm chosen does speed convergence in comparison with the unmodified version. The results also show that the algorithm can successfully solve problems that contain limits on state or control variables, inequality constraints (both independent and dependent, and linear and nonlinear), or stochastic models.

Parrish, R. V.↗

NASA NTRS1972

N-burn optimal analytic trajectories.

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

Jezewski, D. J.↗

NASA NTRS1972

Optimum configurations for bangless sonic booms.

A number of optimization problems are posed and solved for supersonic aircraft flight subject to the condition that a shock wave appears only incipiently in the sonic boom signal at a given point. The principal result is one giving the maximum effective gross weight of an aircraft of given effective length under given flight conditions. The calculus of variations with inequality constraints is used, with the novel features of a non-local isoperimetric relation and of only an upper bound on a control variable.

Hayes, W. D.↗

NASA NTRS1972

Constrained optimal ascent-flyback shuttle trajectories.

An optimal Space Shuttle ascent-flyback trajectory shaping capability is presented which is based on the accelerated gradient parameter optimization technique. A typical atmospheric flight branched optimization problem is analyzed which required the determination of 31 parameters. This parameter set includes the description of the vehicle attitude control angles for three branches of Shuttle flight: first stage ascent, second stage ascent, and first stage flyback. The important in-flight inequality constraints required to maintain the integrity of the vehicle are considered. Results indicate that for a launch into a 55 deg inclined ellipse, a 13% increase in payload can be realized by using optimal control in the first-stage ascent rather than the conventional gravity-turn steering.

Kamm, J. L.↗

NASA NTRS1975

An initial feasibility stage for Stoer's constrained least squares algorithm

A procedure is described for computing an initial feasible vector, x sub 0, for Stoer's algorithm for solving the linear least squares problem subject to linear equality and inequality constraints. The procedure described fits well with Stoer's algorithm since much of the computation performed to determine x sub 0 accomplishes initializing transformations of the problem data, which would otherwise be done in Stoer's algorithm after being given an x sub 0.

Lawson, C. L.↗

NASA NTRS1975

A steepest descents method for reentry optimization

A steepest descents optimization program is applied to the problem of a lifting vehicle entering the earth's atmosphere. The program employs penalty functions representing terminal conditions and inflight inequality constraints. During each iteration, it reduces a single performance measure which is the sum of the performance index and the penalty functions. Therefore, only one set of adjoint equations must be integrated per iteration. Values of weight factors, multiplying the penalty functions, are automatically adjusted before each iteration in order that the penalty functions will approach acceptable values. This method is shown to be a form of the classical Lagrange multiplier methods.

Andrus, J. F.↗

NASA NTRS1975

Primer vector theory and applications

A method developed to compute two-body, optimal, N-impulse trajectories was presented. The necessary conditions established define the gradient structure of the primer vector and its derivative for any set of boundary conditions and any number of impulses. Inequality constraints, a conjugate gradient iterator technique, and the use of a penalty function were also discussed.

Jezewski, D. J.↗

NASA NTRS1976

Optimization of multi-constrained structures based on optimality criteria

A weight-reduction algorithm is developed for the optimal design of structures subject to several multibehavioral inequality constraints. The structural weight is considered to depend linearly on the design variables. The algorithm incorporates a simple recursion formula derived from the Kuhn-Tucker necessary conditions for optimality, associated with a procedure to delete nonactive constraints based on the Gauss-Seidel iterative method for linear systems. A number of example problems is studied, including typical truss structures and simplified wings subject to static loads and with constraints imposed on stresses and displacements. For one of the latter structures, constraints on the fundamental natural frequency and flutter speed are also imposed. The results obtained show that the method is fast, efficient, and general when compared to other competing techniques. Extensions to the generality of the method to include equality constraints and nonlinear merit functions is discussed.

Rizzi, P.↗

NASA NTRS1979

Supercritical tests of a self-optimizing, variable-Camber wind tunnel model

A testing procedure was used in a 16-foot Transonic Propulsion Wind Tunnel which leads to optimum wing airfoil sections without stopping the tunnel for model changes. Being experimental, the optimum shapes obtained incorporate various three-dimensional and nonlinear viscous and transonic effects not included in analytical optimization methods. The method is a closed-loop, computer-controlled, interactive procedure and employs a Self-Optimizing Flexible Technology wing semispan model that conformally adapts the airfoil section at two spanwise control stations to maximize or minimize various prescribed merit functions subject to both equality and inequality constraints. The model, which employed twelve independent hydraulic actuator systems and flexible skins, was also used for conventional testing. Although six of seven optimizations attempted were at least partially convergent, further improvements in model skin smoothness and hydraulic reliability are required to make the technique fully operational.

Levinsky, E. S.↗

NASA NTRS1978

Optimal analytic rendezvous using Clohessy-Wiltshire equations

The optimal solution time that minimizes the sum of the two applied impulses necessary to rendezvous is obtained analytically for the Clohessy-Wiltshire equations with a linear gravity model assumption. A plume impingement inequality constraint on the solution is examined, and an optimal policy is developed. Numerical tests are conducted to verify the analysis and to illustrate the optimal solution algorithm.

Jezewski, D. J.↗

NASA NTRS1979

An analytic approach to optimal rendezvous using Clohessy-Wiltshire equations

An analytic approach is used to obtain the optimal solution time that minimizes the sum of the two applied impulses necessary to rendezvous for the Clohessy-Wiltshire equations. A plume impingement inequality constraint on the solution is examined, and an optimal policy is developed. Numerical tests are conducted to verify the analysis and to illustrate the optimal solution algorithm.

Jezewski, D. J.↗

NASA NTRS1981

Application of constrained optimization to active control of aeroelastic response

Active control of aeroelastic response is a complex in which the designer usually tries to satisfy many criteria which are often conflicting. To further complicate the design problem, the state space equations describing this type of control problem are usually of high order, involving a large number of states to represent the flexible structure and unsteady aerodynamics. Control laws based on the standard Linear-Quadratic-Gaussian (LQG) method are of the same high order as the aeroelastic plant. To overcome this disadvantage of the LQG mode, an approach developed for designing low order optimal control laws which uses a nonlinear programming algorithm to search for the values of the control law variables that minimize a composite performance index, was extended to the constrained optimization problem. The method involves searching for the values of the control law variables that minimize a basic performance index while satisfying several inequality constraints that describe the design criteria. The method is applied to gust load alleviation of a drone aircraft.

Newsom, J. R.↗

NASA NTRS1981

Current research on shear buckling and thermal loads with PASCO: Panel Analysis and Sizing Code

The PASCO computer program to obtain the detailed dimensions of optimum stiffened composite structural panels is described. Design requirements in terms of inequality constraints can be placed on buckling loads or vibration frequencies, lamina stresses and strains, and overall panel stiffness for each of many load conditions. General panel cross sections can be treated. An analysis procedure involving a smeared orthotropic solution was investigated. The conservatism in the VIPASA solution and the danger in a smeared orthotropic solution is explored. PASCO's capability to design for thermal loadings is also described. It is emphasized that design studies illustrate the importance of the multiple load condition capability when thermal loads are present.

Stroud, W. J.↗

NASA NTRS1983

Finite difference numerical methods for boundary control problems governed by hyperbolic partial differential equations

This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.

Chen, G.↗

NASA NTRS1984

A new minimax algorithm

The representation min t s.t. F(I)(x). - t less than or equal to 0 for all i is examined. An active set strategy is designed of functions: active, semi-active, and non-active. This technique will help in preventing zigzagging which often occurs when an active set strategy is used. Some of the inequality constraints are handled with slack variables. Also a trust region strategy is used in which at each iteration there is a sphere around the current point in which the local approximation of the function is trusted. The algorithm is implemented into a successful computer program. Numerical results are provided.

Vardi, A.↗

NASA NTRS1984

Fitting aerodynamic forces in the Laplace domain: An application of a nonlinear nongradient technique to multilevel constrained optimization

A technique which employs both linear and nonlinear methods in a multilevel optimization structure to best approximate generalized unsteady aerodynamic forces for arbitrary motion is described. Optimum selection of free parameters is made in a rational function approximation of the aerodynamic forces in the Laplace domain such that a best fit is obtained, in a least squares sense, to tabular data for purely oscillatory motion. The multilevel structure and the corresponding formulation of the objective models are presented which separate the reduction of the fit error into linear and nonlinear problems, thus enabling the use of linear methods where practical. Certain equality and inequality constraints that may be imposed are identified; a brief description of the nongradient, nonlinear optimizer which is used is given; and results which illustrate application of the method are presented.

Tiffany, S. H.↗

NASA NTRS1985

ISAAC (integrated Structural Analysis and Control) via Continuum Modeling and Distributed Frequency Domain Design Techniques

A methodology is proposed for the integrated design of structures and their controls. The integration of design is accomplished through simultaneous selection, via optimization, of designer-specified parameters governing the structure and its controller. The cost function is chosen to address the primary structural and control goals. Inequality constraints are added to insure that additional design requirements are met. The methodology is illustrated through development of an example involving a cantilever beam. A distributed transfer function is derived from the Bernoulli-Euler beam equation. A controller parameterization is then chosen, and performance requirements specified. A controller is then obtained via optimization, which is shown to give good performance.

Gustafson, C. L.↗

NASA NTRS1985

Optimal impulsive manoeuvres and aerodynamic braking

A method developed for obtaining solutions to the aerodynamic braking problem, using impulses in the exoatmospheric phases is discussed. The solution combines primer vector theory and the results of a suboptimal atmospheric guidance program. For a specified initial and final orbit, the solution determines: (1) the minimum impulsive cost using a maximum of four impulses, (2) the optimal atmospheric entry and exit-state vectors subject to equality and inequality constraints, and (3) the optimal coast times. Numerical solutions which illustrate the characteristics of the solution are presented.

Jezewski, D. J.↗