Walter J. Grantham

RESPONSE SURFACE MODEL DEVELOPMENT FOR GLOBAL/LOCAL OPTIMIZATION USING RADIAL BASIS FUNCTIONS*

This paper considers the optimization of complex multi-parameter systems in which the objective function is not known explicitly, and can only be evaluated either through costly physical experiments or through computationally intensive numerical simulation. Furthermore, the objective function of interest may contain many local extrema. Given a data set consisting of the value of the objective function at a scattered set of parameter values, we are interested in developing a response surface model to reduce dramatically the required computation time for parameter optimization runs.To accomplish these tasks, a response surface model is developed using radial basis functions. Radial basis functions provide a way of creating a model whose objective function values match those of the original system at all sampled data points. Interpolation to any other point is easily accomplished and generates a model which represents the system over the entire parameter space. This paper presents the details of the use

A chaotic limit cycle paradox

Duffings equation with sinusoidal forcing produces chaos for certain combinations of the forcing amplitude and frequency. To determine the most chaotic response achieveable for given bounds on the input force, an optimal control problem was investigated to maximize the largest Lyapunov exponent, which in this case also corresponds to maximizing the Kaplan-Yorke Lyapunov fractal dimension. The resulting bang-bang optimal feedback controller yielded a bounded attractor with a positive largest Lyapunov exponent and a fractional Lyapunov dimension, indicating a chaotic strange attractor. Indeed, the largest Lyapunov exponent was approximately twice as large as that achieved with sinusoidal forcing at the same amplitude. However, the resulting attractor is just a stable limit cycle and is not chaotic or fractal at all! this contradicts the basic idea that a bounded attractor with at least one positive Lyapunov exponent must be chaotic and fractal.This article provides details of this chaotic limit cycle paradox and the resolution of the paradox. In particular for systems of differential equations with only piecewise differentiable right-hand sides, a jump discontinuity condition must be imposed on the state perturbations in order to compute correct Lyapunov exponents.

Estimating Controllability Boundaries for Uncertain Systems

Two methods, combining controllability and Lyapunov stability, are presented for estimating reachability boundaries for nonlinear control systems. The controllability method is exact, but lacks appropriate boundary conditions in certain cases and is effectively restricted to two-dimensional problems. The approximate Lyapunov method combines Lyapunov stability theory with a controllability maximum principle. The Lyapunov estimate of the reachable set generally differs from the actual reachable set, but the estimate is conservative (i.e., guaranteed to contain the actual reachable set), does not require integration of the equations of motion, and is applicable to n-dimensional systems. Both methods are applied to an example of a prey-predator fishery with bounded harvesting efforts.

Neighbouring extremals for nonlinear systems with control constraints

In this paper, we consider a class of nonlinear optimal control problems subject to control constraints. We assume that an initial condition for the dynamical system is specified. Then, we can easily compute an openloop optimal control using any convenient optimal control software package. Now, suppose the optimal trajectory is perturbed due to a change in initial conditions or uncertainty in the model equations. If the perturbations are not too large, it is known that the neighbouring extremal approach can be used to obtain a feedback law which adjusts the open-loop optimal control accordingly provided there are no bounds on the control variables. In this paper, our aim is to develop a computational method for the more general case of fixed-time problems with control constraints.

Discretization chaos - Feedback control and transition to chaos

Problems in the design of feedback controllers for chaotic dynamical systems are considered theoretically, focusing on two cases where chaos arises only when a nonchaotic continuous-time system is discretized into a simpler discrete-time systems (exponential discretization and pseudo-Euler integration applied to Lotka-Volterra competition and prey-predator systems). Numerical simulation results are presented in extensive graphs and discussed in detail. It is concluded that care must be taken in applying standard dynamical-systems methods to control systems that may be discontinuous or nondifferentiable.

DFW microburst model based on AA-539 data

Analysis of the August 2, 1985 crash for an L-1011 jumbo jet (DL-191) on approach to the Dallas-Ft. Worth International Airport (DFW) in a thunderstorm indicates that the severe windshear microburst that caused the crash was composed not only of a strong downflow and outflow but also included several large-scale vortex rings entrained in the flowfield. This paper presents a detailed two-dimensional model of the DFW microburst based on data from the MD-80 (AA-539) that followed behind DL-191 and flew through the microburst about two minutes after the crash of DL-191. The model was developed using wind-vector and flight-path data reconstructed by NASA Ames Research Center and a combination of interactive graphics and least-squares error best fit between the modeled and measured wind vectors along the AA-539 flight path. The model indicates that the flowfield contains some significant elements and vortices not previously reported. The alternating direction of rotation of the vortices in the model suggests a microburst structure based on a von Karman vortex street rather than on a Kelvin-Helmholtz instability. The model also indicates that the reconstructed wind-vector data contain a time lag of at least one second in the horizontal winds.

Gradient Transformation Trajectory Following Algorithms for Determining Stationary Min-Max Saddle Points

For finding a stationary min-max point of a scalar-valued function, we develop and investigate a family of gradient transformation differential equation algorithms. This family includes, as special cases: Min-Max Ascent, Newton’s method, and a Gradient Enhanced Min-Max (GEMM) algorithm that we develop. We apply these methods to a sharp-spined “Stingray” saddle function, in which Min-Max Ascent is globally asymptotically stable but stiff, and Newton’s method is not stiff, but does not yield global asymptotic stability. However, GEMM is both globally asymptotically stable and not stiff. Using the Stingray function we study the stiffness of the gradient transformation family in terms of Lyapunov exponent time histories. Starting from points where Min-Max Ascent, Newton‘s method, and the GEMM method do work, we show that Min-Max Ascent is very stiff. However, Newton’s method is not stiff and is approximately 60 to 440 times as fast as Min-Max Ascent. In contrast, the GEMM method is globally convergent, is not stiff, and is approximately 3 times faster than Newton’s method and approximately 175 to 1000 times faster than Min-Max Ascent.

Controllability conditions and effective controls for nonlinear systems

For nonlinear control systems with control constraints, necessary and sufficient conditions are developed for a feedback control law to transfer the state to a specified target from every initial state at which this is possible. Sufficient conditions are developed, using the Brouwer fixed point theorem, for systems having trajectories that are unique and positively invariant sets that satisfy certain topological conditions.

Notch Filter Feedback Control for k-Period Motion in a Chaotic System ∗

Chaotic motion can sometimes be desirable or undesirable, and hence control over such a phenomenon has become a topic of considerable interest. Currently available methods involve making systematic time-varying small perturbations in the system parameters. A new method is presented here to achieve control over chaotic motion using notch filter output feedback control. The notch filter controller uses an active negative feedback with fixed controller parameters without affecting the original system parameters. The motivation for using a notch filter in the feedback is to disturb the balance of power at the lower end of the participating frequencies in the power spectrum. This results in a truncation of the period-doubling route to chaos. For low-period motions the harmonic balance method is used to show that a single participating frequency can indeed be eliminated. To deal with relatively complex nonlinear plants, and higher-period motions, a numerical optimal parameter selection scheme is presented to choose the notch filter parameters. The procedures are tested on Duffing’s oscillator with a notch filter feedback to achieve desired k-period motion.

A Chaotic System: Discretization and Control

This paper examines chaos in some population models for biological processes. In particular, the paper is concerned with chaos produced by discretization of continuous-time population models, and with the question of whether or not this chaotic behavior can occur when a feedback control management strategy is applied to continuous-time population models. Using two classical population models, it is shown that both exponential discretization and discretization associated with numerical simulation can produce chaos in the resulting discrete-time system. For example, a continuous-time Lotka-Volterra system exhibits periodic trajectories, but a particular discretization procedure is shown to yield discrete-time trajectories that all converge to a very complicated chaotic strange attractor, even if the discretization time steps are small. The results in the paper also show that this chaotic behavior can occur even if a feedback control management strategy is applied to the continuous-time system to stabilize the equilibrium point.