Dale B. McDonald

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

Locally Precise Response Surface Models for the Generalization of Controlled Dynamic Systems and Associated Performance Measures

This treatment demonstrates the utility of response surface models (RSMs) as predictive, companion tools which aid in the development of harvesting (control) strategies applicable to predator-prey dynamic systems. To this end a control algorithm is derived that considers the regulation of a predator-prey natural resource while considering revenue for commercial ventures and regulatory agencies. Numerical simulations provide the mechanism to quantify performance measures associated with control algorithms, yet complicated problems require that all “tools” available be considered. Complex problems may be more tractable when simulation results are combined with alternate, continuous models exhibiting predictive capacities. For this reason, RSMs are appealing; analytic evaluation of the state, the gradient, and the Hessian matrix is possible. From these models we may glean valuable information linked to the gathered data revealing information about the “true nature” of the ecological system. Therefore, we propose to create RSMs based on scattered data obtained from the ordinary differential equation (ODE) dynamic system model. These response surface models are constructed using radial basis functions (RBFs); RSMs so created have the desirable property of matching the objective function value exactly at each sampled data point. Furthermore, they have the ability to interpolate to any desired point throughout the parameter space. This is powerful as the “objective function” may be any function of critical importance to the analyst which in this treatment is the predator biomass time rate of change (ODE) itself. This has the immediate implication of providing a single ODE model, with a “locally” or even perhaps a “globally” precise nature. Since such models are constructed from scattered data, which is consistent with what would be collected from field measurements, a further connection of theory to practice is realized. It will be shown that these RSMs provide greater insight into ecological systems, with special emphasis on parameter estimation.Copyright

Singular Perturbation Trajectory Following Algorithms for Min-Max Differential Games

This chapter examines trajectory following algorithms for differential games subject to simple bounds on player strategy variables. These algorithms are trajectory following in the sense that closed-loop player strategies are generated directly by the solutions to ordinary differential equations. Player strategy differential equations are based upon Lyapunov optimizing control techniques and represent a balance between the current penetration rate for an appropriate descent function and the current cost accumulation rate. This numerical strategy eliminates the need to solve 1) a min-max optimization problem at each point along the state trajectory and 2) nonlinear two-point boundary-value problems. Furthermore, we address “stiff” systems of differential equations that arise during the design process and seriously degrade algorithmic performance. We use standard singular perturbation methodology to produce a numerically tractable algorithm. This results in the Efficient Cost Descent (ECD) algorithm which possesses desirable characteristics unique to the trajectory following method. Equally important as a specification of a new trajectory following algorithm is the observation and resolution of several issues regarding the design and implementation of a trajectory following algorithm in a differential game setting.

Newton Methods to Solve a System of Nonlinear Algebraic Equations

Fundamental insight into the solution of systems of nonlinear equations was provided by Powell. It was found that Newton iterations, with exact line searches, did not converge to a stationary point of the natural merit function, i.e., the Euclidean norm of the residuals. Extensive numerical simulation of Powell’s equations produced the unexpected result that Newton iterations converged to the solution from all initial points, where the function is defined, or from those points where the Jacobian is nonsingular, if no line search is used. The significance of Powell’s example is that an important requirement exists when utilizing Newton’s method to solve such a system of nonlinear equations. Specifically, a merit function, which is used in a line search, must have properties consistent with those of a Lyapunov function to provide sufficient conditions for convergence. This implies that level sets of the merit function are properly nested, either globally, or in some finite local region. Therefore, they are topologically equivalent to concentric spherical surfaces, either globally or in a finite local region. Furthermore, an exact line search at a point, far from the solution, may be counterproductive. This observation, and a primary aim of the present analysis, is to demonstrate that it is desirable to construct new Newton iterations, which do not require a merit function with associated line searches.

Parameter Identification in Ecological Systems via Discontinuous and Singular Control Regimes

Policy decisions regarding commercial harvesting of aquatic species by (typically governmental) regulatory agencies are often based in part upon field data, simulation results, and mathematical models. Regulatory agencies may limit or expand seasons, determine total harvest allowed, increase or decrease licensure fees, and raise or lower taxation rates in response to the state of the ecological system. Ultimately, the regulatory agency uses such measures to ensure viable populations in an attempt to balance ecosystem health and benefits for society. Such decisions impact commercial fishing ventures affecting the nature of harvesting efforts and their intensity. Conclusions drawn from mathematical models of ecological systems, and derived simulation results which affect this reality are highly dependent upon the validity of information available. Knowledge or estimates of critical parameters such as intrinsic growth rate, carrying capacity, etc. and dynamic variables such as biomass levels dictate the usefulness of analytical and numerical analyses. The purpose of this treatment is to illustrate that control laws applied to mathematical models of species dynamics may be used to discern estimates of parameters that inherently exist in such models in an effort to provide more valuable information upon which to base policy decisions. Dynamic models of both single-species evolution and predator-prey interactions are examined.© 2012 ASME

A Classroom Experience in Control Systems Following an Intensive Nonlinear Control Design Research Project: Two Perspectives

It has been well documented in the engineering education literature that introductory linear control systems courses present unique pedagogical challenges. Similarly, it has been reported that engaging undergraduate students in control systems research is challenging. The control of nonlinear systems is the focus of much research; Therefore, a paradox exists; research programs involving undergraduate students are often conducted in a “nonlinear before linear” fashion. Prior research by the authors approached this paradox by investigating whether a meaningful research project in nonlinear control systems could be conducted. While the author and co-author were willing to execute this research (despite the intense time commitment), this type of project is not sustainable long-term unless the significant time invested results in Item 1) illumination of new pedagogical techniques that generalize to each student in the linear control systems course (not just participants in the research project). This manuscript will describe in detail Item 1) . Given the student and faculty mentors experience in the nonlinear controls research project and the student’s experience in the linear control systems course, pedagogical insights gained through this endeavor will be detailed by the faculty member. These are meant to close the loop of instructor assessment of the course during preparation for future offerings. Therefore, the undergraduate student directly provides an accounting of the experiences throughout the research program and introductory control systems course. The result is a proposed pedagogical approach to be implemented by three distinct methodologies. Methodology (1) Traditional lecture format, Methodology (2) Targeted physical or hands-on experiences where students are exposed to a controlled system/process commonly seen in industry, and Methodology (3) Direct student participation in the design of certain aspects of course materials.Copyright

SIMULATION, DESIGN, AND IMPLEMENTATION BASED UNDERGRADUATE RESEARCH IN CONTROL SYSTEMS: NONLINEAR BEFORE LINEAR

A typical undergraduate curriculum introduces linear control systems concepts only, often in a single elective course. This curriculum structure introduces challenges to student involvement in control systems research as nonlinear concepts are the focus of the majority of such efforts. With undergraduate participation in engineering research steadily increasing, nonlinear control concepts must be introduced prior to formal classroom study of linear systems. Given this reality, we propose an intense and relatively brief research program, consisting of three distinct phases. The program objective is to present a targeted educational experience in nonlinear control theory based upon the design and implementation of control laws developed for a particular nonlinear system class. Given significant interaction between the student and the faculty mentor, we believe that an excellent opportunity in undergraduate education and research will be realized, despite the student’s initial unfamiliarity with nonlinear control systems concepts. A research program consisting of three phases is proposed and initial technical results are presented to facilitate a candid discussion of the issues that may prevent undergraduate participation in research and to detail the manner in which many of these obstacles were overcome.Copyright