Analytical Methods for Structural Dominance Analysis in System Dynamics

Abstract

Glossary Behavior mode The traditional meaning of the term is the qualitative nature of the observed system behavior, such as damped or expanding oscillations, overshoot and collapse, exponential growth or adjustment to equilibrium, or limit cycles. In linear systems theory, the term has a more specific meaning, cf. the explanation for eigenvalues. Bode plot (phase and gain plot) A tool used in classical control theory to characterize the frequency response, i.e., the amplification A and phase shift f in the system output variable of interest x(t) = A sin(ot + f) compared to the input variable u(t) = sin(ot), as a function of the frequency o of the input. Chaos A type of behavior exhibited by nonlinear systems that appears to be approximately periodic but with a seemingly random element. A hallmark of chaotic behavior is that it is sensitive to initial conditions. Dominant structure A general term for the feedback loops (or possibly external driving forces) that are "most important" in generating a behavior pattern of interest. In nonlinear models, particularly single-transient models, there is frequently a shift in structural dominance , i.e., in the strength and significance of certain feedback loops. Dynamic decomposition weights (DDW) An application of eigenvector elasticity analysis (EVA) that focuses on how parameter changes influence the relative weights (DDW) of the system behavior modes in a particular variable. Eigenvalue elasticity analysis (EEA) A method of analyzing the significance of a structural element, say a loop or a link in the model with a gain g, in terms of its marginal effect upon the eigenvalues l of the system. There are several such measures, such as the influence measure @l/@g Á g, the elasticity @l/@g Á (g/l), or, in the case of complex-valued eigenvalues, the effect upon the damping ratio, natural frequency , damping time, etc., as illustrated in Fig. 8. See also Loop Eigenvalue Elasticity Analysis (LEEA). Eigenvalue An eigenvalue for a square matrix A is a value l for which the equation Ar = lr has a nonzero solution r 6 ¼ 0. The column vector r is called the (right) eigenvector corresponding to the eigenvalue l. The eigenvalues and eigen-vectors determine the behavior modes (components) in the solution to the linear dynamical system _ x ¼ Ax. A real eigenvalue l leads to an exponential behavior mode exp(lt), while a complex eigenvalue l = t AE io leads to oscillatory behavior modes exp(tt) sin (ot + f).