Feedback loop gains and system behavior ∗

Linking feedback loops and system behavior is part of the foundation of system dynamics, yet the lack of formal tools has so far prevented a systematic application of the concept, except for very simple systems. Having such tools at their disposal would be a great help to analysts in understanding large, complicated simulation models. The paper applies tools from discrete mathematics and graph theory to further develop an idea by Nathan Forrester to formally link individual feedback loop strengths to the system eigenvalues. The significance of a link or a loop gain and an eigenvalue can be expressed in the eigenvalue elasticity, i.e. the relative change of an eigenvalue resulting from a relative change in the gain. The elasticities of individual links and loops may be found through simple matrix operations on the linearized system. Even though the number of feedback loops can grow rapidly with system size, reaching astronomical proportions even for modest systems, a central result of the paper is that one may restrict attention to an independent subset which typically grows only linearly and at most as the square of system size. An algorithm for finding an independent loop set is presented, along with suggestions for how to augment it to select loops with large elasticities. For illustration, the method is applied to a well-known system, the simple longwave model. Because this model exhibits highly nonlinear behavior, it sheds light on the usefulness of linear methods to nonlinear system. The analysis leads to a more thorough and deep understanding of the system and sheds new light on conventional wisdom regarding the role of many of the system’s feedback loops.