Enoch Yeung and Dr. Sean Warnick, Computer Science

The purpose of this project was to investigate systematic methods for finding simplified representations that accurately model the structure and input-output dynamics of high dimensional nonlinear time-invariant systems near equilibrium points. As background, the dynamics of nonlinear time-invariant systems near their equilibrium points can be approximated using linearized state-space systems. As long as the trajectory of the nonlinear system remains close enough to the equilibrium, a linearized system can be a convenient way of analyzing the input-output dynamics and structure of the nonlinear system.

I define structure (loosely, in this article) as the causal dependencies among manifest variables of the linear system, see [3] for background. This notion of structure is, while the linearization remains applicable, also a description of structure of the original nonlinear system. In addition, the behavior of the nonlinear system can be characterized by the input-output properties of the linearized system. The input output dynamics of the linearized system describe how the nonlinear system behaves when perturbed by a variety of inputs, particularly, the trajectory of the output variables of the nonlinear system.

Traditionally, the input-output properties of a linear system also serve as an index to the complexity of the linear system. A linear system’s input-output complexity is characterized by the number of differential equations required to simulate or model a particular input-output behavior. From these notions of complexity emerge the issue of approximation, or model reduction.

While the input-output behavior of the linearization represent one informative aspect of the nonlinear model, the structure of the linearization also represents another valuable piece of information in describing the original nonlinear system. Thus, when considering the problem of model reduction, it is important to preserve two aspects of the complex model using a simplified model: input-output behavior and structural interpretation.

Above, I have defined structure as the causal dependencies among manifest variables in a linear system. However, the literature in controls research, see [1],[2],[3] for a few examples, has demonstrated that a system can exhibit structure in manifold ways. Most often, the structure present in a linearized or nonlinear model reflects the physical realization of the system. However, little work has been done to build a consistent theoretical framework for representing physical structure mathematically.

In contrast, input-output behavior is well defined and simple for a linear system; inputoutput behavior is characterized by a system’s transfer function. The complexity of a linear system’s input-output behavior is characterized in terms of the complexity of the transfer function (the Smith MacMillan degree). Hence, input-output model reduction is well defined and considers the problem of approximating a transfer function with a high degree of complexity with a lower order transfer function.

In order to pose the problem of structure-preserving model reduction; the notion of a system’s structure must be made precise. Thus, as a necessary first step, my research has focused on rigorously defining structure for a linear system. The work in [4],[5],[6], and possibly [7] are representative publications of the results of our research. For a comprehensive treatment, see [4], for a shorter summary and introduction to the issue, see [5], and for the technical underpinnings of the work, see [7]. Here, I conclude with a (very) brief intuitive summary of this research.

A linear system’s structure can be represented in at least four ways. In general, any representative “partition” of the mathematical equations used to describe the system coupled with a language for interconnecting the components of the partition in a manner consistent with the dynamics of the model will lead to a particular representation of structure. I choose to consider four different representations and the subsequent mathematical relationships that relate these four representations.

First, a system’s structure can be represented in terms of its complete computational structure. This representation of structure partitions the system into fundamental units of computation. For linear systems, these fundamental units of computation are assumed to possess the ability to integrate and perform linear algebraic transformations. This notion of structure is complete in the sense that knowledge of a system’s complete computational structure allows for a unique parameterization of the system’s state-space equations. These state-space equations in turn uniquely define the input-output behavior of the system.

Second, a system’s structure can be represented in terms of its subsystem structure. Subsystem structure describes the natural decomposition of a system into subsystems (often with nontrivial complexity) and the appropriate interconnections linking these subsystems to each other. Third, a system’s structure can be represented in terms of the causal relationships among its manifest variables; I have mentioned this notion of structure above and call it signal structure. Finally, a system’s structure can be described in terms of the causal dependencies of the output variables on the input variables. This representation of structure, called zero pattern structure, corresponds to the information present in a system’s transfer function. This rigorous framework of complete computational, subsystem, signal, and zero pattern structure has lead to a host of new research problems (in addition to structure preserving model reduction) outlined in [4]. I plan to address some of these problems while pursuing a PhD at the California Institute of Technology.

References

• H.S. Mortveit and C.M. Reidys, “Discrete, sequential dynamical systems”, Discrete Mathematics, vol. 226, pp. 281–295, 2001.
• Henrik Sandberg and Richard M. Murray, “Model reduction of interconnected linear systems”, Optimal Control Application and Methods, 2009,
• J. Goncalves, R. Howes, and S. Warnick, “Dynamical structure functions for the reverse engineering of LTI networks”, IEEE Transactions of Automatic Control, 2007, August 2007.
• E. Yeung, J. Goncalves, H. Sandberg, and S. Warnick, “The Meaning of Structure in Interconnected Dynamic Systems” IEEE Control Systems Magazine 2011 Special Invited Issue: Designing Controls for Modern Infrastructure Networks, to appear.
• E. Yeung, J. Goncalves, H. Sandberg, and S. Warnick, “Representing Structure in Linear Interconnected Dynamic Systems” Proceedings of the 2010 IEEE Conference on Decision and Control, to appear.
• E. Yeung, LTI Network Modeling: Reconstruction, Reduction, and Representation, Undergraduate Honors Thesis.
• E. Yeung, J. Goncalves, H. Sandberg, S. Warnick, “Mathematical Relationships between Representations of Structure in Linear Interconnected Dynamical Systems,” submitted to the 2011 IEEE Automatic Control Conference.