As we shall see in Parts III and IV, the techniques of proper orthogonal decomposition and Galerkin projection can be powerful tools for obtaining low-order models that capture the qualitative behavior of complex, high-dimensional systems. However, for certain systems, the resulting models can perform poorly: even if a large fraction of energy (over 99%) is captured by the modes used for projection, the resulting low-order models may still have completely different qualitative behavior. The transients may be poorly captured, and the stability types of equilibria can even be different.

In this chapter, we present a method which can dramatically outperform projection onto traditional energy-based empirical eigenfunctions described in Chapter 3.We focus primarily (though not exclusively) on linear systems, for several reasons. Many of the pitfalls of traditional proper orthogonal decomposition can be demonstrated for linear systems, without the additional complexity of nonlinearities. Furthermore, for linear systems, one can use operator norms to quantify the difference between a detailed model and its reduced order approximation. Most importantly, for linear systems, there are established tools for performing model reduction, for instance using balanced truncation, which is described in Section 5.1. In contrast, while some modest extensions to nonlinear systems have been attempted, model reduction of nonlinear systems is still an active area of research.

The techniques described in this chapter also differ from those in Chapters 3 and 4 in that they are formulated for input–output systems. The inputs represent the external influences on the system, for instance from external disturbances, or from actuators in a flow-control setting.