Singular Value Analysis and Balanced Realizations for Nonlinear Systems

For linear control systems minimal realization theory and the related model reduction methods play a crucial role in understanding and handling the system. These methods are well established and have proved to be very successful. In particular the method called balanced truncation gives a good reduced order model with respect to the input-output behavior. This method relies on the relation with the system Hankel operator, which plays a central role in minimal realization theory. Specifically, the Hankel operator supplies a set of similarity invariants, the so called Hankel singular values, which can be used to quantify the importance of each state in the corresponding input-output system. The Hankel operator can also be factored into a composition of observability and controllability operators, from which Gramian matrices can be defined and the notion of balanced realization follows. This linear theory is rather complete and the relations between and interpretations in the state-space and input-output settings are fully understood. This paper gives an overview of the series of research on balanced realization and the related model order reduction method based on nonlinear singular value analysis. Section 2 explains the taken point of view on singular value analysis for nonlinear operators. Section 3 briefly reviews the linear balancing method and balanced truncation in order to show the way of thinking for the nonlinear case. Section 4 treats the state-space balancing method. Then, in Section 5 we continue with balanced realizations based on the singular value analysis of the nonlinear Hankel operator. Furthermore, in Section 6 balanced truncation based on the method of Section 5 is presented. Finally, in Section 7 a numerical simulation illustrates how the proposed model order reduction method works for real-world systems.