D-Optimal Input Design for the Identification of Structured Nonlinear Systems

The goal of system identification is to construct a mathematical model that describes the behavior of a physical system based on data that was obtained during an experiment. Such a model allows us to obtain insight about the inner workings of the system, to make predictions about the future behavior of the system, and to direct the system to a favorable future state. The uncertainty of the obtained model strongly depends on the quality of the measured data. The more noise there is on the data, the more uncertain the model will be. To reduce the model uncertainty, one could measure longer, use more accurate equipment, or excite the system with a more informative input
signal. The first two options increase the total cost of the experiment, while the third option tries to use the experimental resources to their fullest. The field of optimal input design considers the problem of finding the most informative input signal out of the set of possible excitation signals given some prior knowledge about the system. In its most general form, finding the optimal input signal comes down to solving an optimization problem in which a scalar measure of the Fisher information matrix is maximized with respect to the input sequence. The complexity of this optimization strongly depends on the model structure, the input parameterization and noise conditions. For linear dynamic systems and for nonlinear static systems it has been shown in the literature that the input design problem can be formulated as a convex optimization problem. As a result, a vast set of optimization tools can be used to solve these problems efficiently. For nonlinear systems the optimal input design problem is often non-convex making global optimization more difficult, if not impossible. In this work, two methods are studied to design an optimal input
for nonlinear dynamic systems. The first method assumes that system can be described as a discrete finite memory system and that the class of inputs is restricted to digital signals. Under these assumptions it is possible to approximate the optimal input design problem by a convex optimization problem. The second method restricts the class of inputs to band-limed signals, and allows for infinite memory systems. Given these assumptions, the method performs a nonlinear non-convex optimizations with respect to the time samples of the input sequence.