Optimal input design for nonlinear dynamical systems : a graph-theory approach
Abstract: Optimal input design concerns the design of an input sequence to maximize the information retrieved from an experiment. The design of the input sequence is performed by optimizing a cost function related to the intended model application. Several approaches to input design have been proposed, with results mainly on linear models. Under the linear assumption of the model structure, the input design problem can be solved in the frequency domain, where the corresponding spectrum is optimized subject to power constraints. However, the optimization of the input spectrum using frequency domain techniques cannot include time-domain amplitude constraints, which could arise due to practical or safety reasons.In this thesis, a new input design method for nonlinear models is introduced. The method considers the optimization of an input sequence as a realization of the stationary Markov process with finite memory. Assuming a finite set of possible values for the input, the feasible set of stationary processes can be described using graph theory, where de Bruijn graphs can be employed to describe the process. By using de Bruijn graphs, we can express any element in the set of stationary processes as a convex combination of the measures associated with the extreme points of the set. Therefore, by a suitable choice of the cost function, the resulting optimization problem is convex even for nonlinear models. In addition, since the input is restricted to a finite set of values, the proposed input design method can naturally handle amplitude constraints.The thesis considers a theoretical discussion of the proposed input design method for identification of nonlinear output error and nonlinear state space models. In addition, this thesis includes practical applications of the method to solve problems arising in wireless communications, where an estimate of the communication channel with quantized data is required, and application oriented closed-loop experiment design, where quality constraints on the identified parameters must be satisfied when performing the identification step.
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