Optimal Input Design for Nonlinear Block Structured Systems

Designing a good experiment is an important step in the system identification process, since the quality of the estimated model strongly depends on the quality of the experiment data. One aspect of the experiment that can be optimized is the input signal that is used to excite the system.  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 problem strongly depends on the model structure, the input parametrization and the properties of the scalar measure.

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.

Example of an optimal input signal

This work presents two main methods to design an optimal input for nonlinear dynamic systems.

The first method assumes that system can be described as a discreet finite memory system and that the class of inputs is restricted to digital signals. Given these assumptions it is possible to approximate the optimal input design problem by a convex optimization problem. Unfortunately, the numerical optimization of this problem is only tractable for very short memories (only two or three delays).

Example of an optimal input signal for a Wiener system

The second method performs a nonlinear non-convex optimizations with respect to the time samples of the input sequence. This method can be applied for both finite and infinite memory systems but the set of possible excitation signals is restricted to signals that are periodic and bandlimited. The performance of the method is very sensitive with respect to the solver and the design parameters (e.g. sampling time, signal duration, initial values, etc.) since small changes in these parameters can have a high impact onto the design performance. Based on extensive simulation results, general guidelines are derived to tune the aforementioned design parameters

References

[1] Hjalmarsson, J. Martensson, and  B. Ninness. Optimal input design for identification of non-linear systems: Learning from the linear case. In American Control Conference, 2007.ACC ’07, pages 1572–1576, July 2007.

[2] P. E. Valenzuela, C. R. Rojas, and H. Hjalmarsson. A graph theoretical approach to input design for identification of nonlinear dynamical models. Automatica, Vol 51, pp 233–242, January 2015.

[3] A. De Cock, M. Gevers, and J. Schoukens. A preliminary Study on optimal input design for nonlinear systems. Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, pages 4931–4936, Dec 2013.

[4] M. Forgione, X. Bombois, P.M.J. Van den Hof, and H. Hjalmarsson. Experiment design for parameter estimation in nonlinear systems based on multilevel excitation. Control Conference (ECC), 2014 European, pages 25–30, June 2015

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