Optimal Input Design for Identification of Non-linear Systems: Learning From the Linear Case

For linear time-invariant systems, the input influences the accuracy of identified parameters only through its second order properties and its cross-correlation with the noise. A wide range of input design problems for such systems can be recast as semi-definite problems in the auto-correlation coefficients of the input or similar design variables. This allows for efficient numerical solutions of such problems. When the system is non-linear the situation is radically different. Non- linearities can make the parameter accuracy depend on all moments of the input so that the accuracy may depend on the complete distribution of the input sequence. In this contribution we discuss some emerging ways to cope with this situation. In particular we illustrate how to formulate some input design problems as polynomial matrix inequalities for which relaxation methods exist which can generate a sequence of LMI problems with optimal values that under-bound the optimal solution and that converge to a global optimum of the original problem. Both deterministic and stochastic input models are considered. In the stochastic case we discuss how to delineate optimization of the statistical properties from the subsequent signal generation.