Non-parametric methods for L2-gain estimation using iterative experiments

In this paper we develop non-parametric methods to estimate the L2-gain (H∞norm) of a linear dynamical system from iterative experiments. This work is mainly motivated by model error modelling, where the error dynamics are more complex than can be captured by a low order parametric model. The standard system identification approach to the gain estimation problem is to estimate a parametric model of the system, which then is used calculate the gain. When it is possible to update the input signal during the experiment, an alternative way is to iteratively optimize the input signal in order to maximize the estimated input to output gain. A key observation is that the gradient of the gain with respect to the input signal can, without knowing a model, be found from two experiments. Iterative numerical methods for calculation of eigenvalues of positive definite matrices, e.g., the Power Method or the Lanczos Method, can then be applied to update the input signal sequence between experiments to find the maximum gain. The main difficulty compared to the corresponding eigenvalue problem in numerical analysis is the effects of additive measurement noise, which require modified schemes that avoid bias errors. We derive three such related methods and evaluate them by a numerical example. Partial results on convergence and statistical properties of the gain estimate are provided. A constrained stochastic gradient method with local optimization of step-length shows promising results in case of noisy data.