Non-parametric methods for L2L2-gain estimation using iterative experiments

In this paper we develop non-parametric methods to estimate the L2L2-gain (H∞H∞-norm) of a linear dynamical system from iterative experiments. This work is mainly motivated by model error modeling, 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 is then used to calculate the gain. If 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 matrices, e.g., the Power Method or the Lanczos Method, can then be applied to update the input signal sequence between experiments in order 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. Three such related methods are derived and evaluated by a numerical example. Partial results on convergence and statistical properties of the gain estimator are given. A constrained stochastic gradient method with local optimization of step-length gives the best numerical results in the case of noisy data.