An Approximate Bayesian Approach to Optimal Input Signal Design for System Identification

The design of informatively rich input signals is essential for accurate system identification, yet classical Fisher-information-based methods are inherently local and often inadequate in the presence of significant model uncertainty and non-linearity. This paper develops a Bayesian approach that uses the mutual information (MI) between observations and parameters as the utility function. To address the computational intractability of the MI, we maximize a tractable MI lower bound. The method is then applied to the design of an input signal for the identification of quasi-linear stochastic dynamical systems. Evaluating the MI lower bound requires the inversion of large covariance matrices whose dimensions scale with the number of data points <i>N</i>. To overcome this problem, an algorithm that reduces the dimension of the matrices to be inverted by a factor of <i>N</i> is developed, making the approach feasible for long experiments. The proposed Bayesian method is compared with the average D-optimal design method, a semi-Bayesian approach, and its advantages are demonstrated. The effectiveness of the proposed method is further illustrated through four examples, including atomic sensor models, where input signals that generate a large amount of MI are especially important for reducing the estimation error.