Optimal trajectory design for well-conditioned parameter estimation

When attempting to estimate parameters in a dynamical system, it is often beneficial to systematically design the experimental trajectory. This paper presents a method of generating trajectories using an extension of a nonlinear, infinite-dimensional, projection-based trajectory optimization algorithm. A reformulated objective function is derived for the algorithm to minimize the condition number of the Hessian of the batch-least squares identification method. The batch least-squares method is then used to estimate parameters of the nonlinear system. A simulation example is used to demonstrate that an arbitrarily designed trajectory can lead to an ill-conditioned Hessian matrix in the batch-least squares method, which in turn leads to a less precise set of identified parameters. An example using Monte-Carlo simulations of both trajectories shows a reduction in the variance of identified parameters for an example cart-pendulum system.