Optimal Input Design Through Infinity Norm Minimization Using Proximal Mapping

To avoid non-convexity of the criterion, various relaxations are typically used in input design. For example, the input may be assumed to be stationary and the design problem may be formulated in terms of the correlation coefficients. In this contribution, we instead propose a method to directly design the input sequence. This allows to maximize the information obtained from short-time (transient) experiments using non-stationary inputs. We do this by fitting the achieved Fisher matrix to a desired target matrix in a matrix sense, using the infinity norm. The target matrix can either be the desired Fisher matrix, obtained from quality considerations of the intended use of the model, or a matrix directly representing the performance of the application. An often used quantity is the Hessian of the so called the application cost. Thus, the method is formulated as a time domain optimization problem that is non-convex. This optimization problem is solved by alternative minimization and proximal mapping, where we split the problem in three steps where in each step, the cost function is minimized in respect to one of the variables and other variables are kept fix. We repeat these three steps until convergence. The procedure of the algorithm is summarized in a pseudo code. Finally, we illustrate our method in two numerical examples.