Guaranteed parameter estimation of non-linear dynamic systems using high-order bounding techniques with domain and CPU-time reduction strategies

This paper is concerned with guaranteed parameter estimation of nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding—or approximating as closely as possible—the set of all possible parameter values such that the predicted values of certain outputs match their corresponding measurements within prescribed error bounds. A set-inversion algorithm is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a given threshold on the approximation level is met. Such exclusion tests rely on the ability to bound the solution set of the dynamic system for a finite parameter subset, and the tightness of these bounds is therefore paramount; equally important in practice is the time required to compute the bounds, thereby defining a trade-off. In this paper, we investigate such a tradeoff by comparing various bounding techniques based on Taylor models with either interval or ellipsoidal bounds as their remainder terms. We also investigate the use of optimization-based domain reduction techniques in order to enhance the convergence speed of the set-inversion algorithm, and we implement simple strategies that avoid recomputing Taylor models or reduce their expansion orders wherever possible. Case studies of various complexities are presented, which show that these improvements using Taylor-based bounding techniques can significantly reduce the computational burden, both in terms of iteration count and CPU time.