Estimation of model parameters in a dynamic system can be significantly improved with the choice of experimental trajectory. For general nonlinear dynamic systems, finding globally “best” trajectories is typically not feasible; however, given an initial estimate of the model parameters and an initial trajectory, we present a continuous-time optimization method that produces a locally optimal trajectory for parameter estimation in the presence of measurement noise. The optimization algorithm is formulated to find system trajectories that improve a norm on the Fisher information matrix (FIM). A double-pendulum cart apparatus is used to numerically and experimentally validate this technique. In simulation, the optimized trajectory increases the minimum eigenvalue of the FIM by three orders of magnitude, compared with the initial trajectory. Experimental results show that this optimized trajectory translates to an order-of-magnitude improvement in the parameter estimate error in practice.
Further research has resulted in optimal design methods for general nonlinear systems. In work by Emery @cite_17 , least-squares and maximum-likelihood estimation techniques are combined with Fisher information to optimize the experimental trajectories. In this case and a number of others, the dynamics are solved as a discretized, constrained optimization problem @cite_22 @cite_0 .
@cite_0 @cite_22 @cite_17
2033574792 2021436740 2088102099
This paper is concerned with the input design problem for a class of structured nonlinear models. This class contains models described by an interconnection of known linear dynamic systems and unknown static nonlinearities. Many widely used model structures are included in this class. The model class considered naturally accommodates a priori knowledge in terms of signal interconnections. Under certain structural conditions, the identification problem for this model class reduces to standard least squares. We treat the input design problem in this situation. An expression for the expected estimate variance is derived. A method for synthesizing an informative input sequence that minimizes an upper bound on this variance is developed. This reduces to a convex optimization problem. Features of the solution include parameterization of the expected estimate variance by the input distribution, and a graph-based method for input generation. This paper surveys the field of optimal input design for parameter estimation as it has developed over the last two decades. Many of the developments covered are only recent and have not appeared in the open literature elsewhere. After a brief introduction, the paper discusses the historical background of the subject both in the engineering and in the statistical literature. The concepts of optimality and input design are then discussed, followed by a derivation of the Fisher information matrix for multiinput multioutput systems with process noise. The design procedures are divided into the categories of time-domain methods and frequency-domain methods, with the former being more general, but also more time consuming (computationally). Several extensions to state constraints, continuous-time systems, etc., are discussed. A number of examples are given to illustrate the nature of optimal inputs. The results on time-domain synthesis with state constraints and their relationship to "dual control" are new. Optimal experiment design is the definition of the conditions under which an experiment is to be conducted in order to maximize the accuracy with which the results are obtained. This paper summarizes a number of methods by which the parameters of the mathematical model of the system are estimated and describes the application of the Fisher information matrix. Examples are given for thermal property estimation in which the estimation is affected both by measurement noise, which is present during any experiment, but also by uncertainties in the parameters of the model used to describe the system.