Data-Based Stochastic Evaluation of Closed-Loop Stability and Performance

Abstract

Metrics for determining the stability-and performance-robustness of a given controller to variations in open-loop dynamics are examined. Of particular interest are the probability distributions that these metrics have given some amount of uncertainty in the plant. The analysis herein characterizes these distributions using frequency response data collected experimentally for the open-loop plant. In doing so, it avoids using a model for the plant, thereby reducing errors and uncertainty inherent to the modeling of a plant. For a proposed controller, a Monte-Carlo method is used to evaluate the closed-loop performance and stability metrics for a large ensemble of test data and thus create an estimate of their distributions. Since ensembles of test data are typically quite small, a Karhunen-Loeve expansion based method is used to synthetically expand the set of FRFs. Nomenclature = Random process vector = Mean vector for random process = j th experimental realization of random process vector = Orthogonal basis vectors = Weighting matrix = Linear combination vector = Deviation matrix ( ) = Estimated from experimental data = Length of random process vector = Number of contributing basis vectors = Number of experimental realizations = Smoothing parameter = Kernel density estimator = Perturbation of linear combination vector 2 = Candidate combination vector ̃ = l th Metropolis-Hastings Monte-Carlo generated combination vector for j th test realization = Probability of acceptance for candidate combination vector = Acceptance parameter ( ) ( ) = Generated using Metropolis-Hastings Monte Carlo = Performance and stability cost function = Closed-loop, open-loop, and controller transfer functions = k th frequency point in transfer function = Number of outputs in transfer function = Number of inputs in transfer function = Physical Mass, Stiffness and Damping Matrices = Modal damping matrix = Modal damping parameter for r th mode = r th natural frequency = Mode Matrix = State-space matrices = Cholesky decomposition of mass matrix = Random matrix of zero mean, independent, normally distributed variables = Dispersion level = Number of degrees of freedom in model = Second dimension of random matrix