Gradient-, Ensemble-, and Adjoint-Free Data-Driven Parameter Estimation

Abstract

In many applications, models of physical systems have known structure but unknown parameters. By viewing the unknown parameters as constant states, nonlinear estimation methods such as the extended Kalman filter, unscented Kalman filter, and ensemble Kalman filter can be used to estimate the states of the augmented system, thereby providing estimates of the parameters along with the dynamic states. These methods tend to be computationally expensive due to the need for Jacobians, ensembles, or adjoints, especially when the models are high-dimensional. This dissertation presents retrospective cost parameter estimation (RCPE), which does not require gradients, ensembles, or adjoints. Rather, RCPE estimates unknown parameters from a single trajectory, and requires updating an adaptive integrator gain for each unknown parameter. RCPE is applicable to parameter estimation in linear and nonlinear models, where the parameterization may be either affine or nonaffine. The main contribution of this work is to show that the parameter estimates may be permuted in an arbitrary way, and thus a permutation is needed to correctly associate each parameter estimate with the corresponding unknown parameter. RCPE is illustrated through several numerical examples including the Burgers equation and the Global Ionosphere Thermosphere Model (GITM), where the goal is to estimate representational parameters such as eddy diffusion coefficient and thermal conductivity coefficients using measurements of atmospheric variables such as total electron content, density, temperatures etc. The next part of the dissertation focuses on forgetting in the context of recursive least squares (RLS) algorithm. It is a well-known fact that classical RLS with forgetting diverges in the cases where the excitation is not persistent. In this work, an information-driven directional forgetting technique is proposed, which constrains the forgetting to directions in which new information is available, thereby allowing RLS to operate without divergence during periods of loss of persistency. In the last part of this dissertation, retrospective cost adaptive control (RCAC) is extended to the problem of control allocation in overactuated systems. In particular, it is shown that the applied control input lies in the range of the target model used in RCAC, thereby providing a simple technique to constrain the control input to a desired subspace. Finally, RCAC is extended to asymptotically enforce output constraint by formulating the problem as a problem of following conflicting commands, and is used to prevent a scramjet combustor from unstarting using pressure measurements.