Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty

We evaluate the robustness of a probabilistic formulation of system identification (ID) to sparse, noisy, and indirect data. Specifically, we compare estimators of future system behavior derived from the Bayesian posterior of a learning problem to several commonly used least squares-based optimization objectives used in system ID. Our comparisons indicate that the log posterior has improved geometric properties compared with the objective function surfaces of traditional methods that include differentially constrained least squares and least squares reconstructions of discrete time steppers like dynamic mode decomposition (DMD). These properties allow it to be both more sensitive to new data and less affected by multiple minima --- overall yielding a more robust approach. Our theoretical results indicate that least squares and regularized least squares methods like dynamic mode decomposition and sparse identification of nonlinear dynamics (SINDy) can be derived from the probabilistic formulation by assuming noiseless measurements. We also analyze the computational complexity of a Gaussian filter-based approximate marginal Markov Chain Monte Carlo scheme that we use to obtain the Bayesian posterior for both linear and nonlinear problems. We then empirically demonstrate that obtaining the marginal posterior of the parameter dynamics and making predictions by extracting optimal estimators (e.g., mean, median, mode) yields orders of magnitude improvement over the aforementioned approaches. We attribute this performance to the fact that the Bayesian approach captures parameter, model, and measurement uncertainties, whereas the other methods typically neglect at least one type of uncertainty

Robust identification of non-autonomous dynamical systems using stochastic dynamics models

This paper considers the problem of system identification (ID) of linear and nonlinear non-autonomous systems from noisy and sparse data. We propose and analyze an objective function derived from a Bayesian formulation for learning a hidden Markov model with stochastic dynamics. We then analyze this objective function in the context of several state-of-the-art approaches for both linear and nonlinear system ID. In the former, we analyze least squares approaches for Markov parameter estimation, and in the latter, we analyze the multiple shooting approach. We demonstrate the limitations of the optimization problems posed by these existing methods by showing that they can be seen as special cases of the proposed optimization objective under certain simplifying assumptions: conditional independence of data and zero model error. Furthermore, we observe that our proposed approach has improved smoothness and inherent regularization that make it well-suited for system ID and provide mathematical explanations for these characteristics' origins. Finally, numerical simulations demonstrate a mean squared error over 8.7 times lower compared to multiple shooting when data are noisy and/or sparse. Moreover, the proposed approach can identify accurate and generalizable models even when there are more parameters than data or when the underlying system exhibits chaotic behavior

Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Models

This paper proposes a probabilistic Bayesian formulation for system identification (ID) and estimation of nonseparable Hamiltonian systems using stochastic dynamic models. Nonseparable Hamiltonian systems arise in models from diverse science and engineering applications such as astrophysics, robotics, vortex dynamics, charged particle dynamics, and quantum mechanics. The numerical experiments demonstrate that the proposed method recovers dynamical systems with higher accuracy and reduced predictive uncertainty compared to state-of-the-art approaches. The results further show that accurate predictions far outside the training time interval in the presence of sparse and noisy measurements are possible, which lends robustness and generalizability to the proposed approach. A quantitative benefit is prediction accuracy with less than 10% relative error for more than 12 times longer than a comparable least-squares-based method on a benchmark problem

Generalization of System Identification Objective Functions Through Stochastic Hidden Markov Models for Regularization, Smoothness, and Uncertainty Quantification

With the growing availability of computational resources, the interest in learning models of dynamical systems has grown exponentially over the years across many diverse disciplines. As a result of this growth, objective functions for model estimation have been rapidly developed independently across fields such as fluids, control, and machine learning. Theoretical justifications for these objectives, however, have lagged behind. In this dissertation, we provide a unifying theoretical framework for some of the most popular of these objectives, specifically dynamic mode decomposition (DMD), single rollout Markov parameter estimation, sparse identification of nonlinear dynamics (SINDy), and multiple shooting. In this framework, we model a general dynamical system using a hidden Markov model and derive a marginal likelihood that can be used for estimation. The key difference between this and most existing likelihood estimators is that rather than simply modeling the estimation error in the output of the system, we additionally model the error in the dynamics through the inclusion of a process noise term. Not only does this process noise term provide the flexibility needed to generalize many existing objectives, but it also provides three significant advantages in the marginal likelihood. The first is that it generates an explicit regularization term that arises directly from the model formulation without the need for adding heuristic priors onto the parameters. Furthermore, this regularization term is over the output, rather than the parameters, of the model and is therefore applicable to any arbitrary parameterization of the dynamics. Secondly, the process noise term provides smoothing of the marginal likelihood optimization surface without having to discount the information in the data through tempering methods or abbreviated simulation lengths. Lastly, estimation of the process noise term can give a quantification of the uncertainty of the estimated model without necessarily requiring expensive Markov chain Monte Carlo (MCMC) sampling. To evaluate this proposed marginal likelihood, we present an efficient recursive algorithm for linear-Gaussian models and an approximation to this algorithm for all remaining models. We discuss how simplifications to the approximate algorithm can be made when the noise is additive Gaussian and derive simplifications for when it is arbitrary additive/multiplicative noise. Next, we provide theoretical results proving that the considered objectives are special cases of a posterior that uses the proposed marginal likelihood. These results uncover the sets of assumptions needed to transform the negative log posterior into each of the objective functions that we consider. We then present numerical experiments that compare the (approximate) marginal likelihood to each of the considered objectives on a variety of systems. These experiments include linear, chaotic, partial differential equation, limit cycle, and Hamiltonian systems. Additionally, we include a novel comparison of Hamiltonian estimation using symplectic and non-symplectic dynamics propagators. This comparison uses uncertainty quantification both in the form of MCMC sampling and process noise covariance estimation to show that embedding the symplectic propagator into the objective delivers more precise estimates than embedding the objective with the non-symplectic propagator. Overall, the results of this dissertation demonstrate that the marginal likelihood is able to produce more accurate estimates on problems with high amounts of uncertainty in the forms of measurement noise, measurement sparsity, and model expressiveness than comparable objective functions.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/177756/1/ngalioto_1.pd

Ultralow Frequency Electrochemical–Mechanical Strain Energy Harvester Using 2D Black Phosphorus Nanosheets

Advances in piezoelectric or triboelectric materials have enabled high-frequency platforms for mechanical energy harvesting (>10 Hz); however, virtually all human motions occur below 5 Hz and therefore limits application of these harvesting platforms to human motions. Here we demonstrate a device configuration based on sodiated black phosphorus nanosheets, or phosphorene, where mechanoelectrochemical stress–voltage coupling in this material is capable of efficient energy harvesting at frequencies as low as 0.01 Hz. The harvester is tested using both bending and pressing mechanical impulses with peak power delivery of ∼42 nW/cm² and total harvested energy of 0.203 μJ/cm² in the bending mode and ∼9 nW/cm² and 0.792 μJ/cm² in the pressing mode. Our work broadly demonstrates how 2D materials can be effectively leveraged as building blocks in strategies for efficient electrochemical strain energy harvesting

From the Junkyard to the Power Grid: Ambient Processing of Scrap Metals into Nanostructured Electrodes for Ultrafast Rechargeable Batteries

Here we present the first full cell battery device that is developed entirely from scrap metals of brass and steeltwo of the most commonly used and discarded metals. A room-temperature chemical process is developed to convert brass and steel into functional electrodes for rechargeable energy storage that transforms these multicomponent alloys into redox-active iron oxide and copper oxide materials. The resulting steel–brass battery exhibits cell voltages up to 1.8 V, energy density up to 20 Wh/kg, power density up to 20 kW/kg, and stable cycling over 5000 cycles in alkaline electrolytes. Further, we show the versatility of this technique to enable processing of steel and brass materials of different shapes, sizes, and purity, such as screws and shavings, to produce functional battery components. The simplicity of this approach, building from chemicals commonly available in a household, enables a simple pathway to the local recovery, processing, and assembly of storage systems based on materials that would otherwise be discarded