Boris Kramer

Full flux models for optimization and control of heat exchangers

If convection is the dominate mechanism for heat transfer in a heat exchangers, then the devices are often modeled by hyperbolic partial differential equations. One of the difficulties with this approach is that for low (or zero) pipe flows, some of the imperial functions used to model friction can become singular. One way to address low flows is to include the full flux in the model so that the equation becomes a convection-diffusion equation with a “small” diffusion term. We show that solutions of the hyperbolic equation are recovered as limiting (viscosity) solutions of the convection-diffusion model. We employ a composite finite element - finite volume scheme to produce finite dimensional systems for control design. This scheme is known to be unconditionally L2-stable, uniformly with respect to the diffusion term. We present numerical examples to illustrate how the inclusion of a small diffusion term can impact controller design.

Tangential interpolation-based eigensystem realization algorithm for MIMO systems

ABSTRACT The eigensystem realization algorithm (ERA) is a commonly used data-driven method for system identification and reduced-order modelling of dynamical systems. The main computational difficulty in ERA arises when the system under consideration has a large number of inputs and outputs, requiring to compute a singular value decomposition (SVD) of a large-scale dense Hankel matrix. In this work, we present an algorithm that aims to resolve this computational bottleneck via tangential interpolation. This involves projecting the original impulse response sequence onto suitably chosen directions. The resulting data-driven reduced model preserves stability and is endowed with an a priori error bound. Numerical examples demonstrate that the modified ERA algorithm with tangentially interpolated data produces accurate reduced models while, at the same time, reducing the computational cost and memory requirements significantly compared to the standard ERA. We also give an example to demonstrate the limitations of the proposed method.

Combining multiple surrogate models to accelerate failure probability estimation with expensive high-fidelity models

In failure probability estimation, importance sampling constructs a biasing distribution that targets the failure event such that a small number of model evaluations is sufficient to achieve a Monte Carlo estimate of the failure probability with an acceptable accuracy; however, the construction of the biasing distribution often requires a large number of model evaluations, which can become computationally expensive. We present a mixed multifidelity importance sampling (MMFIS) approach that leverages computationally cheap but erroneous surrogate models for the construction of the biasing distribution and that uses the original high-fidelity model to guarantee unbiased estimates of the failure probability. The key property of our MMFIS estimator is that it can leverage multiple surrogate models for the construction of the biasing distribution, instead of a single surrogate model alone. We show that our MMFIS estimator has a mean-squared error that is up to a constant lower than the mean-squared errors of the corresponding estimators that uses any of the given surrogate models alone—even in settings where no information about the approximation qualities of the surrogate models is available. In particular, our MMFIS approach avoids the problem of selecting the surrogate model that leads to the estimator with the lowest mean-squared error, which is challenging if the approximation quality of the surrogate models is unknown. We demonstrate our MMFIS approach on numerical examples, where we achieve orders of magnitude speedups compared to using the high-fidelity model only.

Feedback Control for Systems with Uncertain Parameters Using Online-Adaptive Reduced Models

We consider control and stabilization for large-scale dynamical systems with uncertain, time-varying parameters. The time-critical task of controlling a dynamical system poses major challenges: using large-scale models is prohibitive, and accurately inferring parameters can be expensive, too. We address both problems by proposing an offline-online strategy for controlling systems with time-varying parameters. During the offline phase, we use a high-fidelity model to compute a library of optimal feedback controller gains over a sampled set of parameter values. Then, during the online phase, in which the uncertain parameter changes over time, we learn a reduced-order model from system data. The learned reduced-order model is employed within an optimization routine to update the feedback control throughout the online phase. Since the system data naturally reflects the uncertain parameter, the data-driven updating of the controller gains is achieved without an explicit parameter estimation step. We consider t...

Solving Algebraic Riccati Equations Via Proper Orthogonal Decomposition

Abstract In this paper we present a method to solve algebraic Riccati equations by employing a projection method based on Proper Orthogonal Decomposition. The method only requires simulations of linear systems to compute the solution of a Lyapunov equation. The leading singular vectors are then used to construct a projector which is employed to produce a reduced order system. We compare this approach to an extended Krylov subspace method and a standard Gramian based method.

System Identification via CUR-Factored Hankel Approximation

Subspace-based system identification for dynamical systems is a sound, system-theoretic way to obtain linear, time-invariant system models from data. The interplay of data and systems theory is reflected in the Hankel matrix, a block-structured matrix whose factorization is used for system identification. For systems with many inputs, many outputs, or large time-series of system-response data, established methods based on the singular value decomposition (SVD)---such as the eigensystem realization algorithm (ERA)---are prohibitively expensive. In this paper, we propose an algorithm to reduce the complexity of the ERA from cubic to linear, with respect to the Hankel matrix size. Furthermore, our memory requirements scale at the same rate because we never require loading the entire Hankel matrix into memory. These reductions are realized by replacing the SVD with a CUR decomposition that directly seeks a low-rank approximation of the Hankel matrix. The CUR decomposition is obtained using a maximum-volume--b...

Robust POD model stabilization for the 3D Boussinesq equations based on Lyapunov theory and extremum seeking

We present new results on robust model reduction for partial differential equations. Our contribution is threefold: 1.) The stabilization is achieved via closure models for reduced order models (ROMs), where we use Lyapunov robust control theory to design a new stabilizing closure model that is robust with respect to parametric uncertainties; 2.) The free parameters in the proposed ROM stabilization method are auto-tuned using a data-driven multi-parametric extremum seeking (MES) optimization algorithm; and 3.) The challenging 3D Boussinesq equation numerical test-bed is used to demonstrate the advantages of the proposed method.