emgr - The Empirical Gramian Framework

System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramian are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework - emgr - implements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction

Sample‐based robust model reduction for non‐linear systems biology models

Complex non-linear systems biology models comprise relevant knowledge on processes of pharmacological interest. They are, however, too complex to be used in inferential settings, for example, to allow for the estimation of patient-specific parameters for individual dose optimisation. Thus, there is a need for simple models with interpretable components to infer the drug effect in a clinical setting. In particular, it is essential to accurately quantify and simulate the interindividual variability in the drug response in order to account for covariates like body weight, age and genetic disposition. To this end, non-linear model order reduction and simplification methods can be used if they maintain model interpretability during reduction and consider an entire population rather than just a single reference individual. We present a sample-based approach for robust model order reduction and propose two improvements for efficiency. In particular, we introduce a new sampling method to generate the virtual population based on transformed latin hypercube sampling. Thereby, the sample is stratified in the relevant parameter-space directions, which are identified using empirical observability Gramians. We illustrate our approach in application to a blood coagulation pathway model, where we reduce the complexity from a 62-dimensional highly non-linear to a six-dimensional and a nine-dimensional system of ordinary differential equations for two scenarios, respectively

Empirical differential Gramians for nonlinear model reduction

In this paper, we present an empirical balanced truncation method for nonlinear systems whose input vector fields are constants. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory), and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems

Model order reduction for optimality systems through empirical gramians

In the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE) discretization. This leads to a specific linear but high-dimensional time variant (LTV) dynamical system. To reduce the size of the LTV system, we apply a tailored reduced order modeling technique based on empirical gramians and derived directly from the first-order optimality system. For testing purpose, we focus on two specific examples: a multiobjective optimization and a closed-loop optimal control problem. Our proposed methodology results to be better performing than a standard proper orthogonal decomposition (POD) approach for the above mentioned examples

Model Order Reduction for Gas and Energy Networks

To counter the volatile nature of renewable energy sources, gas networks take a vital role. But, to ensure fulfillment of contracts under these circumstances, a vast number of possible scenarios, incorporating uncertain supply and demand, has to be simulated ahead of time. This many-query gas network simulation task can be accelerated by model reduction, yet, large-scale, nonlinear, parametric, hyperbolic partial differential(-algebraic) equation systems, modeling natural gas transport, are a challenging application for model order reduction algorithms. For this industrial application, we bring together the scientific computing topics of: mathematical modeling of gas transport networks, numerical simulation of hyperbolic partial differential equation, and parametric model reduction for nonlinear systems. This research resulted in the "morgen" (Model Order Reduction for Gas and Energy Networks) software platform, which enables modular testing of various combinations of models, solvers, and model reduction methods. In this work we present the theoretical background on systemic modeling and structured, data-driven, system-theoretic model reduction for gas networks, as well as the implementation of "morgen" and associated numerical experiments testing model reduction adapted to gas network models

GRAMIAN-AWARE CLOSED LOOP FLIGHT CONTROL DESIGN FOR ENERGY HARVESTING THROUGH MODULATING DISTURBANCE SENSITIVITY

Many desired micro aerial vehicle missions are significantly larger than the mission endurance of the vehicles. Due to extreme constraints on size, weight and power available, small scale air vehicles are highly sensitive to atmospheric disturbance. This work introduces a control-theoretic framework that models the magnitude of the vehicle's disturbance sensitivity and observability in conjunction with each other under a gramian-based formulation. To implement atmospheric gust response modulatiom, a ``gramian-aware'' flight control law is designed using open loop plant models across various scales and assuming perfect gust measurement. Time-domain system identification was conducted using data collected from repeatable automated flights in a motion capture arena in order to derive the plant model. Closed-loop simulation results as well as experimental data modulating the plant using cruise speed are presented to illustrate that the gramian-based control laws can be utilized to facilitate atmospheric energy scavenging in gusting environments