On an implicit triangular decomposition of nonlinear control systems that are 1-flat - a constructive approach

We study the problem to provide a triangular form based on implicit differential equations for non-linear multi-input systems with respect to the flatness property. Furthermore, we suggest a constructive method for the transformation of a given system into that special triangular shape, if possible. The well known Brunovsky form, which is applicable with regard to the exact linearization problem, can be seen as special case of this implicit triangular form. A key tool in our investigation will be the construction of Cauchy characteristic vector fields that additionally annihilate certain codistributions. In adapted coordinates this construction allows to single out variables whose time-evolution can be derived without any integration.Comment: submitted to Automatic

On the extraction of the boundary conditions and the boundary ports in second-order field theories

In this paper we consider second-order field theories in a variational setting. From the variational principle the Euler-Lagrange equations follow in an unambiguous way, but it is well known that this is not true for the Cartan form. This has also consequences on the derivation of the boundary conditions when non trivial variations are allowed on the boundary. By posing extra conditions on the set of possible boundary terms we exploit the degree of freedom in the Cartan form to extract physical meaningful boundary expressions. The same mathematical machinery will be applied to derive the boundary ports in a Hamiltonian representation of the partial differential equations which is crucial for energy based control approaches. Our results will be visualized for mechanical systems such as beam and plate models

Linearized Controllability Analysis of Semilinear Partial Differential Equations

It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional nonlinear systems. In this paper, we restrict ourselves to semilinear infinite-dimensional systems, and show that the exact controllability of the linearized system implies exact controllability of the nonlinear system. The restrictions concerning the nonlinear operator are similar to those that can be found in the literature about the linearized stability analysis of semilinear systems.Comment: accepted as full paper for MTNS 202

Analysis and Comparison of Port-Hamiltonian Formulations for Field Theories - demonstrated by means of the Mindlin plate

This paper focuses on the port-Hamiltonian formulation of systems described by partial differential equations. Based on a variational principle we derive the equations of motion as well as the boundary conditions in the well-known Lagrangian framework. Then it is of interest to reformulate the equations of motion in a port-Hamiltonian setting, where we compare the approach based on Stokes-Dirac structures to a Hamiltonian setting that makes use of the involved bundle structure similar to the one on which the variational approach is based. We will use the Mindlin plate, a distributed parameter system with spatial domain of dimension two, as a running example.Comment: 6 pages, submitte

Application of Symmetry Groups to the Observability Analysis of Partial Differential Equations

Symmetry groups of PDEs allow to transform solutions continuously into other solutions. In this paper, we use this property for the observability analysis of nonlinear PDEs with input and output. Based on a differential-geometric representation of the nonlinear system, we derive conditions for the existence of special symmetry groups that do not change the trajectories of the input and the output. If such a symmetry group exists, every solution can be transformed into other solutions with the same input and output trajectories but different initial conditions, and this property can be used to prove that the system is not observable. We also put emphasis on showing how the approach simplifies for linear systems, and how it is related to the well-known observability concepts from infinite-dimensional linear systems theory.Comment: submitted to MTNS 201

Predictive Coarse-Graining

We propose a data-driven, coarse-graining formulation in the context of equilibrium statistical mechanics. In contrast to existing techniques which are based on a fine-to-coarse map, we adopt the opposite strategy by prescribing a probabilistic coarse-to-fine map. This corresponds to a directed probabilistic model where the coarse variables play the role of latent generators of the fine scale (all-atom) data. From an information-theoretic perspective, the framework proposed provides an improvement upon the relative entropy method and is capable of quantifying the uncertainty due to the information loss that unavoidably takes place during the CG process. Furthermore, it can be readily extended to a fully Bayesian model where various sources of uncertainties are reflected in the posterior of the model parameters. The latter can be used to produce not only point estimates of fine-scale reconstructions or macroscopic observables, but more importantly, predictive posterior distributions on these quantities. Predictive posterior distributions reflect the confidence of the model as a function of the amount of data and the level of coarse-graining. The issues of model complexity and model selection are seamlessly addressed by employing a hierarchical prior that favors the discovery of sparse solutions, revealing the most prominent features in the coarse-grained model. A flexible and parallelizable Monte Carlo - Expectation-Maximization (MC-EM) scheme is proposed for carrying out inference and learning tasks. A comparative assessment of the proposed methodology is presented for a lattice spin system and the SPC/E water model

Predictive Collective Variable Discovery with Deep Bayesian Models

Extending spatio-temporal scale limitations of models for complex atomistic systems considered in biochemistry and materials science necessitates the development of enhanced sampling methods. The potential acceleration in exploring the configurational space by enhanced sampling methods depends on the choice of collective variables (CVs). In this work, we formulate the discovery of CVs as a Bayesian inference problem and consider the CVs as hidden generators of the full-atomistic trajectory. The ability to generate samples of the fine-scale atomistic configurations using limited training data allows us to compute estimates of observables as well as our probabilistic confidence on them. The methodology is based on emerging methodological advances in machine learning and variational inference. The discovered CVs are related to physicochemical properties which are essential for understanding mechanisms especially in unexplored complex systems. We provide a quantitative assessment of the CVs in terms of their predictive ability for alanine dipeptide (ALA-2) and ALA-15 peptide

On the Linearization of Flat Two-Input Systems by Prolongations and Applications to Control Design

In this paper we consider $(x,u)$-flat nonlinear control systems with two inputs, and show that every such system can be rendered static feedback linearizable by prolongations of a suitably chosen control. This result is not only of theoretical interest, but has also important implications on the design of flatness based tracking controls. We show that a tracking control based on quasi-static state feedback can always be designed in such a way that only measurements of a (classical) state of the system, and not measurements of a generalized Brunovsky state, as reported in the literature, are required

Energy-Based In-Domain Control of a Piezo-Actuated Euler-Bernoulli Beam

The main contribution of this paper is the extension of the well-known boundary-control strategy based on structural invariants to the control of infinite-dimensional systems with in-domain actuation. The systems under consideration, governed by partial differential equations, are described in a port-Hamiltonian setting making heavy use of the underlying jet-bundle structure, where we restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian. To show the applicability of the proposed approach, we develop a dynamic controller for an Euler-Bernoulli beam actuated with a pair of piezoelectric patches and conclude the article with simulation results.Comment: 10 pages, 1 figur

Differential-Geometric Decomposition of Flat Nonlinear Discrete-Time Systems

We prove that every flat nonlinear discrete-time system can be decomposed by coordinate transformations into a smaller-dimensional subsystem and an endogenous dynamic feedback. For flat continuous-time systems, no comparable result is available. The advantage of such a decomposition is that the complete system is flat if and only if the subsystem is flat. Thus, by repeating the decomposition at most $n-1$ times, where $n$ is the dimension of the state space, the flatness of a discrete-time system can be checked in an algorithmic way. If the system is flat, then the algorithm yields a flat output which only depends on the state variables. Hence, every flat discrete-time system has a flat output which does not depend on the inputs and their forward-shifts. Again, no comparable result for flat continuous-time systems is available. The algorithm requires in each decomposition step the construction of state- and input transformations, which are obtained by straightening out certain vector fields or distributions with the flow-box theorem or the Frobenius theorem. Thus, from a computational point of view, only the calculation of flows and the solution of algebraic equations is needed. We illustrate our results by two examples