On approximate dynamic inversion and proportional-integral control

Approximate dynamic inversion (ADI) has been established as a method to control minimum-phase, nonaffine-in-control systems. Previous results have shown that for single-input nonaffine-in-control systems, every ADI controller admits a linear proportional-integral (PI) realization that is largely independent of the nonlinear function that defines the system. This paper extends these previous results in three ways. First, we present an extension of ADI that renders the closed loop error dynamics independent of the reference model dynamics. It is then shown that the equivalence between the ADI and PI controllers only holds for the time response when applied to the exact system. Finally, key robustness properties of the two control approaches are compared using linear system techniques. These results indicate that the PI realization is preferable when accurate knowledge of the nonlinear system dynamics is not available, and that the ADI realization would be preferred if time delays are the major limitations in the system

Equivalence between Approximate Dynamic Inversion and Proportion-Integral Control

Approximate Dynamic Inversion (ADI) has been established as a method to control minimum-phase, nonaffine-in-control systems. Previous results have shown that for single-input nonaffine-in-control systems, every ADI controller admits a linear Proportional-Integral (PI) realization that is largely independent of the nonlinear function that defines the system. In this report, we first present an extension of the ADI method for single-input nonaffine-in-control systems that renders the closed-loop error dynamics independent of the reference model dynamics. The equivalent PI controller will be derived and both of these results are then extended to multi-input nonaffine-in-control systems.DSO National Laboratories (Singapore), AFOSR grant FA9550-08-1-0086

On Approximate Dynamic Inversion

Approximate Dynamic Inversion has been established as a method to control minimum-phase, nonaffine-in-control systems [1]. In this report, we re-state the main results of [1], clarify some minor notational errors, and prove the same results in an expanded form. In the large, the main results of [1] still stand. The development follows [1] closely, and no novelty is claimed herein. The purpose of this report is mainly to supplement our existing results in [2]–[4] that rely heavily on the results of [1].DSO National Laboratories (Singapore), AFOSR grant FA9550-08-1-008

Model order reduction with novel discrete empirical interpolation methods in space-time

This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin projection onto a linear low-dimensional subspace. In unsteady applications, space-time reduced basis (ST-RB) methods have been developed to achieve a dimension reduction both in space and time, eliminating the computational burden of time marching schemes. However, nonaffine parameterizations dilute any computational speedup achievable by traditional ROMs. Computational efficiency can be recovered by linearizing the nonaffine operators via hyper-reduction, such as the empirical interpolation method in matrix form. In this work, we implement new hyper-reduction techniques explicitly tailored to deal with unsteady problems and embed them in a ST-RB framework. For each of the proposed methods, we develop a posteriori error bounds. We run numerical tests to compare the performance of the proposed ROMs against high-fidelity simulations, in which we combine the finite element method for space discretization on 3D geometries and the Backward Euler time integrator. In particular, we consider a heat equation and an unsteady Stokes equation. The numerical experiments demonstrate the accuracy and computational efficiency our methods retain with respect to the high-fidelity simulations

On the verge : Mechanics in the limit of vanishing strength and stiffness

Material mechanics play a crucial role in a wide variety of scenarios and applications. Here we focused on two central material properties: stiffness and strength. Whereas stiffness characterizes the resistance to deformation for small strains, where the response remains linear, strength describes the resilience of a material to larger deformations and mechanical damage. For conventional materials strength and stiffness are readily described by established mechanical theories. However, many materials in nature, or engineering materials during processing, live in a state where stiffness and/or strength becomes so weak that classical mechanical theories no longer apply. This has been the focal point of this thesis. The exploration of such ultrasoft and/or ultraweak solids faces many challenges, some of which have been addressed in this thesis, including their structure-property relationships and the question howone characterizes these fragile materials where conventional mechanical methods are no longer viable. In chapter 2 we address the challenge of characterizing the mechanical response of solids at the verge of a mechanical instability, where classical approaches fail. We present a new method based on the propagation of infrasonic waves. These waves propagate at low Reynolds numbers, where dissipation is strong. We have not only shown an experimental approach to evaluate wave propagation properties, but also established a theoretical framework to interpret these data and extract quantitative mechanical properties with a unique resolution. In chapter 3 we detail the technical challenges associated with these measurements, performed with the help of optical tweezers to create travelling mechanical waves. When marginal networks are combined with secondary elastic matrices remarkable stiffening is observed. In chapter 4 we present a theoretical model to study the effect of bending rigidity to the mechanics in hybrid materials with simulations. We show how different mechanical regimes arise depending on the bending stiffness and the stiffness of the secondary network. Each of these regimes have different mechanisms that lead to mechanical enhancement of the composite network. Experimental access to these mechanisms is extremely challenging. In chapter 5 we take the first steps to studying these mechanisms experimentally. Here we propose a simple click-chemistry based surface modification method that can help to achieve the complex inter-particle interactions required for establishing hybrid colloidal networks. The second part of this thesis covers hyperweak solids and irreversible deformation. Chapters 6 to 8 deal with colloidal gels that are prototypical examples of hyper weak solids. In chapter 6 we address the structure to dynamics part of the structure-property relation in colloidal gels. We experimentally establish the connection between the intermittent dynamics of individual particles and their local connectivity. We interpret our experimental results with a model that describes single-particle dynamics based on highly cooperative thermal debonding. Our model is in quantitative agreement with experiments and provides a microscopic picture for the structural origin of dynamical heterogeneity and provides a new perspective of the link between structure and the complex mechanics of these heterogeneous solids. Chapter 7 focuses on the dynamics to mechanics part of the structure-property relation by studying fatigue in colloidal gels. Here we combine experiments and computer simulations to show how mechanical loading leads to irreversible strand stretching, which builds slack into the network that softens the solid at small strains and causes strain hardening at larger deformations. We thus find that microscopic plasticity governs fatigue at much larger scales. This sheds new light on fatigue in soft thermal solids and calls for new theoretical descriptions of soft gel mechanics in which local plasticity is taken into account. In chapter 8 we take first steps in investigating the overlooked role of inter-particle friction in colloidal gels. We present a colloidal system with a thermo-responsive trigger for systematically studying the effect of surface properties, grafting density and chain length, on the particle dynamics within colloidal gels. Microscopically, for colloids with a lower grafting density, we observe an increase in the thermal bond angle fluctuations of aggregated colloids. Macroscopically, we observe a clear increase of the linear elastic modulus for gels with increased grafting density and longer chain lengths. These effects are inversely proportional to the magnitude of local bond angle fluctuations. Our model system will allow for further study of the microscopic origins of the complex macroscopic mechanical behavior of hyperweak solids that include bending modes within the network. Fracture and mechanical failure are highly stochastic processes and predicting fracture is highly challenging with conventional theories but crucial to assessing the lifetimes of e.g. buildings, bridges and implants. In chapter 9 we explore new opportunities for predicting fracture in marginal fiber networks. Fracture is the ultimate form of irreversible deformation and, especially in soft materials, characterized with highly non-linear mechanics preempting the moment of failure. We show how machine learning methods can by employed to predict the critical fracture stress solely based on structural and topological input parameters. We show that neural networks, despite their black box behavior, can be used to study the physical mechanisms underlying fracture. By varying the input parameters for our fracture stress predictions we found three parameters for which we can achieve the same prediction quality as for all tested input parameters combined. In the last chapter, the general discussion, we discuss how our results relate to each other and how they fit in a broader context. Furthermore we suggest and describe experiments that can help advance our knowledge of hypersoft and hyperweak materials in the future.</p