Stiffness pathologies in discrete granular systems: bifurcation, neutral equilibrium, and instability in the presence of kinematic constraints

The paper develops the stiffness relationship between the movements and forces among a system of discrete interacting grains. The approach is similar to that used in structural analysis, but the stiffness matrix of granular material is inherently non-symmetric because of the geometrics of particle interactions and of the frictional behavior of the contacts. Internal geometric constraints are imposed by the particles' shapes, in particular, by the surface curvatures of the particles at their points of contact. Moreover, the stiffness relationship is incrementally non-linear, and even small assemblies require the analysis of multiple stiffness branches, with each branch region being a pointed convex cone in displacement-space. These aspects of the particle-level stiffness relationship gives rise to three types of micro-scale failure: neutral equilibrium, bifurcation and path instability, and instability of equilibrium. These three pathologies are defined in the context of four types of displacement constraints, which can be readily analyzed with certain generalized inverses. That is, instability and non-uniqueness are investigated in the presence of kinematic constraints. Bifurcation paths can be either stable or unstable, as determined with the Hill-Bazant-Petryk criterion. Examples of simple granular systems of three, sixteen, and sixty four disks are analyzed. With each system, multiple contacts were assumed to be at the friction limit. Even with these small systems, micro-scale failure is expressed in many different forms, with some systems having hundreds of micro-scale failure modes. The examples suggest that micro-scale failure is pervasive within granular materials, with particle arrangements being in a nearly continual state of instability

Approximately bisimilar symbolic models for incrementally stable switched systems

Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e. existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost DC-DC converter.Comment: 17 page

Towards Scalable Synthesis of Stochastic Control Systems

Formal control synthesis approaches over stochastic systems have received significant attention in the past few years, in view of their ability to provide provably correct controllers for complex logical specifications in an automated fashion. Examples of complex specifications of interest include properties expressed as formulae in linear temporal logic (LTL) or as automata on infinite strings. A general methodology to synthesize controllers for such properties resorts to symbolic abstractions of the given stochastic systems. Symbolic models are discrete abstractions of the given concrete systems with the property that a controller designed on the abstraction can be refined (or implemented) into a controller on the original system. Although the recent development of techniques for the construction of symbolic models has been quite encouraging, the general goal of formal synthesis over stochastic control systems is by no means solved. A fundamental issue with the existing techniques is the known "curse of dimensionality," which is due to the need to discretize state and input sets and that results in an exponential complexity over the number of state and input variables in the concrete system. In this work we propose a novel abstraction technique for incrementally stable stochastic control systems, which does not require state-space discretization but only input set discretization, and that can be potentially more efficient (and thus scalable) than existing approaches. We elucidate the effectiveness of the proposed approach by synthesizing a schedule for the coordination of two traffic lights under some safety and fairness requirements for a road traffic model. Further we argue that this 5-dimensional linear stochastic control system cannot be studied with existing approaches based on state-space discretization due to the very large number of generated discrete states.Comment: 22 pages, 3 figures. arXiv admin note: text overlap with arXiv:1407.273

2000-2003 Real Estate Bubble in the UK but not in the USA

In the aftermath of the burst of the ``new economy'' bubble in 2000, the Federal Reserve aggressively reduced short-term rates yields in less than two years from 6.5% to 1.25% in an attempt to coax forth a stronger recovery of the US economy. But, there is growing apprehension that this is creating a new bubble in real estate, as strong housing demand is fuelled by historically low mortgage rates. Are we going from Charybdis to Scylla? This question is all the more excruciating at a time when many other indicators suggest a significant deflationary risk. Using economic data, Federal Reserve Chairman A. Greenspan and Governor D.L. Kohn dismissed recently this possibility. Using the theory of critical phenomena resulting from positive feedbacks in markets, we confirm this view point for the US but find that mayhem may be in store for the UK: we unearth the unmistakable signatures (log-periodicity and power law super-exponential acceleration) of a strong unsustainable bubble there, which could burst before the end of the year 2003.Comment: Latex, 22 pages including 8 eps figures; A revised version accepted for publication in Physica