Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions

This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. Such maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS

Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle

This paper derives a differential contraction condition for the existence of an orbitally-stable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimization tools such as sum-of-squares programming to be used to search for certificates of the existence of a stable limit cycle. Many desirable properties of contracting dynamics are extended to this context, including preservation of contraction under a broad class of interconnections. In addition, by introducing the concepts of differential dissipativity and transverse differential dissipativity, contraction and transverse contraction can be established for large scale systems via LMI conditions on component subsystems.Comment: 6 pages, 1 figure. Conference submissio

Model uncertainty and monetary policy

Model uncertainty has the potential to change importantly how monetary policy should be conducted, making it an issue that central banks cannot ignore. In this paper, I use a standard new Keynesian business cycle model to analyze the behavior of a central bank that conducts policy with discretion while fearing that its model is misspecified. I begin by showing how to solve linear-quadratic robust Markov-perfect Stackelberg problems where the leader fears that private agents form expectations using the misspecified model. Next, I exploit the connection between robust control and uncertainty aversion to present and interpret my results in terms of the distorted beliefs held by the central bank, households, and firms. My main results are as follows. First, the central bank's pessimism leads it to forecast future outcomes using an expectations operator that, relative to rational expectations, assigns greater probability to extreme inflation and consumption outcomes. Second, the central bank's skepticism about its model causes it to move forcefully to stabilize inflation following shocks. Finally, even in the absence of misspecification, policy loss can be improved if the central bank implements a robust policy.Monetary policy

Optimum PID Control of Multi-wing Attractors in A Family of Lorenz-like Chaotic Systems

Multi-wing chaotic attractors are highly complex nonlinear dynamical systems with higher number of index-2 equilibrium points. Due to the presence of several equilibrium points, randomness of the state time series for these multi-wing chaotic systems is higher than that of the conventional double wing chaotic attractors. A real coded Genetic Algorithm (GA) based global optimization framework has been presented in this paper, to design optimum PID controllers so as to control the state trajectories of three different multi-wing Lorenz like chaotic systems viz. Lu, Rucklidge and Sprott-1 system.Comment: 6 pages, 21 figures; 2012 Third International Conference on Computing, Communication and Networking Technologies (ICCCNT'12), July 2012, Coimbator

Heterogeneous mark-ups, growth and endogenous misallocation

The recent work on misallocation argues that aggregate productivity in poor countries is low because various market frictions prevent marginal products from being equalized. By focusing on such allocative inefficiencies, misallocation is construed as a purely static phenomenon. This paper argues that misallocation also has dynamic consequences because it interacts with firms’ innovation and entry decisions, which determine the economy’s growth rate. To study this link between misallocation and growth, I construct a tractable endogenous growth model with heterogeneous firms, where misallocation stems from imperfectly competitive output markets. The model has an analytical solution and hence makes precise predictions about the relationship between growth, misallocation and welfare. It stresses the importance of entry. An increase in entry reduces misallocation by fostering competition. If entry also increases the economy-wide growth rate, static misallocation and growth are negatively correlated. The welfare consequences of misallocation might therefore be much larger once these dynamic considerations are taken into account. Using firm-level panel data from Indonesia, I present reduced form evidence for the importance of imperfect output market and calibrate the structural parameters. A policy, which reduces existing entry barriers, increases growth and reduces misallocation. The dynamic growth effects are more than four times as large as their static counterpart

Anti-deSitter universe dynamics in LQC

A model for a flat isotropic universe with a negative cosmological constant $\Lambda$ and a massless scalar field as sole matter content is studied within the framework of Loop Quantum Cosmology. By application of the methods introduced for the model with $\Lambda=0$, the physical Hilbert space and the set of Dirac observables are constructed. As in that case, the scalar field plays here the role of an emergent time. The properties of the system are found to be similar to those of the $k=1$ FRW model: for small energy densities, the quantum dynamics reproduces the classical one, whereas, due to modifications at near-Planckian densities, the big bang and big crunch singularities are replaced by a quantum bounce connecting deterministically the large semiclassical epochs. Thus in Loop Quantum Cosmology the evolution is qualitatively cyclic.Comment: Revtex4, 29 pages, 20 figures, typos correcte