A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this is not possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.

We consider analysis and controller synthesis of piecewise-linear systems. The method is based on constructing quadratic and piecewise-quadratic Lyapunov functions that prove stability and performance for the system. It is shown that proving stability and performance, or designing state-feedback controllers, can be cast as convex optimization problems involving linear matrix inequalities that can be solved very e ciently. A couple of simple examples are included to demonstrate applications of the methods described.

Abstract-In this paper we consider dynamical systems which are driven by "events" that occur asynchronously. It is assumed that the event rates are fixed, or at least they can be bounded on any time period of length T . Such systems are becoming increasingly important in control due to the very rapid advances in digital systems, communication systems, and data networks. Examples of such systems include, control systems in which signals are transmitted over an asynchronous network; distributed control systems in which each subsystem has its own objective, sensors, resources and level of decision making; parallelized numerical algorithms in which the algorithm is separated into several local algorithms operating concurrently at different processors; and queuing networks. We present a Lyapunov-based theory for asynchronous dynamical systems and show how Lyapunov functions and controllers can be constructed for such systems by solving linear matrix inequality (LMI) and bilinear matrix inequality (BMI) problems. Examples are also presented to demonstrate the effectiveness of the approach.

We consider parameter estimation in linear models when some of the parameters are known to be integers. Such problems arise, for example, in positioning using phase measurements in the global positioning system GPS. Given a linear model, we address two problems:1. The problem of estimating the parameters. 2. The problem of verifying the parameter estimates. Under Gaussian measurement noise:Maximum likelihood estimates of the parameters are given by solving an integer least-squares problem. Theoretically, this problem is very di cult to solve N P -hard.Verifying the parameter estimates computing the probability of correct integer parameter estimation is related to computing the integral of a Gaussian PDF over the Voronoi cell of a lattice. This problem is also very di cult computationally. However, by using a polynomial-time algorithm due to Lenstra, Lenstra, and Lov asz LLL algorithm:The integer least-squares problem associated with estimating the parameters can be solved efciently in practice. Sharp upper and lower bounds can be found on the probability of correct integer parameter estimation. We conclude the paper with simulation results that are based on a GPS setup.

A general circuit model is derived for the electrical noise of electrode-electrolyte systems, with emphasis on its implications for electrochemical sensors. The noise power spectral densities associated with all noise sources introduced in the model are also analytically calculated. Current and voltage fluctuations in typical electrode-electrolyte systems are demonstrated to originate from either thermal equilibrium noise created by conductors, or nonequilibrium excess noise caused by charge transfer processes produced by electrochemical interactions. The power spectral density of the thermal equilibrium noise is predicted using the fluctuation-dissipation theorem of thermodynamics, while the excess noise is assessed in view of charge transfer kinetics, along with mass transfer processes in the electrode proximity. The presented noise model not only explains previously reported noise spectral densities such as thermal noise in sensing electrodes, shot noise in electrochemical batteries, and 1/f noise in corrosive interfaces, it also provides design-oriented insight into the fabrication of low-noise micro-and nanoelectrochemical sensors.