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Alexandre Megretski

Publications and source records attributed to Alexandre Megretski.

25 records · Page 2Linked to original sources
arXiv2011

Optimization of Lyapunov Invariants in Verification of Software Systems

The paper proposes a control-theoretic framework for verification of numerical software systems, and puts forward software verification as an important application of control and systems theory. The idea is to transfer Lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications. These include but are not limited to absence of overflow, absence of division-by-zero, termination in finite time, presence of dead-code, and certain user-specified assertions. Central to this framework are Lyapunov invariants. These are properly constructed functions of the program variables, and satisfy certain properties-resembling those of Lyapunov functions-along the execution trace. The search for the invariants can be formulated as a convex optimization problem. If the associated optimization problem is feasible, the result is a certificate for the specification.

eess.SY↗
arXiv2010

Convex Optimization In Identification Of Stable Non-Linear State Space Models

A new framework for nonlinear system identification is presented in terms of optimal fitting of stable nonlinear state space equations to input/output/state data, with a performance objective defined as a measure of robustness of the simulation error with respect to equation errors. Basic definitions and analytical results are presented. The utility of the method is illustrated on a simple simulation example as well as experimental recordings from a live neuron.

math.OC↗
arXiv2010

Finite Approximations of Switched Homogeneous Systems for Controller Synthesis

We demonstrate the use of a new, control-oriented notion of finite state approximation for a particular class of hybrid systems. Specifically, we consider the problem of designing a stabilizing binary output feedback switching controller for a pair of unstable homogeneous second order systems. The constructive approach presented in this note, in addition to yielding an explicit construction of a deterministic finite state approximate model of the hybrid plant, allows us to efficiently establish a useable upper bound on the quality of approximation, and leads to a discrete optimization problem whose solution immediately provides a certifiably correct-by-design controller for the original system. The resulting controller consists of a finite state observer for the plant and a corresponding full state feedback switching control law.

math.OC↗
arXiv2006

Input Classes for Identification of Bilinear Systems

This paper asks what classes of input signals are sufficient in order to completely identify the input/output behavior of generic bilinear systems. The main results are that step inputs are not sufficient, nor are single pulses, but the family of all pulses (of a fixed amplitude but varying widths) do suffice for identification.

math.OC↗
arXiv2002

Optimal Detection of Symmetric Mixed Quantum States

We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly nonabelian generating group, and show that if the generator satisfies a certain constraint, then the LSM is optimal. In particular, for pure-state GU ensembles the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.

quant-ph↗
arXiv2002

Designing Optimal Quantum Detectors Via Semidefinite Programming

We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing between a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program followed by the solution of a set of linear equations or, at worst, a standard linear programming problem. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time. Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.

quant-ph↗