Pavel S. Shcherbakov

Fixed-time Consensus Algorithm for Multi-agent Systems with Integrator Dynamics

Abstract The paper addresses the problem of exact average-consensus reaching in a prespecified time. The communication topology is assumed to be defined by a weighted undirected graph and the agents are represented by integrators. A nonlinear control protocol which ensures a finite-time convergence is proposed. With the designed protocol, any a priori specified convergence time can be guaranteed regardless of the initial conditions.

RACT: Randomized Algorithms Control Toolbox for MATLAB

Abstract This paper introduces a new M atlab package, R act , aimed at solving a class of probabilistic analysis and synthesis problems arising in control. The package offers a convenient way for defining various types of structured uncertainties as well as formulating and analyzing the ensuing robustness analysis tasks from a probabilistic point of view. It also provides a full-featured framework for LMI-formulated probabilistic synthesis problems, which includes sequential probabilistic methods as well as scenario methods for robust design. The R act package is freely available at http://ract.sourceforge.net , and only requires the Y almip toolbox to be installed in the M atlab environment.

Nonlinear fixed-time control protocol for uniform allocation of agents on a segment

The paper addresses the problem of row straightening of agents via local interactions. A nonlinear control protocol that ensures finite-time equidistant allocation on a segment is proposed. With the designed protocol, any settling time can be guaranteed regardless of the initial conditions. A robust modification of the control algorithm based on sliding mode control technique is presented. The case of multidimensional agents is also considered. The theoretical results are illustrated via numerical simulations.

Superstable Linear Control Systems. I. Analysis

The notion of superstability of linear control systems was introduced. Superstability which is a sufficient condition for stability was formulated in terms of linear constraints on the entries of a matrix or the coefficients of a characteristic polynomial. In the first part of the paper, the properties of superstable systems were studied. The norms of solutions were proved to decrease exponentially monotonically in the absence of perturbations, and the solutions were proved to be uniformly bounded in the presence of bounded perturbations. A generalization to nonlinear and time varying systems was discussed. Spectral properties of superstable systems were studied. For interval matrices, a complete solution was given to the problem of robust superstability.

A Randomized Cutting Plane Method with Probabilistic Geometric Convergence

We propose a randomized method for general convex optimization problems; namely, the minimization of a linear function over a convex body. The idea is to generate

Petersen's lemma on matrix uncertainty and its generalizations

N

Random spherical uncertainty in estimation and robustness

random points inside the body, choose the best one, and cut the part of the body defined by the linear constraint. We analyze the convergence properties of the algorithm from a theoretical viewpoint, i.e., under the rather standard assumption that an algorithm for uniform generation of random points in the convex body is available. Under this assumption, the expected rate of convergence for such method is proved to be geometric. This analysis is based on new results on the statistical properties of the empirical minimum over a convex body that we obtained in this paper. Moreover, explicit sample size results on convergence are derived. In particular, we compute the minimum number of random points that should be generated at each step in order to guarantee that, in a probabilistic sense, the method performs better than the deterministic center-of-gravity algorithm. From a practical viewpoint, we show how the method can be implemented using hit-and-run versions of Markov-chain Monte Carlo algorithms and exemplify the performance of this implementable modification via a number of illustrative problems. A crucial notion for the hit-and-run implementation is that of boundary oracle, which is available for most optimization problems including linear matrix inequalities and many other kinds of constraints. Preliminary numerical results for semidefinite programs are presented confirming that the randomized approach might be competitive to modern deterministic convex optimization methods.

The D-decomposition technique for linear matrix inequalities

Various generalizations and refinements are proposed for a well-known result on robust matrix sign-definiteness, which is extensively exploited in quadratic stability, design of robustly quadratically stabilizing controllers, robust LQR-problem, etc. The main emphasis is put on formulating the results in terms of linear matrix inequalities.

Superstable Linear Control Systems. II. Design

A theorem is formulated that gives an exact probability distribution for a linear function of a random vector uniformly distributed over a ball in n-dimensional space. This mathematical result is illustrated via applications to a number of important problems of estimation and robustness under spherical uncertainty. These include parameter estimation, characterization of attainability sets of dynamical systems, and robust stability of affine polynomial families.

Fixed-order controller design for SISO systems using Monte Carlo technique

In the framework of the theory of linear matrix inequalities, a method is proposed for determining all the domains in the parameter space having the property that an affine family of symmetric matrices has the same fixed number of like-sign eigenvalues inside each of the domains. The approach leans on the ideas of D-decomposition; it is particularly efficient in the problems involving few parameters. Generalizations of the method are considered along with its modifications to the presence of uncertainty.