Andrey A. Tremba

D-decomposition technique state-of-the-art

It is a survey of recent extensions and new applications for the classical D-decomposition technique. We investigate the structure of the parameter space decomposition into root invariant regions for single-input single-output systems linear depending on the parameters. The D-decomposition for uncertain polynomials is considered as well as the problem of describing all stabilizing controllers of the certain structure (for instance, PID-controllers) that satisfy given H∞-criterion. It is shown that the D-decomposition technique can be naturally linked with M-Δ framework (a general scheme for analysis of uncertain systems) and it is applicable for describing feasible sets for linear matrix inequalities. The problem of robust synthesis for linear systems can be also treated via D-decomposition technique.

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.

Regularization-based solution of the PageRank problem for large matrices

For a column-stochastic matrix, consideration was given to determination of the eigenvector which corresponds to the unit eigenvalue. Such problems are encountered in many applications,—in particular, at ranking the web pages (PageRank). Since the PageRank problem is of special interest for larger matrices, the emphasis was made on the power method for direct iterative calculation of the eigenvector. Several variants of regularization of the power methods were compared, and their relations were considered. The distinctions of their realizations were given.

Large deviations in linear control systems with nonzero initial conditions

Research in the transient response in linear systems with nonzero initial conditions was initiated by A.A. Feldbaum in his pioneering work [1] as early as in 1948. However later, studies in this direction have faded down, and since then, the notion of transient process basically means the response of the system with zero initial conditions to the unit step input. A breakthrough in this direction is associated with the paper [2] by R.N. Izmailov, where large deviations of the trajectories from the origin were shown to be unavoidable if the poles of the closed-loop system are shifted far to the left in the complex plane.In this paper we continue the analysis of this phenomenon for systems with nonzero initial conditions. Namely, we propose a more accurate estimate of the magnitude of the peak and show that the effect of large deviations may be observed for different root locations. We also present an upper bound on deviations by using the linear matrix inequality (LMI) technique. This same approach is then applied to the design of a stabilizing linear feedback aimed at diminishing deviations in the closed-loop system. Related problems are also discussed, e.g., such as analysis of the transient response of systems with zero initial conditions and exogenous disturbances in the form of either unit step function or harmonic signal.

Design of the low-order controllers by the H∞ criterion: A parametric approach

Consideration was given to the problem of describing all stabilizing controllers of a given structure (for example, the the PID-controllers) satisfying the H∞ criterion. Controllers of a certain family were defined by the parameters k, and in the parameter space a domain corresponding to the desired criteria was specified. Two approaches were proposed where (i) the desired domain is represented as an intersection of the admissible sets or (ii) its boundary is determined analytically. The two-parameter case is of special importance because it allows one to make use of the graphical mathematics.

Extension of a saddle point mirror descent algorithm with application to robust PageRank

The paper is devoted to designing an efficient recursive algorithm for solving the robust PageRank problem recently proposed by Juditsky and Polyak (2012) [4]. To this end, we reformulate the problem to a specific convex-concave saddle point problem min_x∈X max_y∈Y q(x, y) with simple convex sets X ∈ ℝ^N and Y ∈ ℝ^N, i.e., standard simplex and Euclidean unit ball, respectively. Aiming this goal we develop an extension of saddle point mirror descent algorithm where additional parameter sequence is introduced, thus providing more degree of freedom and the refined error bounds. Detailed complexity results of this method applied to the robust PageRank problem are given and discussed. Numerical example illustrates the theoretical results proved.

Robust D-decomposition under lp-bounded parametric uncertainties

Consideration was given to stability of an affine family of uncertain polynomials defined by two real or one complex parameter, the rest of the parameters characterizing indeterminacy. On the plane of family parameters, a domain was established where the uncertain polynomials are stable. The method of robust D-decomposition was used. For the cases where the uncertain parameters are real and bounded in the Euclidean norm or are complex and bounded in the lp norm, expressions for the boundary of these domains were obtained.

Application of the Mirror Descent Method to minimize average losses coming by a poisson flow

We treat a convex problem to minimize average loss function for a stochastic system operating in continuous time. The losses on time horizon T arise at the jump times of a Poisson process with intensity being an unknown random process. The oracle gives randomly noised gradients of the loss function; the noises are additive, unbiased, with the bounded dual norm in average square sense. The goal consists in minimizing the average integral loss over a given convex compact set in the N-dimension space. We propose a mirror descent algorithm and prove an explicit upper bound for the average integral loss regret. The bound is of type “square root of T” with an explicit coefficient. Finally, we describe an example of optimization for a server processing a stream of incoming requests, and we discuss simulation results.

Adaptive mirror descent algorithm for the minimization of expected cumulative losses driven by a renewal process

The problem considered in this paper is the minimization of expected cumulative losses in a stochastic system. The losses over time horizon are formed by the values of an unknown loss function at the consecutive jump times of a renewal process. The loss is assumed to be a convex function of a vector parameter, and the only available information is represented by an oracle which provides stochastic sub-gradients of the loss function. The control objective is to minimize the expected cumulative loss over a given convex compact set. We propose an adaptive mirror descent algorithm and prove an explicit upper bound for the related regret, which is the difference between the expected cumulative losses and the minimum. Finally, to exemplify the efficiency of the method, we consider the problem of minimization of the expected cumulative losses over the standard simplex by handling a stream of losses arriving by the Erlang process, and we discuss the simulation results.

A mirror descent algorithm for minimization of mean Poisson flow driven losses

A problem of minimization of integral losses on given horizon is considered for stochastic system in continuous time. The losses occur in jump times of a Poisson process, and represent continuous convex function of control parameter on convex compact finite-dimensional set. At the jump times an oracle provide stochastically perturbed sub-gradient of the loss function, bounded in mean squares; the noise is additive and centered. Control strategy generated by Mirror Descent algorithm is suggested. For the strategy an explicit upper bound for integral loss discrepancy over its minimum is proved. Example of such strategy application to queueing model is examined.