The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Real and Complex Analysis

This paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms. This includes the wide class of NP-hard control problems representable by means of parameter-dependent linear matrix inequalities (LMIs). It is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem (the scenario problem) whose solution is approximately feasible for the original (usually infinite) set of constraints, i.e., the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased. We provide an explicit and efficient bound on the number of samples required to attain a-priori specified levels of probabilistic guarantee of robustness. A rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial-time solution, if robustness is intended in the proposed risk-adjusted sense.

https://www.sheffield.ac.uk/polopoly_fs/1.60408!/file/campi_scenario-optimization.pdf

The scenario approach to robust control design

(1990). ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY. Electric Machines & Power Systems: Vol. 18, No. 2, pp. 209-210.

Energy function analysis for power system stability

Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance’ parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chance-constrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chance-constrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a pre-specified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations. In this paper, we consider an alternative ‘randomized’ or ‘scenario’ approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.

/pdf/uncertain-convex-programs-randomized-solutions-and-3vs4hcxix0.pdf

Uncertain convex programs: randomized solutions and confidence levels

https://hal.inria.fr/hal-00920078/document

Fixed-time consensus algorithm for multi-agent systems with integrator dynamics

Consider the classical state feedback design in the linear system x = Ax + Bu subject to performance specifications with an additional requirement that the control input vector u = Kx has as many zero entries as possible. The corresponding gain K is referred to as a row-sparse controller. We propose an approach to approximate solution of this kind of nonconvex problems by formulating the proper convex surrogate,-the minimization of a certain matrix norm subject to LMI constraints. The novelty of the paper is the problem formulation itself and the construction of the surrogate. The two main contributions are the design of low-dimensional output to be used in static output feedback, and suboptimal design illustrated via LQR. The results of preliminary numerical experiments are twofold. First, in many test problems, the number of controls was considerably reduced without significant loss in performance. Second, the number of nonzero entries obtained by our method is either very close to or coincide with the minimum possible amount. The approach can be further extended to handle numerous problems of optimal and robust control in sparse formulation.

/pdf/an-lmi-approach-to-structured-sparse-feedback-design-in-2uvmci5ene.pdf

An LMI approach to structured sparse feedback design in linear control systems

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.

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

A number of challenging problems in linear control theory are considered which admit simple formulation and yet lack efficient solution methods. These problems relate to the classical theory of linear systems as well as to the robust theory where the system description contains uncertainty. Various solution methods are discussed and the results of numerical simulations are given.

Pavel Shcherbakov

Papers

Fixed-time consensus algorithm for multi-agent systems with integrator dynamics

An LMI approach to structured sparse feedback design in linear control systems

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

Hard Problems in Linear Control Theory: Possible Approaches to Solution

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