A Statistical Learning Theory Approach for Uncertain Linear and Bilinear Matrix Inequalities

In this paper, we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities, which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results in statistical learning theory, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chervonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of these problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality. The effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters.Comment: 19 pages, 2 figures, Accepted for Publication in Automatic

Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples

Sequential Randomized Algorithms for Convex Optimization in the Presence of Uncertainty

In this paper, we propose new sequential randomized algorithms for convex optimization problems in the presence of uncertainty. A rigorous analysis of the theoretical properties of the solutions obtained by these algorithms, for full constraint satisfaction and partial constraint satisfaction, respectively, is given. The proposed methods allow to enlarge the applicability of the existing randomized methods to real-world applications involving a large number of design variables. Since the proposed approach does not provide a priori bounds on the sample complexity, extensive numerical simulations, dealing with an application to hard-disk drive servo design, are provided. These simulations testify the goodness of the proposed solution.Comment: 18 pages, Submitted for publication to IEEE Transactions on Automatic Contro

A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P

Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions

This paper proposes an uncertainty analysis framework based on the characterization of the uncertain parameter space. This characterization enables the identification of worst-case uncertainty combinations and the approximation of the failure and safe domains with a high level of accuracy. Because these approximations are comprised of subsets of readily computable probability, they enable the calculation of arbitrarily tight upper and lower bounds to the failure probability. A Bernstein expansion approach is used to size hyper-rectangular subsets while a sum of squares programming approach is used to size quasi-ellipsoidal subsets. These methods are applicable to requirement functions whose functional dependency on the uncertainty is a known polynomial. Some of the most prominent features of the methodology are the substantial desensitization of the calculations from the uncertainty model assumed (i.e., the probability distribution describing the uncertainty) as well as the accommodation for changes in such a model with a practically insignificant amount of computational effort

Randomized algorithms for control of uncertain systems with application to hand disk drives

Ph.DDOCTOR OF PHILOSOPH

Distributed Random Convex Programming via Constraints Consensus

This paper discusses distributed approaches for the solution of random convex programs (RCP). RCPs are convex optimization problems with a (usually large) number N of randomly extracted constraints; they arise in several applicative areas, especially in the context of decision under uncertainty, see [2],[3]. We here consider a setup in which instances of the random constraints (the scenario) are not held by a single centralized processing unit, but are distributed among different nodes of a network. Each node "sees" only a small subset of the constraints, and may communicate with neighbors. The objective is to make all nodes converge to the same solution as the centralized RCP problem. To this end, we develop two distributed algorithms that are variants of the constraints consensus algorithm [4],[5]: the active constraints consensus (ACC) algorithm, and the vertex constraints consensus (VCC) algorithm. We show that the ACC algorithm computes the overall optimal solution in finite time, and with almost surely bounded communication at each iteration. The VCC algorithm is instead tailored for the special case in which the constraint functions are convex also w.r.t. the uncertain parameters, and it computes the solution in a number of iterations bounded by the diameter of the communication graph. We further devise a variant of the VCC algorithm, namely quantized vertex constraints consensus (qVCC), to cope with the case in which communication bandwidth among processors is bounded. We discuss several applications of the proposed distributed techniques, including estimation, classification, and random model predictive control, and we present a numerical analysis of the performance of the proposed methods. As a complementary numerical result, we show that the parallel computation of the scenario solution using ACC algorithm significantly outperforms its centralized equivalent

Finite-Time Control of Uncertain Linear Systems Using Statistical Learning Methods

In this paper we show how some difficult linear algebra problems can be “approximately” solved using statistical learning methods. We illustrate our results by considering the state and output feedback, finite-time robust stabilization problems for linear systems subject to time-varying norm-bounded uncertainties and to unknown disturbances. In the state feedback case, we have obtained in an earlier paper, a sufficient condition for finite-time stabilization in the presence of time-varying disturbances; such condition requires the solution of a Linear Matrix Inequality (LMI) feasibility problem, which is by now a standard application of linear algebraic methods. In the output feedback case, however, we end up with a Bilinear Matrix Inequality (BMI) problem which we attack by resorting to a statistical approach