IEIIT

In recent years, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information.

This paper presents an alternative approach to design of linear parameter-varying (LPV) control systems. In contrast to previous methods, which are focused on deterministic algorithms, this paper is based on a probabilistic setting. The proposed randomized algorithm provides a sequence of candidate solutions converging with probability one to a feasible solution in a ÿnite number of steps. The main features of this approach are as follows: (i) The randomized algorithm gives a method for general LPV plants with state space matrices depending on scheduling parameters in a nonlinear manner. That is, the probabilistic setting does not need a gridding of the set of scheduling parameters or approximations such as a linear fractional transformation of the state space matrices. (ii) The proposed algorithm is sequential and, at each iteration, it does not require heavy computational e ort such as solving simultaneously a large number of linear matrix inequalities. ?

Many robust control problems can be formulated in abstract form as convex feasibility programs, where one seeks a solution x that satisfies a set of inequalities of the form F . = {f (x, ) 0, ∈ D}. This set typically contains an infinite and uncountable number of inequalities, and it has been proved that the related robust feasibility problem is numerically hard to solve in general.

The objective of this paper is twofold. First, the problem of generation of real random matrix samples with uniform distribution in structured (spectral) norm bounded sets is studied. This includes an analysis of the distribution of the singular values of uniformly distributed real matrices, and an e cient (i.e. polynomial-time) algorithm for their generation. Second, it is shown how the developed techniques may be used to solve in a probabilistic setting several hard problems involving systems subject to real structured uncertainty. ?

The probabilistic approach to analysis and design of robust control systems is an emerging philosophy that gained increasing interest in recent years, see e.g. the book [24]. Opposed to the so-far dominating paradigm of deterministic worst-case robustness, the probabilistic approach presents itself as a natural tool to deal with the random character of uncertainties affecting control systems. In this paper, we discuss randomized algorithms for probabilistic robustness, with particular attention to recently developed methodologies for controller synthesis.

This paper presents an alternative approach to solve robust performance problems for linear parametervarying (LPV) control systems. In contrast to previous methods, which are focused on deterministic algorithms, this paper is based on a probabilistic setting. The proposed randomized algorithm provides a sequence of candidate solutions converging with probability one to a feasible solution in a finite number of steps. The main advantages of this method over the previous literature are as follows: (i) The randomized algorithm gives a direct method for general LPV plants with state space matrices depending on scheduling parameters in a nonlinear manner. That is, the probabilistic setting does not require any approximation, such as a linear fractional transformation (LFT), of the state space matrices or a gridding on the set of scheduling parameters. (ii) The proposed algorithm is sequential and, at each iteration, it does not require heavy computational effort such as solving simultaneously a large number of linear matrix inequalities.

In this paper we study the probabilistic real stability radius, which gives a description of the degradation of the probability of stability b e y ond the classical real stability radius. The main result of the paper is to determine the probability density function of the singular values of the perturbation matrix . Subsequently, w e propose an algorithm for uniform generation of matrix samples, so that the probability of stability can be e ectively estimated via a randomized algorithm.

Environmental systems often involve phenomena that are continuous functions not only of time, but also of other independent variables, such as space coordinates. Typical examples are transportation phenomena of mass or energy, such as heat transmission and/or exchange, humidity diffusion or concentration distributions. These systems are intrinsically distributed parameter systems whose description usually requires the introduction of partial differential equations (PDE). Therefore, their modelling can be quite complex, both for what concerns the model construction and its identification. Indeed, a typical approach for the simulation of such systems is the use of finite elements techniques. However, this kind of description usually involves a huge number of parameters and requires time-consuming computations while not being suited for identification. For this reason, such models are generally not suitable for control purposes. In many cases, however, the involved phenomena depend on the independent (space) variables in a smooth way, and for fixed values of the independent variables, input-output relations can be satisfactorily represented by linear time-invariant models. In such conditions, a possible alternative to PDE consists in representing the physical system with a Linear Parameter Varying (LPV) model whose parameters are functions of the independent variables. The advantage of this approach is the relatively simple model obtained, which is directly suitable for control purposes and can be easily identified from input-output data by means of classical techniques. Moreover, optimal identification schemes can be derived for such models, allowing the optimization of the number of measurements. This can be particularly useful in several environmental applications for which the cost of measurements represents a severe constraint.

In this paper, we present an overview of probabilistic techniques based on randomized algorithms for solving "hard" problems arising in performance verification and control of complex systems. This area is fairly recent, even though its roots lie in the robustness techniques for handling uncertain control systems developed in the 1980s. In contrast to these deterministic techniques, the main ingredient of the methods discussed in this survey is the use of probabilistic concepts. The introduction of probability and random sampling permits overcoming the fundamental tradeoff between numerical complexity and conservatism that lie at the roots of the worst-case deterministic methodology. The simplicity of implementation of randomized techniques may also help bridging the gap between theory and practical applications.

In this paper, we consider robust controller design problems where the design objectives are divided into two categories: hard specifications and soft specifications. Hard specifications need to be met robustly against uncertainties, and are addressed using a classical H 1 =-synthesis approach. Soft specifications are given as average performance requirements with respect to uncertainty, and are addressed using a probabilistic design method. The key element in this approach is an algorithm for random generation of stable transfer matrices, with a bound on the H 1 norm.

This paper follows the line of research aimed to develop randomized algorithms for probabilistic analysis and design of control systems. In particular, a result for the generation of real matrix samples uniformly distributed in the spectral norm ball is presented. To this end, the distribution of the singular values of random uniform matrices is rst studied. The sample generation is then based on the conditional densities method. A key result of this paper is the computation in closed form of the marginal densities of the singular values, leading to a polynomial-time method for the real matrix sample generation problem.

Many real-world engineering design problems are naturally cast in the form of optimization programs with uncertainty-contaminated data. In this context, a reliable design must be able to cope in some way with the presence of uncertainty. In this paper, we consider two standard philosophies for finding optimal solutions for uncertain convex optimization problems. In the first approach, classical in the stochastic optimization literature, the optimal design should minimize the expected value of the objective function with respect to uncertainty (average approach), while in the second one it should minimize the worst-case objective (worst-case or min-max approach). Both approaches are briefly reviewed in this paper and are shown to lead to exact and numerically efficient solution schemes when the uncertainty enters the data in simple form. For general uncertainty dependence however, these problems are numerically hard. In this paper, we present two techniques based on uncertainty randomization that permit to solve efficiently some suitable probabilistic relaxation of the indicated problems, with full generality with respect to the way in which the uncertainty enters the problem data. In the specific context of truss topology design, uncertainty in the problem arises, for instance, from imprecise knowledge of material characteristics and/or loading configurations. In this paper, we show how reliable structural design can be obtained using the proposed techniques based on the interplay of convex optimization and randomization.

In this paper, fast recursive algorithms for the approximation of an n-dimensional convex polytope by means of an inscribed ellipsoid are presented. These algorithms consider at each step a single inequality describing the polytope and, under mild assumptions, they are guaranteed to converge in a ÿnite number of steps. For their recursive nature, the proposed algorithms are better suited to treat a quite large number of constraints than standard o -line solutions, and have their natural application to problems where the set of constraints is iteratively updated, as on-line estimation problems, nonlinear convex optimization procedures and set membership identiÿcation. ?

This paper introduces a new Matlab package, Ract, 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 Ract package is freely available at http://ract.sourceforge.net, and only requires the Yalmip toolbox to be installed in the Matlab environment.

The so-called PageRank algorithm has been used at Google for properly ranking search results. It quantifies the importance of each page by the structure of links in the web. In our recent work, we have proposed a distributed randomized approach for the PageRank computation, where the pages find their own values by communicating with linked pages. This paper builds upon this approach to improve the computation and communication load for the algorithm by systematically aggregating the webpages while maintaining performance.

In this paper, we consider a robust controller design problem, where the design objectives are divided into two categories: Hard specifications and soft specifications. Hard specifications need to be met robustly against uncertainties, and are addressed using a classical H ∞ /µ-synthesis approach. Soft specifications are given as average performance requirements with respect to uncertainty, and are addressed using a probabilistic design method.

In this paper, we address the problem of analyzing the performance of an electrical circuit in the presence of uncertainty in the network components. In particular, we consider the case when the uncertainties are known to be bounded and have probabilistic nature, and aim at evaluating the probability that a given system property holds. In contrast to the standard Monte Carlo approach, which utilizes random samples of the uncertainty to estimate "soft" bounds on this probability, we present a methodology providing "hard" (deterministic) upper and lower bounds. To this aim, we develop an iterative algorithm, based on a property oracle, which is shown to converge asymptotically to the true probability of property satisfaction.

The focal point of this paper is the well known problem of polynomial positivity over a given domain. More specifically, we consider a multivariate polynomial f (x) with parameter vector x restricted to a hypercube X ⊂ R n . The objective is to determine if f (x) > 0 for all x ∈ X . Motivated by NP-Hardness considerations, we introduce the so-called dilation integral method. Using this method, a "softening" of this problem is described. That is, rather than insisting that f (x) be positive for all x ∈ X , we consider the notions of practical positivity and practical non-positivity. As explained in the paper, these notions involve the calculation of a quantity ǫ > 0 which serves as an upper bound on the percentage volume of violation in parameter space where f (x) ≤ 0. Whereas checking the polynomial positivity requirement may be computationally prohibitive, using our ǫ-softening and associated dilation integrals, computations are typically straightforward. One highlight of this paper is that we obtain a sequence of upper bounds ǫ k which are shown to be "sharp" in the sense that they converge to zero whenever the positivity requirement is satisfied. Since for fixed n, computational difficulties generally increase with k, this paper also focuses on results which reduce the size of the required k in order to achieve an acceptable percentage volume certification level. For large classes of problems, as the dimension of parameter space n grows, the required k value for acceptable percentage volume violation may be quite low. In fact, it is often the case that low volumes of violation can be achieved with values as low as k = 2.