A probabilistic framework for problems with real structured uncertainty in systems and control

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 efficient (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.     

Moving least-squares approximations for linearly-solvable stochastic optimal control problems

Nonlinear stochastic optimal control problems are fundamental in control theory.A general class of such problems can be reduced to computing the principal eigenfunction of a linear operator.Here,we describe a new method for finding this eigenfunction using a moving least-squares function approximation.We use efficient iterative solvers that do not require matrix factorization,thereby allowing us to handle large numbers of basis functions.The bases are evaluated at collocation states that change over iterations of the algorithm,so as to provide higher resolution at the regions of state space that are visited more often.The shape of the bases is automatically defined given the collocation states,in a way that avoids gaps in the coverage.Numerical results on test problems are provided.     

Iterative Learning Control of Iteration-Varying Systems via Robust Update Laws with Experimental Implementation

Iterative learning control (ILC) is an efficient way of improving the tracking performance of repetitive systems. While ILC can offer significant improvement to the transient response of complex dynamical systems, the fundamental assumption of iteration invariance of the process limits potential applications. Utilizing abstract Banach spaces as our problem setting, we develop a general approach that is applicable to the various frameworks encountered in ILC. Our main result is that robust invariant update laws lead to stable behavior in ILC systems, where iteration-varying systems converge to bounded neighborhoods of their nominal counterparts when uncertainties are bounded. Furthermore, if the uncertainties are convergent along the iteration axis, convergence to the nominal case can be guaranteed.     

Optimization under uncertainty with applications to design of truss structures

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, the 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.     

Discrete time mean-field stochastic linear-quadratic optimal control problems

This paper firstly presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Secondly, the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method. Thirdly, by introducing several sequences of bounded linear operators, the problem is formulated as an operator stochastic linear-quadratic optimal control problem. By the kernel-range decomposition representation of the expectation operator and its pseudo-inverse, the optimal control is derived using solutions to two algebraic Riccati difference equations. Finally, by completing the square, the two Riccati equations and the optimal control are also obtained.     

On the infinite time solution to state-constrained stochastic optimal control problems

A method is presented for solving the infinite time Hamilton-Jacobi-Bellman (HJB) equation for certain state-constrained stochastic problems. The HJB equation is reformulated as an eigenvalue problem, such that the principal eigenvalue corresponds to the expected cost per unit time, and the corresponding eigenfunction gives the value function (up to an additive constant) for the optimal control policy. The eigenvalue problem is linear and hence there are fast numerical methods available for finding the solution.     

Characterizing optimal solutions to weighted mixed sensitivity problems

Optimal weighted mixed sensitivity designs for single-input/single-output, linear, time-invariant systems are characterized by qualitative properties. Conditions for optimality based on flatness of the objective function and the relative numbers of closed right half-plane poles and zeros of the plant and optimal controller are examined.     

Approaches to uncertainties in facility layout problems: Perspectives at the beginning of the 21<Superscript>st</Superscript> Century

Production uncertainty is one of the most challenging aspects in manufacturing environments in the 21st century. The next generation of intelligent manufacturing is dynamically depending on the production requirements, and success in designing agile facilities is closely related to what extent these requirements are satisfied. This paper presents the most recent advancements in designing robust and flexible facilities under uncertainty. The focus is on exploring the way uncertainty is incorporated in facility design, namely dynamic and stochastic facility layout problems. Recent approaches are explored and categorized in detail, and previous approaches are briefly reviewed in the related categories. Furthermore, research avenues warranting exploration in the emerging field of facility design are also discussed.     

Solutions to a class of linear-quadratic-Gaussian (LQG) stochastic team problems with nonclassical information

We consider a stochastic dynamic team problem with two controllers and nonclassical information, which can be viewed as the transmission of a garbled version of a Gaussian message over a number of noisy channels under a fidelity criterion. We show that the optimum solution (under a quadratic loss functional) consists of linearly transforming the garbled message to a certain (optimum) power level P^* and then optimally decoding it by using a linear transformation at the receiving end. The power level P^* is determined by the solution of a fifth order algebraic equation. The paper also discusses an extension of this result to the case when the channel noise is correlated with the input random variable, and shows that for the single channel case the optimum solution is again linear.     

Probabilistic prediction methods for nonlinear systems with application to stochastic model predictive control

The performance of modern control methods, such as model predictive control, depends significantly on the accuracy of the system model. In practice, however, stochastic uncertainties are commonly present, resulting from inaccuracies in the modeling or external disturbances, which can have a negative impact on the control performance. This article reviews the literature on methods for predicting probabilistic uncertainties for nonlinear systems. Since a precise prediction of probability density functions comes along with a high computational effort in the nonlinear case, the focus of this article is on approximating methods, which are of particular relevance in control engineering practice. The methods are classified with respect to their approximation type and with respect to the assumptions about the input and output distribution. Furthermore, the application of these prediction methods to stochastic model predictive control is discussed including a literature review for nonlinear systems. Finally, the most important probabilistic prediction methods are evaluated numerically. For this purpose, the estimation accuracies of the methods are investigated first and the performance of a stochastic model predictive controller with different prediction methods is examined subsequently using multiple nonlinear systems, including the dynamics of an autonomous vehicle.     

Using evolution to improve neural network learning: pitfalls and solutions

Autonomous neural network systems typically require fast learning and good generalization performance, and there is potentially a trade-off between the two. The use of evolutionary techniques to improve the learning abilities of neural network systems is now widespread. However, there are a range of different evolutionary approaches that could be applied, and no systematic investigation has been carried out to find which work best. In this paper, such an investigation is presented, and it is shown that a range of evolutionary techniques can generate high performance networks, but they often lead to unwanted side effects, such as occasional instances of very poor performance. The nature of these problems are explored further, and it is shown how the evolution of age dependent plasticities and/or the use of ensemble techniques can alleviate them. A range of techniques are thus identified, with differing properties, that can be matched to the specific requirements of each application.     

A rolling horizon approach for stochastic mixed complementarity problems with endogenous learning: Application to natural gas markets

In this paper we present a new approach for solving energy market equilibria that is an extension of the classical Nash-Cournot approach. Specifically, besides allowing the market participants to decide on their own decision variables such as production, flows or the like, we allow them to compete in terms of adjusting the data in the problem such as scenario probabilities and costs, consistent with a dynamic, more realistic approach to these markets. Such a problem in its original form is very hard to solve given the product of terms involving decision-dependent data and the variables themselves. Moreover, in its more general form, the players can affect not only each others׳ objective functions but also the constraint sets of opponents making such a formulation a more complicated instance of generalized Nash problems. This new approach involves solving a sequence of stochastic mixed complementarity (MCP) problems where only partial foresight is used, i.e., a rolling horizon. Each stochastic MCP or roll, involves a look-ahead for a fixed number of time periods with learning on the part of the players to approximate the extended Nash paradigm. Such partial foresight stochastic MCPs also offer a realism advantage over more traditional perfect foresight formulations. Additionally, the rolling-horizon approach offers a computational advantage over scenario-reduction methods as is demonstrated with numerical tests on a natural gas market stochastic MCP. Lastly, we introduce a new concept, the Value of the Rolling Horizon (VoRH) to measure the closeness of different rolling horizon schemes to a perfect foresight benchmark and provide some numerical tests on it using a stylized natural gas market.     

Solution to a class of stochastic LQ problems with bounded control

A new approach for finding an exact analytical solution to the modified Hamilton-Jacobi-Bellman equation is proposed. Together with the recently developed hybrid solution method, the proposed strategy allows to find a solution to a whole class of stochastic optimal control problems with bounded in magnitude control force.     

Learning automata-based solutions to the optimal web polling problem modelled as a nonlinear fractional knapsack problem

We consider the problem of polling web pages as a strategy for monitoring the world wide web. The problem consists of repeatedly polling a selection of web pages so that changes that occur over time are detected. In particular, we consider the case where we are constrained to poll a maximum number of web pages per unit of time, and this constraint is typically dictated by the governing communication bandwidth, and by the speed limitations associated with the processing. Since only a fraction of the web pages can be polled within a given unit of time, the issue at stake is one of determining which web pages are to be polled, and we attempt to do it in a manner that maximizes the number of changes detected. We solve the problem by first modelling it as a stochastic nonlinear fractional knapsack problem. We then present an online learning automata (LA) system, namely, the hierarchy of twofold resource allocation automata (H-TRAA), whose primitive component is a twofold resource allocation automaton (TRAA). Both the TRAA and the H-TRAA have been proven to be asymptotically optimal. Finally, we demonstrate empirically that the H-TRAA provides orders of magnitude faster convergence compared to the learning automata knapsack game (LAKG) which represents the state-of-the-art for this problem. Further, in contrast to the LAKG, the H-TRAA scales sub-linearly. Based on these results, we believe that the H-TRAA has also tremendous potential to handle demanding real-world applications, particularly those which deal with the world wide web.     

Global maximum principle for the forward–backward stochastic optimal control problem with poisson jumps

This paper is concerned with the forward–backward stochastic optimal control problem with Poisson jumps. A necessary condition of optimality in the form of a global maximum principle as well as a sufficient condition of optimality are presented under the assumption that the diffusion and jump coefficients do not contain the control variable, and the control domain need not be convex. The case where there are some state constraints is also discussed. A financial example is discussed to illustrate the application of our result. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On optimal control of mean-field stochastic systems driven by Teugels martingales via derivative with respect to measures

ABSTRACT

This paper deals with partial information stochastic optimal control problem for general controlled mean-field systems driven by Teugels martingales associated with some Lévy process having moments of all orders, and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. The proof of our result is based on the derivative with respect to the probability law by applying Lions derivatives and a corresponding Itô formula. As an application, conditional mean-variance portfolio selection problem in incomplete market, where the system is governed by some Gamma process is studied to illustrate our theoretical results.     

Learning control of process systems with hard input constraints

In this paper, a novel and simple learning control strategy based on using a bounded nonlinear controller for process systems with hard input constraints is proposed. To enable the bounded nonlinear controller to learn to control a changing plant by merely observing the process output errors, a simple learning algorithm for parameter updating is derived based on the Lyapunov stability theorem. The learning scheme is easy to implement, and does not require any a priori process knowledge except the system output response direction. For demonstrating the effectiveness and applicability of the learning control strategy, the control of a once-through boiler, as well as an open-loop unstable continuously stirred tank reactor (CSTR), were investigated. Furthermore, extensive comparisons of the proposed scheme with the conventional PI controller and with some existing model-free intelligent controllers were also performed. Due to significant features of simple structure, efficient algorithm and good performance, the proposed learning control strategy appears to be a promising and practical approach to the intelligent control of process systems subject to hard input constraints.     

Computing the Shapley value in allocation problems: approximations and bounds,with an application to the Italian VQR research assessment program

Abstract

In allocation problems with indivisible goods, money compensation is used to distribute worth in a fair way. Coalitional games provide a formal mathematical framework to model such problems, and the Shapley value is a solution concept widely used to realise a fair distribution. To overcome its intractability, we describe how to simplify allocation problems and we propose algorithms for computing lower bounds and upper bounds of the Shapley value that can be combined with approximation algorithms. The proposed techniques have been implemented and tested on a real-world application of allocation problems, namely, the Italian research assessment program known as VQR.     

Robust control with uncertainty estimation for feedback linearizable systems: application to control of distillation columns

The goal of this paper is to describe a linearizing feedback adaptive control structure which leads to a high quality regulation of the output error in the presence of uncertainties and external disturbances. The controller consists of three elements: a nominal input–output linearizing compensator, a state observer and an uncertainty estimator, which provides the adaptive part of the control structure. In this way, the feedback controller, based on the disturbance observer, compensates for external disturbances and plant uncertainties. The effectiveness of the controller is demonstrated on a distillation column via numerical simulations. ©