On Repetitive Scenario Design

Repetitive Scenario Design (RSD) is a randomized approach to robust design based on iterating two phases: a standard scenario design phase that uses $N$ scenarios (design samples), followed by randomized feasibility phase that uses $N_o$ test samples on the scenario solution. We give a full and exact probabilistic characterization of the number of iterations required by the RSD approach for returning a solution, as a function of $N$, $N_o$, and of the desired levels of probabilistic robustness in the solution. This novel approach broadens the applicability of the scenario technology, since the user is now presented with a clear tradeoff between the number $N$ of design samples and the ensuing expected number of repetitions required by the RSD algorithm. The plain (one-shot) scenario design becomes just one of the possibilities, sitting at one extreme of the tradeoff curve, in which one insists in finding a solution in a single repetition: this comes at the cost of possibly high $N$. Other possibilities along the tradeoff curve use lower $N$ values, but possibly require more than one repetition

Robust Model Predictive Control via Scenario Optimization

This paper discusses a novel probabilistic approach for the design of robust model predictive control (MPC) laws for discrete-time linear systems affected by parametric uncertainty and additive disturbances. The proposed technique is based on the iterated solution, at each step, of a finite-horizon optimal control problem (FHOCP) that takes into account a suitable number of randomly extracted scenarios of uncertainty and disturbances, followed by a specific command selection rule implemented in a receding horizon fashion. The scenario FHOCP is always convex, also when the uncertain parameters and disturbance belong to non-convex sets, and irrespective of how the model uncertainty influences the system's matrices. Moreover, the computational complexity of the proposed approach does not depend on the uncertainty/disturbance dimensions, and scales quadratically with the control horizon. The main result in this paper is related to the analysis of the closed loop system under receding-horizon implementation of the scenario FHOCP, and essentially states that the devised control law guarantees constraint satisfaction at each step with some a-priori assigned probability p, while the system's state reaches the target set either asymptotically, or in finite time with probability at least p. The proposed method may be a valid alternative when other existing techniques, either deterministic or stochastic, are not directly usable due to excessive conservatism or to numerical intractability caused by lack of convexity of the robust or chance-constrained optimization problem.Comment: This manuscript is a preprint of a paper accepted for publication in the IEEE Transactions on Automatic Control, with DOI: 10.1109/TAC.2012.2203054, and is subject to IEEE copyright. The copy of record will be available at http://ieeexplore.ieee.or

Repetitive Scenario Design

Repetitive Scenario Design (RSD) is a randomized approach to robust design based on iterating two phases: a standard scenario design phase that uses N scenarios (design samples), followed by randomized feasibility phase that uses No test samples on the scenario solution. We give a full and exact probabilistic characterization of the number of iterations required by the RSD approach for returning a solution, as a function of N, No, and of the desired levels of probabilistic robustness in the solution. This novel approach broadens the applicability of the scenario technology, since the user is now presented with a clear tradeoff between the number N of design samples and the ensuing expected number of repetitions required by the RSD algorithm. The plain (one-shot) scenario design becomes just one of the possibilities, sitting at one extreme of the tradeoff curve, in which one insists in finding a solution in a single repetition: this comes at the cost of possibly high N. Other possibilities along the tradeoff curve use lower N values, but possibly require more than one repetition

Stochastic model predictive control of LPV systems via scenario optimization

A stochastic receding-horizon control approach for constrained Linear Parameter Varying discrete-time systems is proposed in this paper. It is assumed that the time-varying parameters have stochastic nature and that the system's matrices are bounded but otherwise arbitrary nonlinear functions of these parameters. No specific assumption on the statistics of the parameters is required. By using a randomization approach, a scenario-based finite-horizon optimal control problem is formulated, where only a finite number M of sampled predicted parameter trajectories (‘scenarios') are considered. This problem is convex and its solution is a priori guaranteed to be probabilistically robust, up to a user-defined probability level p. The p level is linked to M by an analytic relationship, which establishes a tradeoff between computational complexity and robustness of the solution. Then, a receding horizon strategy is presented, involving the iterated solution of a scenario-based finite-horizon control problem at each time step. Our key result is to show that the state trajectories of the controlled system reach a terminal positively invariant set in finite time, either deterministically, or with probability no smaller than p. The features of the approach are illustrated by a numerical example

Convex Relaxations for Pose Graph Optimization with Outliers

Pose Graph Optimization involves the estimation of a set of poses from pairwise measurements and provides a formalization for many problems arising in mobile robotics and geometric computer vision. In this paper, we consider the case in which a subset of the measurements fed to pose graph optimization is spurious. Our first contribution is to develop robust estimators that can cope with heavy-tailed measurement noise, hence increasing robustness to the presence of outliers. Since the resulting estimators require solving nonconvex optimization problems, we further develop convex relaxations that approximately solve those problems via semidefinite programming. We then provide conditions under which the proposed relaxations are exact. Contrarily to existing approaches, our convex relaxations do not rely on the availability of an initial guess for the unknown poses, hence they are more suitable for setups in which such guess is not available (e.g., multi-robot localization, recovery after localization failure). We tested the proposed techniques in extensive simulations, and we show that some of the proposed relaxations are indeed tight (i.e., they solve the original nonconvex problem 10 exactly) and ensure accurate estimation in the face of a large number of outliers.Comment: 10 pages, 5 figures, accepted for publication in the IEEE Robotics and Automation Letters, 201

Robust dynamic traffic assignment for single destination networks under demand and capacity uncertainty

In this article, we discuss the system-optimum dynamic traffic assignment (SO-DTA) problem in the presence of time-dependent uncertainties on both traffic demands and road link capacities. Building on an earlier formulation of the problem based on the cell transmission model, the SO-DTA problem is robustly solved, in a probabilistic sense, within the framework of random convex programs (RCPs). Different from traditional robust optimization schemes, which find a solution that is valid for all the values of the uncertain parameters, in the RCP approach we use a fixed number of random realizations of the uncertainty, and we are able to guarantee a priori a desired upper bound on the probability that a new, unseen realization of the uncertainty would make the computed solution unfeasible. The particular problem structure and the introduction of an effective domination criterion for discarding a large number of generated samples enables the computation of a robust solution for medium- to large-scale networks, with low desired violation probability, with a moderate computational effort. The proposed approach is quite general and applicable to any problem that can be formulated through a linear programing model, where the stochastic parameters appear in the constraint constant terms only. Simulation results corroborate the effectiveness of our approach

Sparse identification of posynomial models

Posynomials are nonnegative combinations of monomials with possibly fractional and both positive and negative exponents. Posynomial models are widely used in various engineering design endeavors, such as circuits, aerospace and structural design, mainly due to the fact that design problems cast in terms of posynomial objectives and constraints can be solved efficiently by means of a convex optimization technique known as geometric programming (GP). However, while quite a vast literature exists on GP-based design, very few contributions can yet be found on the problem of identifying posynomial models from experimental data. Posynomial identification amounts to determining not only the coefficients of the combination, but also the exponents in the monomials, which renders the identification problem hard. In this paper, we propose an approach to the identification of multivariate posynomial models based on the expansion on a given large-scale basis of monomials. The model is then identified by seeking coefficients of the combination that minimize a mixed objective, composed by a term representing the fitting error and a term inducing sparsity in the representation, which results in a problem formulation of the “square-root LASSO” type, with nonnegativity constraints on the variables. We propose to solve the problem via a sequential coordinate-minimization scheme, which is suitable for large-scale implementations. A numerical example is finally presented, dealing with the identification of a posynomial model for a NACA 4412 airfoil