76,172 research outputs found
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
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
Fully probabilistic control for uncertain nonlinear stochastic systems
This paper develops a novel probabilistic framework for stochastic nonlinear and uncertain control problems. The proposed framework exploits the Kullback–Leibler divergence to measure the divergence between the distribution of the closed-loop behavior of a dynamical system and a predefined ideal distribution. To facilitate the derivation of the analytic solution of the randomized controllers for nonlinear systems, transformation methods are applied such that the dynamics of the controlled system becomes affine in the state and control input. Additionally, knowledge of uncertainty is taken into consideration in the derivation of the randomized controller. The derived analytic solution of the randomized controller is shown to be obtained from a generalized state-dependent Riccati solution that takes into consideration the state-and control-dependent functional uncertainty of the controlled system. The pro-posed framework is demonstrated on an inverted pendulum on a cart problem, and the results are obtaine
On the Sample Size of Random Convex Programs with Structured Dependence on the Uncertainty (Extended Version)
The "scenario approach" provides an intuitive method to address chance
constrained problems arising in control design for uncertain systems. It
addresses these problems by replacing the chance constraint with a finite
number of sampled constraints (scenarios). The sample size critically depends
on Helly's dimension, a quantity always upper bounded by the number of decision
variables. However, this standard bound can lead to computationally expensive
programs whose solutions are conservative in terms of cost and violation
probability. We derive improved bounds of Helly's dimension for problems where
the chance constraint has certain structural properties. The improved bounds
lower the number of scenarios required for these problems, leading both to
improved objective value and reduced computational complexity. Our results are
generally applicable to Randomized Model Predictive Control of chance
constrained linear systems with additive uncertainty and affine disturbance
feedback. The efficacy of the proposed bound is demonstrated on an inventory
management example.Comment: Accepted for publication at Automatic
A scenario approach for non-convex control design
Randomized optimization is an established tool for control design with
modulated robustness. While for uncertain convex programs there exist
randomized approaches with efficient sampling, this is not the case for
non-convex problems. Approaches based on statistical learning theory are
applicable to non-convex problems, but they usually are conservative in terms
of performance and require high sample complexity to achieve the desired
probabilistic guarantees. In this paper, we derive a novel scenario approach
for a wide class of random non-convex programs, with a sample complexity
similar to that of uncertain convex programs and with probabilistic guarantees
that hold not only for the optimal solution of the scenario program, but for
all feasible solutions inside a set of a-priori chosen complexity. We also
address measure-theoretic issues for uncertain convex and non-convex programs.
Among the family of non-convex control- design problems that can be addressed
via randomization, we apply our scenario approach to randomized Model
Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro
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