21 research outputs found
Randomized Solutions to Convex Programs with Multiple Chance Constraints
The scenario-based optimization approach (`scenario approach') provides an
intuitive way of approximating the solution to chance-constrained optimization
programs, based on finding the optimal solution under a finite number of
sampled outcomes of the uncertainty (`scenarios'). A key merit of this approach
is that it neither assumes knowledge of the uncertainty set, as it is common in
robust optimization, nor of its probability distribution, as it is usually
required in stochastic optimization. Moreover, the scenario approach is
computationally efficient as its solution is based on a deterministic
optimization program that is canonically convex, even when the original
chance-constrained problem is not. Recently, researchers have obtained
theoretical foundations for the scenario approach, providing a direct link
between the number of scenarios and bounds on the constraint violation
probability. These bounds are tight in the general case of an uncertain
optimization problem with a single chance constraint. However, this paper shows
that these bounds can be improved in situations where the constraints have a
limited `support rank', a new concept that is introduced for the first time.
This property is typically found in a large number of practical
applications---most importantly, if the problem originally contains multiple
chance constraints (e.g. multi-stage uncertain decision problems), or if a
chance constraint belongs to a special class of constraints (e.g. linear or
quadratic constraints). In these cases the quality of the scenario solution is
improved while the same bound on the constraint violation probability is
maintained, and also the computational complexity is reduced.Comment: This manuscript is the preprint of a paper submitted to the SIAM
Journal on Optimization and it is subject to SIAM copyright. SIAM maintains
the sole rights of distribution or publication of the work in all forms and
media. If accepted, the copy of record will be available at
http://www.siam.or
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
Error Bounds in Nonlinear Model Predictive Control with Linear Differential Inclusions of Parametric-Varying Embeddings
In this work, we provide deterministic error bounds for the actual state
evolution of nonlinear systems embedded with the linear parametric variable
(LPV) formulation and steered by model predictive control (MPC). The main
novelty concerns the explicit derivation of these deterministic bounds as
polytopic tubes using linear differential inclusions (LDIs), which provide
exact error formulations compared to linearization schemes that introduce
additional error and deteriorate conservatism. The analysis and method are
certified by solving the regulator problem of an unbalanced disk that stands as
a classical control benchmark example.Comment: 7 pages, 3 figure
Input-to-state stabilization of feasible model predictive controllers for linear systems
Research on sub-optimal Model Predictive Control (MPC) has led to a variety of optimization methods based on explicit or online approaches, or combinations thereof. Its foremost aim is to guarantee essential controller properties, i.e. recursive feasibility, stability, and robustness, at reduced and predictable computational cost, i.e. computation time and storage space. This paper shows how the input sequence of any (not necessarily stabilizing) sub-optimal controller and the shifted input sequence from the previous time step can be used in an optimal convex combination, which is easy to determine online, in order to guarantee input-to-state stability for the closed-loop system. The presented method is thus able to stabilize a wide range of existing sub-optimal MPC schemes that lack a formal stability guarantee, if they can be considered as a continuous map from the state space to the space of feasible input sequences
Dissertatio Iuridica De Rerum Communione
Quam Autoritate ac Indultu Magnifici ICtorum Ordinis in Illustri hac LL. nutrice Salana, Praesidente ... Dn. Georg-Adamo Struvio ... Publico Eruditorum examini sistit Ernestus Sigismundus Schildbach. Autor. Ad d. Maii ...Nicht identisch mit VD17 14:020890R (dort ohne Widmungsempfänger auf der Rücks. des Titelbl.
The scenario approach for stochastic model predictive control with bounds on closed-loop constraint violations,” Automatica (provisionally accepted). Available online at: http://arxiv. org/pdf/1307.5640v1.pdf
Abstract Many practical applications of control require that constraints on the inputs and states of the system be respected, while optimizing some performance criterion. In the presence of model uncertainties or disturbances, for many control applications it suffices to keep the state constraints at least for a prescribed share of the time, as e.g. in building climate control or load mitigation for wind turbines. For such systems, a new control method of Scenario-Based Model Predictive Control (SCMPC) is presented in this paper. It optimizes the control inputs over a finite horizon, subject to robust constraint satisfaction under a finite number of random scenarios of the uncertainty and/or disturbances. While previous approaches have shown to be conservative (i.e. to stay far below the specified rate of constraint violations), the new method is the first to account for the special structure of the MPC problem in order to significantly reduce the number of scenarios. In combination with a new framework for interpreting the probabilistic constraints as average-in-time, rather than pointwise-in-time, the conservatism is eliminated. The presented method retains the essential advantages of SCMPC, namely the reduced computational complexity and the handling of arbitrary probability distributions. It also allows for adopting sample-and-remove strategies, in order to trade performance against computational complexity