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

Calafiore, Giuseppe Carlo; Dabbene, Fabrizio

doi:10.1016/s0005-1098(02)00015-8

Cited by 38 publications

(28 citation statements)

References 32 publications

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“…In other cases, it more simply represents the different importance we place on different instances. Extracting δ samples according to a given probability measure P is not always a simple task to accomplish, see [8] for a discussion of this topic and polynomial-time algorithms for the sample generation in some matrix norm-bounded sets.…”

Section: 11mentioning

confidence: 99%

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Uncertain convex programs: randomized solutions and confidence levels

2004

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Abstract. Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or 'instance' parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chance-constrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chance-constrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a pre-specified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations.In this paper, we consider an alternative 'randomized' or 'scenario' approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.

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Section: 11mentioning

confidence: 99%

“…On the other hand, to pursue the randomized approach, we assume that each vector δ i is uniformly distributed over the ball δ i ≤ 1, and, for fixed , β, we determine N according to (8) and draw N samples δ (i) , . .…”

Section: Robust Linear Programsmentioning

confidence: 99%

Uncertain convex programs: randomized solutions and confidence levels

2004

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“…In the robustness analysis of control systems, the definition of uncertainty is very significant (Calafiore and Dabbene, 2002). To design an effective control system, a complex dynamic plant should be described as a relative simple model.…”

Section: Basic Ideas On Robust Controller Designmentioning

confidence: 99%

U-model based LMI robust controller design

Liu

Yuan

Zhu

et al. 2014

Proceedings of the 33rd Chinese Control Conference

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Abstract:In this study, an LMI (Linear Matrix Inequality) based H∞ robust controller design approach is proposed to improve the performance of the designed U-pole placement control systems. Unlike the classical design procedures, the control-oriented U-model based nonlinear control systems cancel the nonlinearity of the nonlinear models. Therefore, the closed loop transfer function of U-pole placement control system can be regarded as a linear block. The solvability and sub-optimality of discrete-time H ∞ robust control are converted to find feasible solutions for LMIs. Once the internal parameters changed, the LMI based H∞ controllers have a higher level of robustness compared with traditional controllers. A nonlinear dynamic model is selected to test the performance of the LMI robust controller to demonstrate the proposed approach effective.

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“…In particular, while the cases p = 1 and ∞ can be immediately reduced to multiple random vector generation for which the techniques described in [13] can be used, the solution for the induced 2 norm ball requires the development of specific methods. In particular, the algorithms presented in [9,14], see also [52], are based on the singular value decomposition of the complex (real) matrix in the matrix product…”

Section: Sample Generation Problemmentioning

confidence: 99%

“…Hence, one of the advantages of the probabilistic approach is to provide a rethinking of the relation between the stochastic and the robust paradigms, utilizing classical worst-case bounds of robust control together with probabilistic information, which is often neglected in the deterministic context. The interplay of probability and robustness also leads to innovative concepts such as the probabilistic robustness margin and the probability degradation function, see, for instance, [9,14].…”

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confidence: 99%

A survey of randomized algorithms for control synthesis and performance verification

Calafiore¹,

Dabbene

Tempo

2007

Journal of Complexity

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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. Preliminaries and motivationsInformation-based complexity (IBC) is a general theory, developed within computer science, that studies the complexity of approximately solving problems in the presence of partial and/or contaminated information, see [54,55] and references therein. Typical applications of IBC are ଁ Preliminary versions of this paper have been published in the Proceedings of the IEEE Conference of Decision and Control, Maui,

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

Cited by 38 publications

References 32 publications

Uncertain convex programs: randomized solutions and confidence levels

Uncertain convex programs: randomized solutions and confidence levels

U-model based LMI robust controller design

A survey of randomized algorithms for control synthesis and performance verification

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