We introduce a new methodology for the design of cautious adaptive controllers based on the following two-step procedure: (i) a probability measure describing the likelihood of different models is updated on-line base...
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We introduce a new methodology for the design of cautious adaptive controllers based on the following two-step procedure: (i) a probability measure describing the likelihood of different models is updated on-line based on observations, and (ii) a controller with certain robust control specifications is tuned to the updated probability by means of randomized algorithms. The robust control specifications are assigned as average specifications with respect to the estimated probability measure, and randomized algorithms are used to make the controller tuning computationally tractable. This paper provides a general overview of the proposed new methodology. Still, many issues remain open and represent interesting topics for future research. (C) 2003 Elsevier Science B.V. All rights reserved.
In this paper, we present a novel development of randomized algorithms for quadratic stability analysis of sampled-data systems with memoryless quantizers. The specific randomized algorithm employed generates a quadra...
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In this paper, we present a novel development of randomized algorithms for quadratic stability analysis of sampled-data systems with memoryless quantizers. The specific randomized algorithm employed generates a quadratic Lyapunov function and leads to realistic bounds on the performance of such systems. A key feature of this method is that the characteristics of quantizers are exploited to make the algorithm computationally efficient. (C) 2003 Elsevier Ltd. All rights reserved.
We present some randomized algorithms for computing multilinear rank-(mu(1),mu(2),mu(3)) approximations of tensors by combining the sparse subspace embedding and the singular value decomposition. The error bound for t...
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We present some randomized algorithms for computing multilinear rank-(mu(1),mu(2),mu(3)) approximations of tensors by combining the sparse subspace embedding and the singular value decomposition. The error bound for this algorithm with the high probability is obtained by the properties of sparse subspace embedding. Furthermore, combining the power scheme and the proposed randomized algorithm, we derive a three-stage randomized algorithm and make a probabilistic analysis for its error bound. The efficiency of the proposed algorithms is illustrated via numerical examples.
This paper demonstrates the development of purely data-driven, nonintrusive parametric reduced-order models for the emulation of high-dimensional field outputs using randomized linear algebra techniques. Typically, lo...
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This paper demonstrates the development of purely data-driven, nonintrusive parametric reduced-order models for the emulation of high-dimensional field outputs using randomized linear algebra techniques. Typically, low-dimensional representations are built using the proper orthogonal decomposition combined with interpolation/regression in the latent space via supervised learning. However, even moderately large simulations can lead to data sets on which the cost of computing the proper orthogonal decomposition becomes intractable due to storage and computational complexity of the numerical procedure. In an attempt to reduce the offline cost, the proposed method demonstrates the application of randomized singular value decomposition and sketching-based randomized singular value decomposition to compute the proper orthogonal decomposition basis. The predictive capability of reduced-order models resulting from regular singular value decomposition and randomized/sketching-based algorithms are compared with each other to ensure that the decrease in computational cost does not result in a loss in accuracy. Demonstrations on canonical and practical fluid flow problems show that the reduced-order models constructed using randomized methods are competitive in their predictive accuracy with reduced-order models that employ the conventional deterministic method. Through this new method, it is expected that truly large-scale parametric reduced-order models can be constructed under a significantly limited computational resource budget.
By now it is known that several problems in the robustness analysis and synthesis of control systems are NP-complete or NP-hard. These negative results force us to modify our notion of "solving" a given prob...
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By now it is known that several problems in the robustness analysis and synthesis of control systems are NP-complete or NP-hard. These negative results force us to modify our notion of "solving" a given problem. An approach that is recently gaining popularity is that of using randomized algorithms, which can be used to solve a problem approximately, most of the time. We begin with the premise that many problems in robustness analysis and synthesis can be formulated as the minimization of an objective function with respect to the controller parameters. It is argued that, in order to assess the performance of a controller as the plant varies over a prespecified family, it is better to use the average performance of the controller as the objective function to be minimized, rather than its worst-case performance, as the worst-case objective function usually leads to rather conservative designs. Then it is shown that a property from statistical learning theory known as uniform convergence of empirical means (UCEM) plays an important role in allowing us to construct efficient randomized algorithms for a wide variety of controller synthesis problems. In particular, whenever the UCEM property holds, there exists an efficient (i.e., polynomial-time) randomized algorithm. Using very recent results in statistical learning theory, it is shown that the UCEM property holds in any problem in which the satisfaction of a performance constraint can be expressed in terms of a finite number of polynomial inequalities, In particular, several problems such as robust stabilization and weighted H-2/H-infinity-norm minimization are amenable to the randomized approach. (C) 2001 Elsevier Science Ltd. All rights reserved.
A randomized algorithm for finding a hyperplane separating two finite point sets in the Euclidean space Rd and a randomized algorithm for solving linearly constrained general convex quadratic problems are proposed. Th...
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A randomized algorithm for finding a hyperplane separating two finite point sets in the Euclidean space Rd and a randomized algorithm for solving linearly constrained general convex quadratic problems are proposed. The expected running time of the separating algorithm is O(dd! (m + n)), where m and n are cardinalities of sets to be separated. The expected running time of the algorithm for solving quadratic problems is O(dd! s) where s is the number of inequality constraints. These algorithms are based on the ideas of Seidel's linear programming algorithm [6]. They are closely related to algorithms of [8], [2], and [9] and belong to an abstract class of algorithms investigated in [1]. The algorithm for solving quadratic problems has some features of the one proposed in [7].
In recent Sears, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems, Unlike classical worst case methods, these algorithms provide probabili...
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In recent Sears, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems, Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information, The applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres, Sampling in more general uncertainty sets is generally performed through overbounding, at the expense of an exponential rejection rate. In this paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general M-Delta configuration are studied, In particular, efficient polynomial-time algorithms for uncertainty structures Delta consisting of an arbitrary number of full complex blocks and uncertain parameters are developed.
In this paper, we consider the design of globally asymptotically stabilizing state-dependent switching rules for multimodal systems, first restricting attention to linear time-invariant (LTI) systems with only two sta...
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In this paper, we consider the design of globally asymptotically stabilizing state-dependent switching rules for multimodal systems, first restricting attention to linear time-invariant (LTI) systems with only two states for the switch, and then generalizing the results to multimodal LTI systems and to nonlinear systems. In all cases, the systems considered do not allow the construction of a single quadratic Lyapunov function and, hence, fall in the class of problems that require multiple Lyapunov functions and thus are nonconvex. To address the challenge of nonconvexity, we introduce probabilistic algorithms, and prove their probability-one convergence under a new notion of convergence. Then, to reduce complexity, we develop modified versions of the algorithm. We also present a class of more general nonconvex problems to which this approach can be applied. The results are illustrated using two- and three-dimensional systems with multiple switch states.
We present a randomized algorithm which generalizes ideas of K. L. CLARKSON, R. SEIDEL and E. WELZL for LP to problems with a convex objective function and affine constraints. An analysis of the expected running time ...
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We present a randomized algorithm which generalizes ideas of K. L. CLARKSON, R. SEIDEL and E. WELZL for LP to problems with a convex objective function and affine constraints. An analysis of the expected running time of the main algorithm shows a linear dependency on the number of constraints, but an exponential one on the dimension which can be somewhat improved by applying several modifications. We finally report some numerical results.
We present three randomized pseudo-polynomial algorithms for the problem of finding a base of specified value in a weighted represented matroid subject to parity conditions. These algorithms, the first two being an im...
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We present three randomized pseudo-polynomial algorithms for the problem of finding a base of specified value in a weighted represented matroid subject to parity conditions. These algorithms, the first two being an improved version of those presented by P. M. Camerini et al. (1992, J. algorithms 13, 258-273) use fast arithmetic working over a finite field chosen at random among a set of appropriate fields. We show that the choice of a best algorithm among those presented depends on a conjecture related to the best value of the so-called Linnik constant concerning the distribution of prime numbers in arithmetic progressions. This conjecture, which we call the C-conjecture, is a strengthened version of a conjecture formulated in 1934 by S. Chowla. If the C-conjecture is true, the choice of a best algorithm is simple, since the last algorithm exhibits the best performance;either when the performance is measured in arithmetic operations, or when it is measured in bit operations and mild assumptions hold. If the C-conjecture is false we are still able to identify a best algorithm, but in this case the choice is between the first two algorithms and depends on the asymptotic growth of m with respect to those of U and n, where 2n, 2m, U are the rank, the number of elements;and the maximum weight assigned to the elements of the matroid, respectively. (C) 1999 Academic Press.
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