Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance ’ parameter. A recently emerged successful paradigm for attacking these problems is robust optimization, where one seeks a solution which simultaneously satisfies all possible constraint instances. In practice, however, the robust approach is effective only for problem families with rather simple dependence on the instance parameter (such as affine or polynomial), and leads in general to conservative answers, since the solution is usually computed by transforming the original semi-infinite problem into a standard one, by means of relaxation techniques. In this paper, we take an alternative ‘randomized ’ or ‘scenario ’ approach: by randomly sampling the uncertainty parameter, we substitute the original infinite constraint set with a finite set of N constraints. 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 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.

In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the S-procedure. This formulation leads to convex optimization problems that can be essentially solved in O(n 3)—n being the size of unknown vector — by means of suitable interior point barrier methods, as well as to closed form results in some particular cases. We further show that the uncertain linear equations paradigm can be directly applied to various state-bounding problems for dynamical systems subject to set-valued noise and model uncertainty.

Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-butbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control. Keywords optimization under uncertainty · robust optimization · convex programming · chance constraints · robust linear control

Abstract: The problem of robustness in fault detection using observers has been treated basically using the active approach, based on decoupling the effects of the uncertainty from the effects of the faults on the residual. On the other hand, the passive approach is based on propagating the effect of the uncertainty to the residuals and then using adaptive thresholds. In this paper, the passive approach based on adaptive thresholds produced using a model with uncertain parameters bounded in intervals, also known as an "interval model", will be presented in the context of linear observer methodology, deriving their corresponding interval version. Finally, an example based on an industrial actuator used as an FDI benchmark in the European project DAMADICS will be used for testing the proposed approach. Copyright © 2003 IFA C

In this paper, we consider the problem of parameter estimation in a linear stochastic model, where the observations are affected by noise with uncertain variance. In particular, we discuss a linear estimator which minimizes a worst-case measure of the a-posteriori covariance of the parameters. The estimate is efficiently computed by means of convex programming, and may be updated with upcoming observations in a recursive setting. 1

In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations (ULE). The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the S- procedure. The resulting bounding condition is expressed as a Linear Matrix Inequality (LMI) constraint on the ellipsoid parameters and the additional scaling variables. This formulation leads to a convex optimization problem that can be e#ciently solved by means of interior point barrier methods. 1

In this paper, the robust H 2 and H# filter design problems are considered, where the uncertainties, unstructured or structured, are norm bounded and represented by linear fractional transformation (LFT). The main result is that after upper-bounding the objectives, the problems of minimizing the upper bounds are converted to finite dimensional convex optimization problems involving linear matrix inequalities (LMIs). These are extensions of the results for systems with polytopic uncertainty. It is also shown that for the unstructured, norm bounded uncertainty case, the results here are less conservative than former results, where Riccati equation approach are used. A numerical example is given to illustrate the results.

have close zero frequency response. The model is truncated to 20 states by means of the described quasi-convex optimization technique (QCO method), Hankel model reduction. We implement QCO method on the frequency grid with 84 samples with tolerance in bisection procedure 10 06. The optimization together with calculating frequency samples took 74 seconds and the resulting approximation error is 2:9 1 10 05. Hankel model reduction took around 20 minutes providing the error 7:98 1 10 05. Results, see in the Fig. 1. For the given frequency interval QCO provided a better model than Hankel reduction. However, in general we do not expect QCO approximations to be better than Hankel reduction approximations. This example shows, that for large/medium scale systems we win sufficiently in time and do not really lose in approximation quality. VII. CONCLUSION In this technical note we have discussed multi-input-multi-output extension

Abstract—This paper is concerned with the set-membership filtering (SMF) problem for discrete-time nonlinear systems. We employ the Takagi–Sugeno (T-S) fuzzy model to approximate the nonlinear systems over the true value of state and to overcome the difficulty with the linearization over a state estimate set rather than a state estimate point in the set-membership framework. Based on the T-S fuzzy model, we develop a new nonlinear SMF estimation method by using the fuzzy modeling approach and the S-procedure technique to determine a state estimation ellipsoid that is a set of states compatible with the measurements, the unknown-but-bounded process and measurement noises, and the modeling approximation errors. A recursive algorithm is derived for computing the ellipsoid that guarantees to contain the true state. A smallest possible estimate set is recursively computed by solving the semidefinite programming problem. An illustrative example shows the effectiveness of the proposed method for a class of discrete-time nonlinear systems via fuzzy switch. Index Terms—Convex optimization, linear set-membership filtering (SMF), nonlinear SMF, unknown-but-bounded noise, Takagi–Sugeno (T-S) fuzzy model. I.

In this paper, the problem of set-membership filtering is considered for discrete-time systems with equality and inequality constraints between their state variables. We formulate the problem of set-membership filtering as finding the set of estimates that belong to an ellipsoid. A centre and a shape matrix of the ellipsoid are used to describe the set of estimates and the solution to the set of estimates is obtained in terms of matrix inequality. Unknown but bounded process and measurement noises are handled under the inequality constraints by using S-procedure. We apply Finsler’s Lemma to project the set of estimates onto the constrained surface. A recursive algorithm is developed for computing the ellipsoid that guarantees to contain the true state under the state constraints, which is easily implemented by semi-definite programming via interior-point approach. A vehicle tracking example is provided to demonstrate the effectiveness of the proposed set-membership filtering with state equality constraints. I.