In this note we construct minimax observers for linear stationary DAEs with bounded uncertain inputs, given noisy measurements. We prove a new duality principle and show that a finite (infinite) horizon minimax observer exists if and only if the DAE is @math -impulse observable ( @math -detectable) . Remarkably, the regularity of the DAE is not required.
To the best of our knowledge, the results of this paper are new. Its preliminary version appeared in @cite_15 . With respect to @cite_15 the main differences are: (i) new (necessary and sufficient) conditions for existence of minimax observers (Theorem ), (ii) detailed proofs. Duality principle for non-stationary DAEs was introduced in @cite_6 , provided @math . It was then used to derive a sub-optimal observer. In contrast, our duality theorems hold for DAEs with uncertain @math and the constructed observers are optimal in that the worst-case estimation errors associated with the observers are minimal (see Definitions -). The algorithm of @cite_20 constructing a finite horizon minimax observer by using projectors @math is a special case of the one of this paper. Minimax observers for discrete time DAEs were considered in @cite_0 .
@cite_0 @cite_15 @cite_20 @cite_6
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In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAE. In this paper, we propose and evaluate algorithms for solving the worst-case-coverage deployment problem in ad-hoc wireless sensor net works. The worst-case-coverage deployment problem is to deploy additional sensors in the wireless sensor field to optimize the worst-case coverage. We derive a duality theorem that reveals the close relation between the maximum breach path and the minimum Delaunay cut. The duality theorem is similar to the well-known max-flow-mm-cut theorem in the field of network optimization. The major difference lies in the fact that the object function we study in this paper is nonlinear rather than linear. Based on the duality theorem, we propose an efficient dual algorithm to solve the worst- case-coverage deployment problem. In addition, we propose a genetic algorithm for deploying a number of additional sensors simultaneously. We use analytical proofs and simulation results to justify the usage of the proposed approaches We present a practical, stratified autocalibration algorithm with theoretical guarantees of global optimality. Given a projective reconstruction, the first stage of the algorithm upgrades it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, the algorithm upgrades this affine reconstruction to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step. For each stage, we construct and minimize tight convex relaxations of the highly non-convex objective functions in a branch and bound optimization framework. We exploit the problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data. MATLAB code for the implementation is made available to the community. In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence ( u _ ) for the dual problem applying Tikhonov method. Finally we represent ( u _ ) in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.