Control and Estimation in Linear Time-Varying Systems Based on Ellipsoidal Reachability Sets
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PICAL ISSUE
Control and Estimation in Linear Time-Varying Systems Based on Ellipsoidal Reachability Sets D. V. Balandin∗,a and M. M. Kogan∗∗,b ∗
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Lobachevsky Nizhny Novgorod State University, Nizhny Novgorod, Russia Nizhny Novgorod State University of Architecture and Civil Engineering, Nizhny Novgorod, Russia e-mail: a [email protected], b [email protected] Received July 23, 2019 Revised October 3, 2019 Accepted January 30, 2020
Abstract—Linear continuous or discrete time-varying systems in which the sum of a quadratic form of the initial state and the integral or sum of quadratic forms of a disturbance on a finite horizon is bounded above by a given value are considered. It is demonstrated that the reachability set of such a continuous- or discrete-time system is an evolving ellipsoid, and its ellipsoid matrix satisfies a linear matrix differential or difference equation, respectively. The optimal ellipsoidal observer and identification algorithm that yield the best ellipsoidal estimates of the system’s state and unknown parameters are designed. In addition, the optimal controllers ensuring that the system’s state will fall into a target set or that the system’s trajectory will stay within the ellipsoidal tube are designed. A connection between the optimal ellipsoidal observer and the Kalman filter is established. Some illustrative examples for the Mathieu equation, which describes the parametric oscillations of a linear oscillator, are given. Keywords: linear time-varying system, ellipsoidal reachability set, optimal control, optimal estimation DOI: 10.1134/S0005117920080019
1. INTRODUCTION The authors dedicate this paper to the memory of Ju.I. Neimark, as they are his (direct or indirect) scholars. The authors would like to repeat the words written by Juri Isaakovich in the epigraph of his last monograph Mathematical Modeling as Science and Art [1, 2]: “In science and its applications, like in life, the most important thing is understanding. It is always simple, but difficult to get.” In the monograph [3] Neimark presented the results of his wide-range research on stability, control, and optimization in dynamic systems. One of the significant topics of those studies was control problems under uncertainties in the mathematical model of a controlled object and disturbances affecting it. This topic was further developed in the subsequent works by Neimark; for example, see [4–7]. It will be considered below as well. In the problems of estimation and control in dynamic systems without complete information about the initial conditions, disturbances, and measurement noises, a key role is played by the reachability set of the system, interpreted as the set of all states in which the system can be at a given time instant under all possible values of the uncertain factors. The characterization of reachability sets and their dependence on system parameters allows designing optimal estimation and control systems under which the reachability sets of the resulting system are included, at a given time instant or on some time inte
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