Linear Matrix Inequalities in Control Problems
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L THEORY
Linear Matrix Inequalities in Control Problems M. V. Khlebnikov1∗ 1
Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997 Russia e-mail: ∗ [email protected] Received March 17, 2020; revised June 25, 2020; accepted June 26, 2020
Abstract—Many contemporary automatic control problems are characterized by large dimensions, the presence of uncertainty in the description of the system, the presence of uncontrolled exogenous disturbances, the need to analyze large amounts of information online, decentralization/simplification of control systems in multi-agent systems, and a number of other factors that complicate the application of classical methods of control theory. Therefore, the problem of developing new efficient methods that take into account these specific features becomes topical. In this regard, the technique of linear matrix inequalities is very promising. This paper presents the results of new studies that develop the technique of linear matrix inequalities and use it to solve applied control theory problems. DOI: 10.1134/S0012266120110105
INTRODUCTION The first linear matrix inequalities (in implicit form) appeared at the end of the 19th century in Lyapunov’s classical work [1] on stability theory. The very same term of linear matrix inequalities was introduced by V.A. Yakubovich in his papers dating back to the 1960s. A significant development of the theory of linear matrix inequalities is associated with the ellipsoidal technique for describing uncertainties, which was developed by Kurzhanskii [2], Schweppe [3], and Chernous’ko [4]. The technique of linear matrix inequalities acquired its modern form in the fundamental monograph [5], supported by the presence of powerful computational tools for solving problems for linear matrix inequalities with the use of numerical interior point methods [6, 7]. The first monographs in Russian dedicated to linear matrix inequalities are the monographs [8, 9]. In control problems, in particular, for linear systems, there are always uncertainties. We note two related key problems: if an uncertainty is contained in the description of the system, then the problem of robustness arises; if an uncertainty is contained in the inputs of the system, then we are dealing with exogenous disturbances. The problem of suppressing exogenous disturbances is one of the main problems in control theory and is studied in its various subdivisions. For example, problems with random Gaussian noise are considered in linear quadratic optimization; the problem of H∞ -optimization is associated either with harmonic exogenous disturbances or with random Gaussian ones or with disturbances in the class L2 . However, in many cases of practical importance, exogenous disturbances are simply bounded; there is no further information about them. Scientists began to get interested in the problem of suppressing nonrandom bounded exogenous disturbances back in the middle of the 20th century. In the mid-1940s, Bulgakov [10] posed the problem of accumulation of disturbances
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