Universal controllers of V.A. Yakubovich: a systematic approach to LQR problems with uncertain external signals∗∗The paper was partially supported by RFBR, grants 13-08-01014 and 14-08-01015, and St. Petersburg State University, grant 6.38.230.2015. Theorems 6 and 13 are obtained at Institute for Problems of Mechanical Engineering RAS and supported solely by Russian Scientific Foundation (RSF), grant 14-29-00142

Any practical control system is affected by external signals. Some of them are “informative” (e.g. reference signals), and the others (disturbances, measurement noises etc.) are undesired as they can visibly deteriorate the system behavior. General approaches to disturbance rejection and reference tracking have been elaborated in the framework of geometric control theory, among them are disturbance decoupling control and internal model principles. These approaches, however, assume the state and control of the system to be unconstrained. To take such constraints into account, one usually has to consider optimization problems where the performance index penalizes in some way the system process. Optimization problems in presence of uncertain signals have been addressed in the context of robust control and stochastic control. Standard methods like usually provide only suboptimality of the process; the optimal value can be found for either “worst-case” signal (like in minimax H 8 -and L 1 -optimization approaches) or “on average”, assuming the external signals to be stochastic with known spectral density. In a series of his papers published in 1992-2012, Vladimir A. Yakubovich promoted the approach of universal controllers for linear-quadratic regulation (LQR) problems under uncertain signals. The term “universal” emphasizes that the controller renders the solution of the closed-loop system optimal for any external signal. Although this is not possible for arbitrary signals, in some special classes of signals the universal controllers not only exist but may even be chosen linear. The existence of linear optimal controllers has been proved for two important classes of uncertain signals, that is, polyharmonic signals with known spectrum and signals with fast decreasing spectral density. In this paper we extend the results of V.A. Yakubovich to more general classes of systems and quadratic performance indices, arising in problems of optimal oscillation damping, reference tracking and model matching.