The Irregular Linear Quadratic Control Problem for Deterministic Case with Time Delay
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The Irregular Linear Quadratic Control Problem for Deterministic Case with Time Delay∗ LI Tongxing · LI Lin · LEI Jing · JIN Nana · JU Peijun
DOI: 10.1007/s11424-020-9136-6 Received: 24 April 2019 / Revised: 24 October 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract This study deals with the irregular linear quadratic control problem governed by continuous time system with time delay. Linear quadratic (LQ) control for irregular Riccati equation with time delay remains challenging since the controller could not be solved from the equilibrium condition directly. The merit of this paper is that based on a new approach of ‘two-layer optimization’, the controller entries of irregular case with time delay are deduced from two equilibrium conditions in two different layers, which is fundamentally different from the classical regular LQ control. The authors prove that the irregular LQ with time delay is essentially different from the regular case. Specifically, the predictive controller bases on the feedback gain matrix and the state is given in the last part. The presented conclusions are completely new to our best knowledge. Examples is presented to show the effectiveness of the proposed approach. Keywords Forward and backward differential equations (FBDEs), irregular Riccati equation, time delay system. LI Tongxing (Corresponding author) School of Mathematics and Statistics, Taishan University, Taian 271000, China. Email: tongxing [email protected]. LI Lin School of Information Science and Engineering, Shandong Agricultural University, Taian 271000, China. Email: lin [email protected]. LEI Jing School of Mathematics and Statistics, Taishan University, Taian 271000, China. Email: [email protected]. JIN Nana School of Control Science and Engineering, Shandong University, Jinan 250061, China. Email: [email protected]. JU Peijun School of Mathematics and Statistics, Taishan University, Taian 271000, China. Email: [email protected]. ∗ This research was supported by the Natural Science Foundation of Shandong Province under Grant Nos. ZR2018MF019 and ZR2019MF052, the National Natural Science Foundation of China under Grant No. 61873179. This paper was recommended for publication by Editor SUN Jian.
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Introduction
It is known that time delays usually degrade the system performance and are key points of instability. In particular, time delays give rise to chaos phenomenon. Control problem for time delay system has received considerable attention since the 1950’s owing to their widely used background in engineering, finance, communication and physic[1–9] . The good news is that some fundamental progress has been made on stabilization and the optimal control problems for time delay systems, especially for deterministic systems[10] . In [10], Artstein proposed ‘Reduction Technique’. Under some reasonable conditions, linear systems with delayed control action are transformed into systems without delays. It is generally known that LQ control is one of the most basic and fashionable too
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