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New forms of Riccati equations and the further results of the optimal control for linear discrete-time systems

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  • Control Theory
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Abstract

For linear discrete-time systems, the traditional finite horizon optimal controller is proved to render the closed-loop systems asymptotically stable under some assumptions in literature. In this paper, a new form of finite horizon discrete-time Riccati equation is proposed. It is proved that the new form of finite horizon discrete-time Riccati equation is equivalent to the other three old ones. Based on this new form of finite horizon discrete-time Riccati equation, the finite horizon optimal controller of linear discrete time systems is proved to render the closed-loop system exponentially stable without any assumptions. At the same time, a new form of infinite horizon discrete-time Riccati equation is proposed when the discrete system is controllable or stabilizable. It is proved that the new form of infinite horizon discrete-time Riccati equation is equivalent to the other three old ones too. Based on this new form of infinite horizon discrete-time Riccati equation, the infinite horizon optimal controller of linear discrete-time systems is proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally an unstable batch reactor and an unstable inverted pendulum are used to verify the theory results of this paper.

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Suzhou University of Science and Technology, No.8 of Changjiang Road, Suzhou, China

    Hongli Liu & Qixin Zhu

  2. School of Electrical and Electronic Engineering, East China Jiaotong University, No. 808 of Shuanggang Road, Nanchang, China

    Qixin Zhu

Authors
  1. Hongli Liu
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  2. Qixin Zhu
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Corresponding author

Correspondence to Qixin Zhu.

Additional information

Recommended by Associate Editor Juhoon Back under the direction of Editor Poo-Gyeon Park.

This work was supported by National Nature Science Foundation of China (51375323, 61164014).

Hongli Liu received her B.S. degree in Mechanical Manufacturing process and equipment from Xi’an Polytechnic University, Xi’an, China, in 1996. She received her M.S. degree in Mechanical Engineering from East China Jiaotong University, China, in 2010. She was with Nantong University, Nantong, China, from July 1996 to July 2006. She was with East China Jiaotong University, Nanchang, China, from August 2006 to February 2013. She is currently a lecturer at School of Mechanical Engineering, Suzhou University of Science and Technology, Suzhou, China. Her research interests include optimal control, servo control, networked control system and the applications of control theory in mechanical engineering.

Qixin Zhu received his B.S. and M.S. degrees in Industrial Automation from Xi’an Polytechnic University, Xi’an, China, in 1994 and 1997, respectively. He received his Ph.D. degree in Control Theory and Control Engineering from Nanjing University of Aeronautics and Astronautics in 2003. He was with Nantong University, Nantong, China, from April 1997 to July 2004. He was with ASM Assembly Automation Ltd., Hong Kong, China, from July 2004 to August 2006. He was with East China Jiaotong University, Nanchang, China, from August 2006 to February 2013. He is currently a professor at School of Mechanical Engineering, Suzhou University of Science and Technology, Suzhou, China. His research interests include servo control, networked control systems, robot and the applications of control theory in engineering.

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Liu, H., Zhu, Q. New forms of Riccati equations and the further results of the optimal control for linear discrete-time systems. Int. J. Control Autom. Syst. 12, 1160–1166 (2014). https://doi.org/10.1007/s12555-013-0202-x

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  • DOI: https://doi.org/10.1007/s12555-013-0202-x

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