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Mathematical Problems in Engineering
Volume 2013, Article ID 582879, 22 pages
Research Article

Design of Optimal PID Controller with -Routh Stability for Different Processes

1Department of Information Service and Intelligent Control, Shenyang Institute of Automation Chinese Academy of Sciences, Shenyang 110016, China
2University of Chinese Academy of Sciences (Graduate School of Chinese Academy of Sciences), Beijing 100039, China

Received 30 November 2012; Revised 27 February 2013; Accepted 11 March 2013

Academic Editor: Shane Xie

Copyright © 2013 XianHong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents a design method of the optimal proportional-integral-derivative (PID) controller with -Routh stability for different processes through Lyapunov approach. The optimal PID controller could be acquired by minimizing an augmented integral squared error (AISE) performance index which contains control error and at least first-order error derivative, or even may contain nth-order error derivative. The optimal control problem could be transformed into a nonlinear constraint optimization (NLCO) problem via Lyapunov theorems. Therefore, optimal PID controller could be obtained by solving NLCO problem through interior method or other optimization methods. The proposed method can be applied for different processes, and optimal PID controllers under various control weight matrices and -Routh stability are presented for different processes. Control weight matrix and -Routh stability’s effects on system performances are studied, and different tuning methods’ system performances are also discussed. -Routh stability’s effects on disturbance rejection ability are investigated, and different tuning methods’ disturbances rejection ability is studied. To further illustrate the proposed method, experimental results of coupled water tank system (CWTS) under different set points are presented. Both simulation results and experiment results show the effectiveness and usefulness of the proposed method.