Controller design for delay systems via eigenvalue assignment – on a new result in the distribution of quasi-polynomial roots

Abstract

This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment. First, a new result on the root distribution for a class of quasi-polynomials is developed based on the extension of the Hermite–Biehler theorem. Then, such result is applied to proportional–integral (PI) controller parameter design for a first-order plant with time delay through pole placement. The complete region of PI gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed-loop system with delay plant are assigned to desired positions in the complex plane. Furthermore, on the basis of the previous result, this paper also extended the PI control to the proportional–integral–derivative (PID) control. It is worth pointing out that this work aims to improve the performance of the closed-loop system on the premise of guaranteeing the stability.

Keywords:

• Hermite–Biehler theorem
• eigenvalue assignment
• time delay
• proportional–integral (PI) control
• proportional–integral–derivative (PID) control

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (NSFC) (No. 61374137, No. 61403311), the IAPI Fundamental Research Funds (2013ZCX02-03), and the Fundamental Research Funds for the Central Universities (No. 3102014JCQ01068, No. N130810001).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China (NSFC) [grant number 61374137], [grant number 61403311], the IAPI Fundamental Research Funds [grant number 2013ZCX02-03], and the Fundamental Research Funds for the Central Universities [grant number 3102014JCQ01068], [grant number N130810001].