On pole assignment of high-order discrete-time linear systems with multiple state and input delays

Lixuan Zhang; Xuefei Yang

doi:10.3934/dcdss.2022022

Article Contents

2022, Volume 15, Issue 11: 3351-3368. Doi: 10.3934/dcdss.2022022

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On pole assignment of high-order discrete-time linear systems with multiple state and input delays

Lixuan Zhang and
Xuefei Yang^,

Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, 150001, China

^*Corresponding author: Xuefei Yang
^*Corresponding author: Xuefei Yang

Received: October 2021

Revised: December 2021

Early access: February 2022

Published: September 2022

This work was supported in part by the National Science Foundation of China (61903102, 61773387), and the Fundamental Research Funds for the Central Universities

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Abstract

This paper studies the problem of pole assignment for high-order discrete-time linear systems with multiple state and input delays. When the number of state delays is larger than or equal to that of input delays, an effective predictor feedback controller is proposed based on the augmented technique, and the design process for the feedback gain is also presented. In addition, it is proved that the pole assignment problem is solvable if and only if the solutions to a linear matrix equation are such that a matrix is nonsingular. When the number of state delays is smaller than that of input delays, the original system is first transformed into a delay-free system with keeping the system controllability invariant, and then, the corresponding controller with designable feedback gain is established. To characterize all of the feedback gains, a factorization approach is introduced which can provide full degree of freedom. Numerical examples are employed to illustrate the effectiveness of the proposed approaches.

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Mathematics Subject Classification: Primary: 93B52, 15A18, 93B05; Secondary: 93B11, 15A23.

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Figure 1. Polar plot of the closed-loop eigenvalues

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Figure 2. State response of time-delay system (32) by predictor feedback

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Figure 3. Polar plot of the closed-loop eigenvalues

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Figure 4. State response of time-delay system (35) by state feedback

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