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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

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

*Corresponding author: Xuefei Yang

Received  October 2021 Revised  December 2021 Early access February 2022

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

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.

Citation: Lixuan Zhang, Xuefei Yang. On pole assignment of high-order discrete-time linear systems with multiple state and input delays. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022022
##### References:
 [1] Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Trans. Automat. Control, 27 (1982), 869-879.  doi: 10.1109/TAC.1982.1103023. [2] O. Bachelier, J. Bosche and D. Mehdi, On pole placement via eigenstructure assignment approach, IEEE Trans. Automat. Control, 51 (2006), 1554-1558.  doi: 10.1109/TAC.2006.880809. [3] J. Chiasson and J. J. Loiseau, Applications of Time Delay Systems, Vol. 352, Springer, 2007. doi: 10.1007/978-3-540-49556-7. [4] G. R. Duan, Solutions of the equation $AV+BW = VF$ and their application to eigenstructure assignment in linear systems, IEEE Trans. Automat. Control, 38 (1993), 276-280.  doi: 10.1109/9.250470. [5] G. R. Duan, Eigenstructure assignment by decentralized output feedback-A complete parametric approach, IEEE Trans. Automat. Control, 39 (1994), 1009-1014.  doi: 10.1109/9.284882. [6] G. R. Duan, Generalized Sylvester Equations: Unified Parametric Solutions, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18589. [7] S. Fang, J. Huang and J. Ma, Stabilization of a discrete-time system via nonlinear impulsive control, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1803-1811.  doi: 10.3934/dcdss.2020106. [8] M. Fu, Pole placement via static output feedback is NP-hard, IEEE Trans. Automat. Control, 49 (2004), 855-857.  doi: 10.1109/TAC.2004.828311. [9] K. Gu, J. Chen and V. L. Kharitonov, Stability of Time-Delay Systems, Control Engineering. Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0039-0. [10] T. Kailath, Linear Systems, Prentice-Hall Information and System Sciences Series. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. [11] J. Klamka, Relative and absolute controllability of discrete systems with delays in control, Internat. J. Control, 26 (1977), 65-74.  doi: 10.1080/00207177708922289. [12] X. Le and J. Wang, Neurodynamics-based robust pole assignment for high-order descriptor systems, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 2962-2971.  doi: 10.1109/TNNLS.2015.2461553. [13] X. Li, T. Caraballo, R. Rakkiyappan and X. Han, On the stability of impulsive functional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 3130-3140.  doi: 10.1002/mma.3303. [14] G. P. Liu and R. Patton, Eigenstructure Assignment for Control System Design, New York, NY, USA: Wiley, 1998. [15] Q. S. Liu and B. Zhou, Regulation of linear systems with both pointwise and distributed input delays by memoryless feedback, J. Franklin Inst., 356 (2019), 5172-5192.  doi: 10.1016/j.jfranklin.2018.12.032. [16] Y. M. Liu and I. K. Fong, On the controllability and observability of discrete-time linear time-delay systems, Internat. J. Systems Sci., 43 (2012), 610-621.  doi: 10.1080/00207721.2010.543490. [17] N. Mcdonald, Time Lags in Biological Models, Springer-Verlag, 1978. doi: 10.1007/978-3-642-93107-9. [18] E. S. M. Mostafa, A. W. Aboutahoun and F. F. Omar, On the solution of the eigenvalue assignment problem for discrete-time systems, J. Appl. Math., 2017 (2017), Art. ID 7256769, 12 pp. doi: 10.1155/2017/7256769. [19] Y. M. Ram, J. E. Mottershead and M. G. Tehrani, Partial pole placement with time delay in structures using the receptance and the system matrices, Linear Algebra Appl., 434 (2011), 1689-1696.  doi: 10.1016/j.laa.2010.07.014. [20] X. T. Wang and L. Zhang, Partial eigenvalue assignment of high order systems with time delay, Linear Algebra Appl., 438 (2013), 2174-2187.  doi: 10.1016/j.laa.2012.10.011. [21] T. Wei, X. Xie and X. Li, Input-to-state stability of delayed reaction-diffusion neural networks with multiple impulses, AIMS Math., 6 (2021), 5786-5800.  doi: 10.3934/math.2021342. [22] X. Yang, B. Zhou, F. Mazenc and J. Lam, Global stabilization of discrete-time linear systems subject to input saturation and time delay, IEEE Trans. Automat. Control, 66 (2021), 1345-1352.  doi: 10.1109/TAC.2020.2989791. [23] Z. X. Yang, G. B. Zhang, G. Tian and Z. Feng, Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 581-603.  doi: 10.3934/dcdss.2017029. [24] J. Zhai, L. Gao and S. Li, Robust eigenvalue placement optimization for high-order descriptor systems in a union region with disjoint discs based on harmony search algorithm, Neural Computing and Applications, 28 (2017), 1207-1220.  doi: 10.1007/s00521-016-2422-5. [25] L. Zhang, Multi-input partial eigenvalue assignment for high order control systems with time delay, Mechanical Systems and Signal Processing, 72 (2016), 376-382.  doi: 10.1016/j.ymssp.2015.09.033. [26] L. Zhang and X. T. Wang, Partial eigenvalue assignment for high order system by multi-input control, Mechanical Systems and Signal Processing, 42 (2014), 129-136.  doi: 10.1016/j.ymssp.2013.06.026. [27] Y. Zhao, X. Li and J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467, 10 pp. doi: 10.1016/j.amc.2020.125467. [28] B. Zhou and G. R. Duan, A new solution to the generalized Sylvester matrix equation $AV-EVF=BW$, Systems Control Lett., 55 (2006), 193-198.  doi: 10.1016/j.sysconle.2005.07.002. [29] B. Zhou and G. R. Duan, Pole assignment of high-order linear systems with high-order time-derivatives in the input, J. Franklin Inst., 357 (2020), 1437-1456.  doi: 10.1016/j.jfranklin.2019.10.030. [30] B. Zhou, J. Lam and G. R. Duan, Full delayed state feedback pole assignment of discrete-time time-delay systems, Optimal Control Appl. Methods, 31 (2010), 155-169.  doi: 10.1002/oca.899.

show all references

##### References:
 [1] Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Trans. Automat. Control, 27 (1982), 869-879.  doi: 10.1109/TAC.1982.1103023. [2] O. Bachelier, J. Bosche and D. Mehdi, On pole placement via eigenstructure assignment approach, IEEE Trans. Automat. Control, 51 (2006), 1554-1558.  doi: 10.1109/TAC.2006.880809. [3] J. Chiasson and J. J. Loiseau, Applications of Time Delay Systems, Vol. 352, Springer, 2007. doi: 10.1007/978-3-540-49556-7. [4] G. R. Duan, Solutions of the equation $AV+BW = VF$ and their application to eigenstructure assignment in linear systems, IEEE Trans. Automat. Control, 38 (1993), 276-280.  doi: 10.1109/9.250470. [5] G. R. Duan, Eigenstructure assignment by decentralized output feedback-A complete parametric approach, IEEE Trans. Automat. Control, 39 (1994), 1009-1014.  doi: 10.1109/9.284882. [6] G. R. Duan, Generalized Sylvester Equations: Unified Parametric Solutions, CRC Press, Boca Raton, FL, 2015.  doi: 10.1201/b18589. [7] S. Fang, J. Huang and J. Ma, Stabilization of a discrete-time system via nonlinear impulsive control, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 1803-1811.  doi: 10.3934/dcdss.2020106. [8] M. Fu, Pole placement via static output feedback is NP-hard, IEEE Trans. Automat. Control, 49 (2004), 855-857.  doi: 10.1109/TAC.2004.828311. [9] K. Gu, J. Chen and V. L. Kharitonov, Stability of Time-Delay Systems, Control Engineering. Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0039-0. [10] T. Kailath, Linear Systems, Prentice-Hall Information and System Sciences Series. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. [11] J. Klamka, Relative and absolute controllability of discrete systems with delays in control, Internat. J. Control, 26 (1977), 65-74.  doi: 10.1080/00207177708922289. [12] X. Le and J. Wang, Neurodynamics-based robust pole assignment for high-order descriptor systems, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 2962-2971.  doi: 10.1109/TNNLS.2015.2461553. [13] X. Li, T. Caraballo, R. Rakkiyappan and X. Han, On the stability of impulsive functional differential equations with infinite delays, Math. Methods Appl. Sci., 38 (2015), 3130-3140.  doi: 10.1002/mma.3303. [14] G. P. Liu and R. Patton, Eigenstructure Assignment for Control System Design, New York, NY, USA: Wiley, 1998. [15] Q. S. Liu and B. Zhou, Regulation of linear systems with both pointwise and distributed input delays by memoryless feedback, J. Franklin Inst., 356 (2019), 5172-5192.  doi: 10.1016/j.jfranklin.2018.12.032. [16] Y. M. Liu and I. K. Fong, On the controllability and observability of discrete-time linear time-delay systems, Internat. J. Systems Sci., 43 (2012), 610-621.  doi: 10.1080/00207721.2010.543490. [17] N. Mcdonald, Time Lags in Biological Models, Springer-Verlag, 1978. doi: 10.1007/978-3-642-93107-9. [18] E. S. M. Mostafa, A. W. Aboutahoun and F. F. Omar, On the solution of the eigenvalue assignment problem for discrete-time systems, J. Appl. Math., 2017 (2017), Art. ID 7256769, 12 pp. doi: 10.1155/2017/7256769. [19] Y. M. Ram, J. E. Mottershead and M. G. Tehrani, Partial pole placement with time delay in structures using the receptance and the system matrices, Linear Algebra Appl., 434 (2011), 1689-1696.  doi: 10.1016/j.laa.2010.07.014. [20] X. T. Wang and L. Zhang, Partial eigenvalue assignment of high order systems with time delay, Linear Algebra Appl., 438 (2013), 2174-2187.  doi: 10.1016/j.laa.2012.10.011. [21] T. Wei, X. Xie and X. Li, Input-to-state stability of delayed reaction-diffusion neural networks with multiple impulses, AIMS Math., 6 (2021), 5786-5800.  doi: 10.3934/math.2021342. [22] X. Yang, B. Zhou, F. Mazenc and J. Lam, Global stabilization of discrete-time linear systems subject to input saturation and time delay, IEEE Trans. Automat. Control, 66 (2021), 1345-1352.  doi: 10.1109/TAC.2020.2989791. [23] Z. X. Yang, G. B. Zhang, G. Tian and Z. Feng, Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 581-603.  doi: 10.3934/dcdss.2017029. [24] J. Zhai, L. Gao and S. Li, Robust eigenvalue placement optimization for high-order descriptor systems in a union region with disjoint discs based on harmony search algorithm, Neural Computing and Applications, 28 (2017), 1207-1220.  doi: 10.1007/s00521-016-2422-5. [25] L. Zhang, Multi-input partial eigenvalue assignment for high order control systems with time delay, Mechanical Systems and Signal Processing, 72 (2016), 376-382.  doi: 10.1016/j.ymssp.2015.09.033. [26] L. Zhang and X. T. Wang, Partial eigenvalue assignment for high order system by multi-input control, Mechanical Systems and Signal Processing, 42 (2014), 129-136.  doi: 10.1016/j.ymssp.2013.06.026. [27] Y. Zhao, X. Li and J. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467, 10 pp. doi: 10.1016/j.amc.2020.125467. [28] B. Zhou and G. R. Duan, A new solution to the generalized Sylvester matrix equation $AV-EVF=BW$, Systems Control Lett., 55 (2006), 193-198.  doi: 10.1016/j.sysconle.2005.07.002. [29] B. Zhou and G. R. Duan, Pole assignment of high-order linear systems with high-order time-derivatives in the input, J. Franklin Inst., 357 (2020), 1437-1456.  doi: 10.1016/j.jfranklin.2019.10.030. [30] B. Zhou, J. Lam and G. R. Duan, Full delayed state feedback pole assignment of discrete-time time-delay systems, Optimal Control Appl. Methods, 31 (2010), 155-169.  doi: 10.1002/oca.899.
Polar plot of the closed-loop eigenvalues
State response of time-delay system (32) by predictor feedback
Polar plot of the closed-loop eigenvalues
State response of time-delay system (35) by state feedback
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