2013, 3(3): 445-462. doi: 10.3934/naco.2013.3.445

Direct model reference adaptive control of linear systems with input/output delays

1. 

Department of Electrical and Computer Engineering, College of Engineering and Applied Science, University of Wyoming, Laramie, WY 82071, United States, United States

Received  April 2013 Revised  June 2013 Published  July 2013

In this paper, we develop a Direct Model Reference Adaptive Tracking Controller for linear systems with unknown time varying input delays. This controller can also reject bounded disturbances of known waveform but unknown amplitude, e.g. steps or sinusoids. In this paper a robustness result is developed for DMRAC of linear systems with unknown small constant or time varying input delays using the concept of un-delayed ideal trajectories. We will show that the adaptively controlled system is globally stable, but the adaptive tracking error is no longer guaranteed to approach the origin. However, exponential convergence to a neighborhood can be achieved as a result of the control design. A simple example will be provided to illustrate this adaptive control method.
Citation: James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 445-462. doi: 10.3934/naco.2013.3.445
References:
[1]

M. Balas, R. Erwin and R. Fuentes, Adaptive control of persistent disturbances for aerospace structures, Proceedings of the AIAA Guidance, Navigation and Control Conference, Denver, CO, 2000.

[2]

M. Balas, S. Gajendar and L. Robertson, Adaptive tracking control of linear systems with unknown delays and persistent disturbances (or who you callin retarded?), Proceedings of the AIAA Guidance, Navigation and Control Conference, Chicago, IL, Aug 2009.

[3]

M. Balas, J. Nelson, S. Gajendar and L. Robertson, Robust adaptive control of nonlinear systems with input/output delays, Proceedings of the AIAA Guidance, Navigation and Control Conference, Portland, OR, Aug 2011.

[4]

R. Fuentes and M. Balas, Direct adaptive rejection of persistent disturbances, Journal of Mathematical Analysis and Applications, 251 (2000), 28-39. doi: 10.1006/jmaa.2000.7017.

[5]

R. Fuentes and M. Balas, Disturbance accommodation for a class of tracking control systems, Proceedings of the AIAA Guidance, Navigation and Control Conference, Denver, CO, 2000.

[6]

R. Fuentes and M. Balas, Robust model reference adaptive control with disturbance rejection, Proceedings of the American Control Conference, Anchorage, AK, 2002.

[7]

K. Gu, V. L. Kharitonov and J. Chen, "Stability of Time Delay Systems," Bikhauser, 2003.

[8]

M. Krstic, Compensation of infinite-dimensional actuator and sensor dynamics : nonlinear delay-adaptive systems, IEEE Control Systems Magazine, 01 (2010), 22-41. doi: 10.1109/MCS.2009.934990.

[9]

J. Luna, Time delay systems and applications on satellite control,, University of New Mexico/ARFL Space Vechicles, (). 

[10]

J. Nelson, M. Balas and R. Erwin, Direct model reference adaptive control of linear systems with unknown time varying input/output delays, Proceedings of the AIAA Guidance, Navigation and Control Conference, Minneapolis, MN, Aug 2012.

[11]

S.-I. Niculescu, "Delay Effects on Stability," Springer, 2001.

[12]

J.-P. Richard, Time delay systems: an overview of some recent advances and open problems, Science Direct Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5.

[13]

A. Seuret, Networked control under synchronization errors, Proceedings of the American Control Conference, Seattle, WA, 2008.

[14]

R. Sipahi, S.-I. Nculescu, C. T. Abdallah, W. Michiels and K. Gu., Stability and stabalization of systems with time delay, IEEE Control Systems Magazine, 31 (2001), 38-65.

[15]

J. Wen, Time domain and frequency domain conditions for strict positive realness, IEEE Transactions on Automatic Control, 33 (1988), 988-992. doi: 10.1109/9.7263.

show all references

References:
[1]

M. Balas, R. Erwin and R. Fuentes, Adaptive control of persistent disturbances for aerospace structures, Proceedings of the AIAA Guidance, Navigation and Control Conference, Denver, CO, 2000.

[2]

M. Balas, S. Gajendar and L. Robertson, Adaptive tracking control of linear systems with unknown delays and persistent disturbances (or who you callin retarded?), Proceedings of the AIAA Guidance, Navigation and Control Conference, Chicago, IL, Aug 2009.

[3]

M. Balas, J. Nelson, S. Gajendar and L. Robertson, Robust adaptive control of nonlinear systems with input/output delays, Proceedings of the AIAA Guidance, Navigation and Control Conference, Portland, OR, Aug 2011.

[4]

R. Fuentes and M. Balas, Direct adaptive rejection of persistent disturbances, Journal of Mathematical Analysis and Applications, 251 (2000), 28-39. doi: 10.1006/jmaa.2000.7017.

[5]

R. Fuentes and M. Balas, Disturbance accommodation for a class of tracking control systems, Proceedings of the AIAA Guidance, Navigation and Control Conference, Denver, CO, 2000.

[6]

R. Fuentes and M. Balas, Robust model reference adaptive control with disturbance rejection, Proceedings of the American Control Conference, Anchorage, AK, 2002.

[7]

K. Gu, V. L. Kharitonov and J. Chen, "Stability of Time Delay Systems," Bikhauser, 2003.

[8]

M. Krstic, Compensation of infinite-dimensional actuator and sensor dynamics : nonlinear delay-adaptive systems, IEEE Control Systems Magazine, 01 (2010), 22-41. doi: 10.1109/MCS.2009.934990.

[9]

J. Luna, Time delay systems and applications on satellite control,, University of New Mexico/ARFL Space Vechicles, (). 

[10]

J. Nelson, M. Balas and R. Erwin, Direct model reference adaptive control of linear systems with unknown time varying input/output delays, Proceedings of the AIAA Guidance, Navigation and Control Conference, Minneapolis, MN, Aug 2012.

[11]

S.-I. Niculescu, "Delay Effects on Stability," Springer, 2001.

[12]

J.-P. Richard, Time delay systems: an overview of some recent advances and open problems, Science Direct Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5.

[13]

A. Seuret, Networked control under synchronization errors, Proceedings of the American Control Conference, Seattle, WA, 2008.

[14]

R. Sipahi, S.-I. Nculescu, C. T. Abdallah, W. Michiels and K. Gu., Stability and stabalization of systems with time delay, IEEE Control Systems Magazine, 31 (2001), 38-65.

[15]

J. Wen, Time domain and frequency domain conditions for strict positive realness, IEEE Transactions on Automatic Control, 33 (1988), 988-992. doi: 10.1109/9.7263.

[1]

K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050

[2]

Le Viet Cuong, Thai Son Doan. Assignability of dichotomy spectra for discrete time-varying linear control systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3597-3607. doi: 10.3934/dcdsb.2020074

[3]

Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Linear optimal control of time delay systems via Hermite wavelet. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 143-156. doi: 10.3934/naco.2019044

[4]

Tingwen Huang, Guanrong Chen, Juergen Kurths. Synchronization of chaotic systems with time-varying coupling delays. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1071-1082. doi: 10.3934/dcdsb.2011.16.1071

[5]

Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control and Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827

[6]

Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial and Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175

[7]

Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193

[8]

Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653

[9]

Hamid Maarouf. Local Kalman rank condition for linear time varying systems. Mathematical Control and Related Fields, 2022, 12 (2) : 433-446. doi: 10.3934/mcrf.2021029

[10]

Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098

[11]

David Schley, S.A. Gourley. Linear and nonlinear stability in a diffusional ecotoxicological model with time delays. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 575-590. doi: 10.3934/dcdsb.2002.2.575

[12]

Elimhan N. Mahmudov. Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 353-371. doi: 10.3934/naco.2021010

[13]

Wei Feng, Xin Lu. Global stability in a class of reaction-diffusion systems with time-varying delays. Conference Publications, 1998, 1998 (Special) : 253-261. doi: 10.3934/proc.1998.1998.253

[14]

Hongbiao Fan, Jun-E Feng, Min Meng. Piecewise observers of rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1535-1556. doi: 10.3934/jimo.2016.12.1535

[15]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

[16]

Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559

[17]

Abdelfettah Hamzaoui, Nizar Hadj Taieb, Mohamed Ali Hammami. Practical partial stability of time-varying systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3585-3603. doi: 10.3934/dcdsb.2021197

[18]

Dinh Cong Huong, Mai Viet Thuan. State transformations of time-varying delay systems and their applications to state observer design. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 413-444. doi: 10.3934/dcdss.2017020

[19]

Carlos Nonato, Manoel Jeremias dos Santos, Carlos Raposo. Dynamics of Timoshenko system with time-varying weight and time-varying delay. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 523-553. doi: 10.3934/dcdsb.2021053

[20]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3837-3849. doi: 10.3934/dcdss.2020444

 Impact Factor: 

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]