Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method

Fernando Agustín Pazos, Anibal Zanini, Amit Bhaya. Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method. International Journal of Automation and Computing, 2020, 17(3): 453-463. doi: 10.1007/s11633-019-1205-8
 Citation: Fernando Agustín Pazos, Anibal Zanini, Amit Bhaya. Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method. International Journal of Automation and Computing, 2020, 17(3): 453-463.

## Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method

###### Corresponding author:email: quini.coppe@gmail.com (corresponding author), ORCID: 0000-0001-9951-9790
• pn1
1 For a given Hilbert space $H$ and a closed interval $J$, we denote $L_2(J, H)$ as the Hilbert space of all measurable functions ${{f}}: J\rightarrow H$ such that $\int_J\| {{f}}(t)\|^2\partial t <\infty$.
• Figure  1.  Block diagram of the discrete-time linear system

Figure  2.  Components of the regularized control vector ${ u}_{k,\lambda}$ versus time

Figure  3.  State vectors versus time. The solid lines are the components of the discrete target state. The dotted lines are the output of the continuous-time plant if the unregularized control signal ${ u}_k$ is applied. The dashed lines are the output of the continuous-time plant if the regularized control signal ${ u}_{k,\lambda}$ is applied.(Color versions of the figures in this paper are available online.)

Figure  4.  Mean error given by (21) versus the regularization parameter $\lambda$

Figure  5.  Components of the regularized control vector versus time

Figure  6.  State vectors versus time. The solid lines are the components of the discrete target state, the dotted lines are the output of the continuous-time plant if the unregularized control signal ${ u}_k$ is applied, the dashed lines are the output of the continuous-time plant if the regularized control signal ${ u}_{k,\lambda}$ is applied.

Figure  7.  Mean error of the regularized state vector $\bar{\epsilon}$ versus the regularization parameter $\lambda$

•  [1] B. Wittenmark, J. Nilsson, M. Törngren. Timing problems in real-time control systems. In Proceedings of American Control Conference, IEEE, Seattle, USA, pp. 2000–2004, 1995. DOI: 10.1109/ACC.1995.531240. [2] A. Rabello, A. Bhaya. Stability of asynchronous dynamical systems with rate constraints and applications. IEE Proceedings-Control Theory and Applications, vol. 150, no. 5, pp. 546–550, 2003. DOI:  10.1049/ip-cta:20030704. [3] A. Bhaya, P. R. Medeiros. On the stability of asynchronous multirate linear systems. In Proceedings of the 36th IEEE Conference on Decision and Control, IEEE, San Diego, USA, pp. 2041–2042, 1997. DOI: 10.1109/CDC.1997.657066. [4] V. S. Ritchey, G. F. Franklin. A stability criterion for asynchronous multirate linear systems. IEEE Transactions on Automatic Control, vol. 34, no. 5, pp. 529–535, 1989. DOI:  10.1109/9.24205. [5] L. Hetel, J. Daafouz, C. Iung. Analysis and control of LTI and switched systems in digital loops via an event-based modelling. International Journal of Control, vol. 81, no. 7, pp. 1125–1138, 2008. DOI:  10.1080/00207170701670442. [6] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King. Stability criteria for switched and hybrid systems. SIAM Review, vol. 49, no. 4, pp. 545–592, 2007. DOI:  10.1137/05063516X. [7] V. D. Blondel, Y. Nesterov, J. Theys, Approximations of the rate of growth of switched linear systems. In Proceedings of the 7th International Workshop on Hybrid Systems: Computation and Control, Springer, Philadelphia, USA, vol. 2993, pp. 173–186, 2004. DOI: 10.1007/978-3-540-24743-2_12. [8] M. Schinkel, W. H. Chen, A. Rantzer. Optimal control for systems with varying sampling rate. In Proceedings of American Control Conference, IEEE, Anchorage, USA, pp. 2979–2984, 2002. DOI: 10.1109/ACC.2002.1025245. [9] F. E. Felicioni, S. J. Junco. A Lie algebraic approach to design of stable feedback control systems with varying sampling rate. In Proceedings of the 17th World Congress of the International Federation of Automatic Control, IFAC, Seoul, South Korea, pp. 4881–4886, 2008. [10] M. Schinkel, W. H. Chen. Control of sampled-data systems with variable sampling rate. International Journal of Systems Science, vol. 37, no. 9, pp. 609–618, 2006. DOI:  10.1080/00207720600681161. [11] F. Felicioni. Stabilitéet Performance des Systèmes Distribués de Contrôle-Commande, Ph. D. dissertation, Nancy University, Lorraine, France, 2011. (in Spanish) [12] A. Sala. Computer control under time-varying sampling period: An LMI gridding approach. Automatica, vol. 41, no. 12, pp. 2077–2082, 2005. DOI:  10.1016/j.automatica.2005.05.017. [13] F. Pazos, A. Zanini, A. Bhaya. Stabilization of discrete-time linear systems with varying sampling rates via a state feedback controller. In Proceedings of the 14th Brazilian Conference on Dynamics, Control and Applications, São Carlos, Brazil, 2019. (to be published) [14] D. Liberzon, R. Tempo. Common Lyapunov functions and gradient algorithms. IEEE Transactions on Automatic Control, vol. 49, no. 6, pp. 990–994, 2004. DOI:  10.1109/TAC.2004.829632. [15] B. Tang, Q. J. Zeng, D. F. He, Y. Zhang. Random stabilization of sampled-data control systems with nonuniform sampling. International Journal of Automation and Computing, vol. 9, no. 5, pp. 492–500, 2012. DOI:  10.1007/s11633-012-0672-y. [16] C. C. Hua, S. C. Yu, X. P. Guan. Finite-time control for a class of networked control systems with short time-varying delays and sampling jitter. International Journal of Automation and Computing, vol. 12, no. 4, pp. 448–454, 2015. DOI:  10.1007/s11633-014-0849-7. [17] C. C. Sun. Stabilization for a class of discrete-time switched large-scale systems with parameter uncertainties. International Journal of Automation and Computing, vol. 16, no. 4, pp. 543–552, 2019. DOI:  10.1007/s11633-016-0966-6. [18] L. G. Wu, Y. B. Gao, J. X. Liu, H. Y. Li. Event-triggered sliding mode control of stochastic systems via output feedback. Automatica, vol. 82, pp. 79–92, 2017. DOI:  10.1016/j.automatica.2017.04.032. [19] J. X. Liu, L. G. Wu, C. W. Wu, W. S. Luo, L. G. Franquelo. Event-triggering dissipative control of switched stochastic systems via sliding mode. Automatica, vol. 103, pp. 261–273, 2019. DOI:  10.1016/j.automatica.2019.01.029. [20] M. T. Nair, N. Sukavanam, R. Katta. Computation of control for linear approximately controllable system using Tikhonov regularization. Numerical Functional Analysis and Optimization, vol. 39, no. 3, pp. 308–321, 2018. DOI:  10.1080/01630563.2017.1361440. [21] R. Katta, G. D. Reddy, N. Sukavanam. Computation of control for linear approximately controllable system using weighted Tikhonov regularization. Applied Mathematics and Computation, vol. 317, pp. 252–263, 2018. DOI:  10.1016/j.amc.2017.08.012. [22] R. Katta, N. Sukavanam. Computation of control for nonlinear approximately controllable system using Tikhonov regularization. Cogent Mathematics, vol. 3, no. 1, Article number 1216241, 2016 [23] M. Loaiza. A short introduction to Hilbert space theory. Journal of Physics: Conference Series, vol. 839, no. 1, Article number 012002, 2017. [24] M. T. Nair. On controllability of linear systems, Department of Mathematics, IIT Madras, Madras, India, Technical Report, [Online], Available: https://math.iitm.ac.in/public_html/mtnair/IIST-control-mtn.pdf, November 28–29, 2012. [25] R. F. Curtain, H. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory, New York, USA: Springer-Verlag, 1995. DOI:  10.1007/978-1-4612-4224-6. [26] C. T. Chen. Linear System Theory and Design, 3rd ed., New York, USA: Oxford University Press, 1999. [27] H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, London, UK: Kluwer Academic Publishers, 1996. [28] A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems, Washington, USA: Winston and Sons, 1977. [29] V. A. Morozov, Regularization Methods for Ill-Posed Problems, Boca Raton, USA: CRC Press, 1993. [30] G. H. Golub, M. Heath, G. Wahba. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, vol. 21, no. 2, pp. 215–223, 1979. DOI:  10.1080/00401706.1979.10489751. [31] P. C. Hansen. The L-curve and its use in the numerical treatment of inverse problems. Computational Inverse Problems in Electrocardiology, P. Johnston, Ed., Southampton, UK: IMM, 2001. [32] P. C. Hansen, D. P. O′Leary. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM Journal on Scientific Computing, vol. 14, no. 6, pp. 1487–1503, 1993. DOI:  10.1137/0914086. [33] F. Pazos, A. Bhaya. Adaptive choice of the Tikhonov regularization parameter to solve ill-posed linear algebraic equations via Liapunov optimizing control. Journal of Computational and Applied Mathematics, vol. 279, pp. 123–132, 2015. DOI:  10.1016/j.cam.2014.10.022. [34] F. Bauer, M. A. Lukas. Comparingparameter choice methods for regularization of ill-posed problems. Mathematics and Computers in Simulation, vol. 81, no. 9, pp. 1795–1841, 2011. DOI:  10.1016/j.matcom.2011.01.016. [35] K. Kunisch, J. Zou. Iterative choices of regularization parameters in linear inverse problems. Inverse Problems, vol. 14, no. 5, pp. 1247–1264, 1998. DOI:  10.1088/0266-5611/14/5/010. [36] H. Gfrerer. An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal convergence rates. Mathematics of Computation, vol. 49, no. 180, pp. 507–522, 1987. DOI:  10.2307/2008325. [37] L. Reichel, G. Rodriguez. Old and new parameter choice rules for discrete ill-posed problems. Numerical Algorithms, vol. 63, no. 1, pp. 65–87, 2013. DOI:  10.1007/s11075-012-9612-8. [38] T. Regińska. A regularization parameter in discrete ill-posed problems. SIAM Journal on Scientific Computing, vol. 17, no. 3, pp. 740–749, 1996. DOI:  10.1137/S1064827593252672. [39] D. Zhang, T. Z. Huang. Generalized Tikhonov regularization method for large-scale linear inverse problems. Journal of Computational Analysis and Applications, vol. 15, no. 7, pp. 1317–1331, 2013. [40] K. J. Åström, B. Wittenmark, Computer Controlled Systems: Theory and Design, Englewood Cliffs, USA: Prentice-Hall, 1984. [41] G. F. Franklin, J. D. Powell, M. L. Workman, Digital Control of Dynamic Systems, 3rd ed., Menlo Park, USA: Addison-Wesley, 1997. [42] P. C. Hansen. Regularization tools version 4.0 for Matlab 7.3. Numerical Algorithms, vol. 46, no. 2, pp. 189–194, 2007. DOI:  10.1007/s11075-007-9136-9.
•  [1] Azzedine Yahiaoui.  A Practical Approach to Representation of Real-time Building Control Applications in Simulation . International Journal of Automation and Computing, 2020, 17(3): 464-478. doi: 10.1007/s11633-018-1131-1 [2] Noussaiba Gasmi, Assem Thabet, Mohamed Aoun.  New LMI Conditions for Reduced-order Observer of Lipschitz Discrete-time Systems: Numerical and Experimental Results . International Journal of Automation and Computing, 2019, 16(5): 644-654. doi: 10.1007/s11633-018-1160-9 [3] Chang-Chun Sun.  Stabilization for a Class of Discrete-time Switched Large-scale Systems with Parameter Uncertainties . International Journal of Automation and Computing, 2019, 16(4): 543-552. doi: 10.1007/s11633-016-0966-6 [4] Ana Paula Batista, Fábio Gonçalves Jota.  Analysis of the Most Likely Regions of Stability of an NCS and Design of the Corresponding Event-driven Controller . International Journal of Automation and Computing, 2018, 15(1): 39-51. doi: 10.1007/s11633-017-1099-2 [5] Nabiha Touijer, Samira Kamoun.  Robust Self-tuning Control Based on Discrete-time Sliding Mode for Auto-regressive Mathematical Model in the Presence of Unmodelled Dynamics . International Journal of Automation and Computing, 2016, 13(3): 277-284. doi: 10.1007/s11633-015-0921-y [6] Yuan Ge, Yaoyiran Li.  SCHMM-based Compensation for the Random Delays in Networked Control Systems . International Journal of Automation and Computing, 2016, 13(6): 643-652. doi: 10.1007/s11633-016-1001-7 [7] Wei Zhou, Miao Yu, De-Qing Huang.  A High-order Internal Model Based Iterative Learning Control Scheme for Discrete Linear Time-varying Systems . International Journal of Automation and Computing, 2015, 12(3): 330-336. doi: 10.1007/s11633-015-0886-x [8] Chang-Chun Hua, Shao-Chong Yu, Xin-Ping Guan.  Finite-time Control for a Class of Networked Control Systems with Short Time-varying Delays and Sampling Jitter . International Journal of Automation and Computing, 2015, 12(4): 448-454. doi: 10.1007/s11633-014-0849-7 [9] Yuan-Qing Xia, Yu-Long Gao, Li-Ping Yan, Meng-Yin Fu.  Recent Progress in Networked Control Systems-A Survey . International Journal of Automation and Computing, 2015, 12(4): 343-367. doi: 10.1007/s11633-015-0894-x [10] Lan Zhou, Jin-Hua She, Min Wu.  Design of a Discrete-time Output-feedback Based Repetitive-control System . International Journal of Automation and Computing, 2013, 10(4): 343-349. doi: 10.1007/s11633-013-0730-0 [11] Meng Zhao,  Bao-Cang Ding.  A Contractive Sliding-mode MPC Algorithm for Nonlinear Discrete-time Systems . International Journal of Automation and Computing, 2013, 10(2): 167-172. doi: 10.1007/s11633-013-0709-x [12] Ting Wang,  Ming-Xiang Xue,  Chun Zhang,  Shu-Min Fei.  Improved Stability Criteria on Discrete-time Systems with Time-varying and Distributed Delays . International Journal of Automation and Computing, 2013, 10(3): 260-266. doi: 10.1007/s11633-013-0719-8 [13] Song-Gui Yuan, Min Wu, Bao-Gang Xu, Rui-Juan Liu.  Design of Discrete-time Repetitive Control System Based on Two-dimensional Model . International Journal of Automation and Computing, 2012, 9(2): 165-170. doi: 10.1007/s11633-012-0629-1 [14] Saïda Bedoui,  Majda Ltaief,  Kamel Abderrahim.  New Results on Discrete-time Delay Systems Identification . International Journal of Automation and Computing, 2012, 9(6): 570-577 . doi: 10.1007/s11633-012-0681-x [15] Yong-Yun Shao, Xiao-Dong Liu, Xin Sun, Zhan Su.  A New Admissibility Condition of Discrete-time Singular Systems with Time-varying Delays . International Journal of Automation and Computing, 2012, 9(5): 480-486. doi: 10.1007/s11633-012-0670-0 [16] K. Ramakrishnan,  G. Ray.  Stability Criterion with Less LMI Variables for Linear Discrete-time Systems with Additive Time-delays . International Journal of Automation and Computing, 2011, 8(4): 490-492. doi: 10.1007/s11633-011-0608-y [17] Qing-Zheng Gao,  Xue-Jun Xie.  Robustness Analysis of Discrete-time Indirect Model Reference Adaptive Control with Normalized Adaptive Laws . International Journal of Automation and Computing, 2010, 7(3): 381-388. doi: 10.1007/s11633-010-0518-4 [18] Xian-Ming Tang,  Jin-Shou Yu.  Stability Analysis of Discrete-time Systems with Additive Time-varying Delays . International Journal of Automation and Computing, 2010, 7(2): 219-223. doi: 10.1007/s11633-010-0219-z [19] Yong-Gang Chen, Wen-Lin Li.  Improved Results on Robust H∞ Control of Uncertain Discrete-time Systems with Time-varying Delay . International Journal of Automation and Computing, 2009, 6(1): 103-108. doi: 10.1007/s11633-009-0103-x [20] Bao-Feng Wang,  Ge Guo.  Kalman Filtering with Partial Markovian Packet Losses . International Journal of Automation and Computing, 2009, 6(4): 395-400. doi: 10.1007/s11633-009-0395-x

##### 计量
• 文章访问数:  713
• HTML全文浏览量:  402
• PDF下载量:  31
• 被引次数: 0
##### 出版历程
• 收稿日期:  2019-07-16
• 录用日期:  2019-10-08
• 网络出版日期:  2019-12-17
• 刊出日期:  2020-06-01

## Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method

pn1

### English Abstract

Fernando Agustín Pazos, Anibal Zanini, Amit Bhaya. Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method. International Journal of Automation and Computing, 2020, 17(3): 453-463. doi: 10.1007/s11633-019-1205-8
 Citation: Fernando Agustín Pazos, Anibal Zanini, Amit Bhaya. Performance Improvement of Discrete-time Linear Control Systems Subject to Varying Sampling Rates Using the Tikhonov Regularization Method. International Journal of Automation and Computing, 2020, 17(3): 453-463.

/

• 分享
• 用微信扫码二维码

分享至好友和朋友圈