# American Institute of Mathematical Sciences

• Previous Article
Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays
• NACO Home
• This Issue
• Next Article
Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay
September  2020, 10(3): 381-389. doi: 10.3934/naco.2020009

## Passive control for a class of Nonlinear systems by using the technique of Adding a power integrator

 1 School of Mathematics, Liaoning University, Shenyang, 110036, China 2 College of Science, Liaoning Shihua University, Fushun, 113001, China

* Corresponding author: Li Yang

Received  May 2019 Revised  October 2019 Published  February 2020

Fund Project: This work is supported by the scientific research fund LQN201712 of Liaoning Provincial Education Department

This paper studies the problem of passive control for a class of uncertain nonlinear lower-triangle systems. We extend the feedback designing tool named adding a power integrator. By using it repeatedly, the passive controller is given. Under this designing method, we don't need the system to be feedback linearizable. Moreover, comparing with the backstepping technique, the coordinate in the controller designing process of this method does not need to be transformed.

Citation: Jinglai Qiao, Li Yang, Jiawei Yao. Passive control for a class of Nonlinear systems by using the technique of Adding a power integrator. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 381-389. doi: 10.3934/naco.2020009
##### References:
 [1] W. Chen and M. Saif, Passivity and passivity based controller design of a class of switched control systems, in Proc. 16th IFAC World Congress, (2005), 143–147. [2] T. Deng, H. Yao, J. Du and J. Jiang, Neural network dynamic control for high-order nonlinear time-delay systems, Systems Engineering and Electronics, 36 (2014), 1152-1161. [3] P. Doerffer, O. Szulc and R. Bohing, Shock wave smearing by passive control, Journal of Thermal Science, 15 (2006), 43-47. [4] A. T. EL-Sayed, Suppression of nonlinear vibration system described by nonlinear differential equations using passive controller, Nonlinear Dynamics, 78 (2014), 1683-1694.  doi: 10.1007/s11071-014-1550-7. [5] U. K. Kocamaz and Y. Uyaroglu, Controlling Rucklidge chaotic system with a single controller using linear feedback and passive control methods, Nonlinear Dynamics, 75 (2014), 63-72.  doi: 10.1007/s11071-013-1049-7. [6] Y. Li, Y. Fu and G. Duan, Robust passive control for T-S fuzzy systems, in ICIC: Computational Intelligence, (2006), 146–151. [7] L. Liu and L. Sun, Coordinated passivity controller design for SVC and generator excitation with state constraint, Control Engineering of China, 15 (2018), 63-72. [8] Y. Liu and J. Zhao, Stabilization of switched nonlinear systems with passive and non-passive subsystems, Nonlinear Dynamics, 67 (2012), 1709-1716.  doi: 10.1007/s11071-011-0098-z. [9] W. Lin and C. Qian, Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems, Systems & Control Letters, 39 (2000), 339-351.  doi: 10.1016/S0167-6911(99)00115-2. [10] W. Lin and C. Qian, Adaptive regulation of high-order lower-triangular systems: an adding a power integrator technique, Systems & Control Letters, 39 (2000), 353-364.  doi: 10.1016/S0167-6911(99)00114-0. [11] T. Reis and J. C. Willems, A balancing approach to the realization of systems with internal passivity and reciprocity, Systems and Control Letters, 60 (2011), 69-74.  doi: 10.1016/j.sysconle.2010.10.009. [12] C. Sun, H. Sun and X. Diao, Continuous feedback control for a class of nonhomogeneous high-order nonlinear systems, Acta Automatica Sinca, 40 (2014), 149-155. [13] J. Tian and X. Xie, Adaptive state-feedback stabilization for more general high-order stochastic nonlinear systems, Acta Autonatica Sinca, 34 (2008), 1188-1191.  doi: 10.3724/SP.J.1004.2008.01188. [14] Z. Wu, G. Su, P. Shi and J. Chu, Reliable control for SSs with time-varying delays, Lecture Notes in Control and Information Sciences, 443 (2013), 37-52. [15] Z. Q. Wu, M. H. Song and L. Y. Fu, Passivity-based backstepping control for multimachine power system, Advanced Technology of Electrical Engineering and Energy, 32 (2013), 28-31. [16] H. Xiao and L. Zhao, Robust passive control of uncertain switched time-delay systems: a sliding mode control design, Journal of Control Theory and Applications, 11 (2013), 96-102.  doi: 10.1007/s11768-013-1033-2. [17] L. Yang, X. Liu et al., Exponentially dissipative control for singular impulsive dynamical systems, Journal of Dynamic systems, Measurement and Control, 139 (2017), 041008-1–041008-6. [18] L. Yang, X. Liu and Z. Zhang, Dissipative control for singular impulsive dynamical systems, Electronic Journal of Qualitative Theory of Differential Equations, 32 (2012), 1-11.  doi: 10.14232/ejqtde.2012.1.32.

show all references

##### References:
 [1] W. Chen and M. Saif, Passivity and passivity based controller design of a class of switched control systems, in Proc. 16th IFAC World Congress, (2005), 143–147. [2] T. Deng, H. Yao, J. Du and J. Jiang, Neural network dynamic control for high-order nonlinear time-delay systems, Systems Engineering and Electronics, 36 (2014), 1152-1161. [3] P. Doerffer, O. Szulc and R. Bohing, Shock wave smearing by passive control, Journal of Thermal Science, 15 (2006), 43-47. [4] A. T. EL-Sayed, Suppression of nonlinear vibration system described by nonlinear differential equations using passive controller, Nonlinear Dynamics, 78 (2014), 1683-1694.  doi: 10.1007/s11071-014-1550-7. [5] U. K. Kocamaz and Y. Uyaroglu, Controlling Rucklidge chaotic system with a single controller using linear feedback and passive control methods, Nonlinear Dynamics, 75 (2014), 63-72.  doi: 10.1007/s11071-013-1049-7. [6] Y. Li, Y. Fu and G. Duan, Robust passive control for T-S fuzzy systems, in ICIC: Computational Intelligence, (2006), 146–151. [7] L. Liu and L. Sun, Coordinated passivity controller design for SVC and generator excitation with state constraint, Control Engineering of China, 15 (2018), 63-72. [8] Y. Liu and J. Zhao, Stabilization of switched nonlinear systems with passive and non-passive subsystems, Nonlinear Dynamics, 67 (2012), 1709-1716.  doi: 10.1007/s11071-011-0098-z. [9] W. Lin and C. Qian, Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems, Systems & Control Letters, 39 (2000), 339-351.  doi: 10.1016/S0167-6911(99)00115-2. [10] W. Lin and C. Qian, Adaptive regulation of high-order lower-triangular systems: an adding a power integrator technique, Systems & Control Letters, 39 (2000), 353-364.  doi: 10.1016/S0167-6911(99)00114-0. [11] T. Reis and J. C. Willems, A balancing approach to the realization of systems with internal passivity and reciprocity, Systems and Control Letters, 60 (2011), 69-74.  doi: 10.1016/j.sysconle.2010.10.009. [12] C. Sun, H. Sun and X. Diao, Continuous feedback control for a class of nonhomogeneous high-order nonlinear systems, Acta Automatica Sinca, 40 (2014), 149-155. [13] J. Tian and X. Xie, Adaptive state-feedback stabilization for more general high-order stochastic nonlinear systems, Acta Autonatica Sinca, 34 (2008), 1188-1191.  doi: 10.3724/SP.J.1004.2008.01188. [14] Z. Wu, G. Su, P. Shi and J. Chu, Reliable control for SSs with time-varying delays, Lecture Notes in Control and Information Sciences, 443 (2013), 37-52. [15] Z. Q. Wu, M. H. Song and L. Y. Fu, Passivity-based backstepping control for multimachine power system, Advanced Technology of Electrical Engineering and Energy, 32 (2013), 28-31. [16] H. Xiao and L. Zhao, Robust passive control of uncertain switched time-delay systems: a sliding mode control design, Journal of Control Theory and Applications, 11 (2013), 96-102.  doi: 10.1007/s11768-013-1033-2. [17] L. Yang, X. Liu et al., Exponentially dissipative control for singular impulsive dynamical systems, Journal of Dynamic systems, Measurement and Control, 139 (2017), 041008-1–041008-6. [18] L. Yang, X. Liu and Z. Zhang, Dissipative control for singular impulsive dynamical systems, Electronic Journal of Qualitative Theory of Differential Equations, 32 (2012), 1-11.  doi: 10.14232/ejqtde.2012.1.32.
 [1] Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control and Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1 [2] Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations and Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072 [3] Liqiang Jin, Yanyan Yin, Kok Lay Teo, Fei Liu. Event-triggered mixed $H_\infty$ and passive control for Markov jump systems with bounded inputs. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1343-1355. doi: 10.3934/jimo.2020024 [4] Rod Cross, Victor Kozyakin. Double exponential instability of triangular arbitrage systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 349-376. doi: 10.3934/dcdsb.2013.18.349 [5] Wenxue Huang, Xiaofeng Li, Yuanyi Pan. Increase statistical reliability without losing predictive power by merging classes and adding variables. Big Data & Information Analytics, 2016, 1 (4) : 341-347. doi: 10.3934/bdia.2016014 [6] Qilong Zhai, Ran Zhang. Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 403-413. doi: 10.3934/dcdsb.2018091 [7] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [8] Ruth F. Curtain, George Weiss. Strong stabilization of (almost) impedance passive systems by static output feedback. Mathematical Control and Related Fields, 2019, 9 (4) : 643-671. doi: 10.3934/mcrf.2019045 [9] Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407 [10] Mei Luo, Jinrong Wang, Yumei Liao. Bounded consensus of double-integrator stochastic multi-agent systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022088 [11] Hans-Joachim Kroll, Sayed-Ghahreman Taherian, Rita Vincenti. Optimal antiblocking systems of information sets for the binary codes related to triangular graphs. Advances in Mathematics of Communications, 2022, 16 (1) : 169-183. doi: 10.3934/amc.2020107 [12] M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137 [13] Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063 [14] Chunjiang Qian, Wei Lin, Wenting Zha. Generalized homogeneous systems with applications to nonlinear control: A survey. Mathematical Control and Related Fields, 2015, 5 (3) : 585-611. doi: 10.3934/mcrf.2015.5.585 [15] Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579 [16] Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3967-4002. doi: 10.3934/dcdsb.2017204 [17] Jifeng Chu, Jinzhi Lei, Meirong Zhang. Lyapunov stability for conservative systems with lower degrees of freedom. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 423-443. doi: 10.3934/dcdsb.2011.16.423 [18] Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control and Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002 [19] Suoqin Jin, Fang-Xiang Wu, Xiufen Zou. Domain control of nonlinear networked systems and applications to complex disease networks. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2169-2206. doi: 10.3934/dcdsb.2017091 [20] V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223

Impact Factor:

## Metrics

• HTML views (495)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar