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doi: 10.3934/jimo.2020076

Probabilistic robust anti-disturbance control of uncertain systems

Peng Cheng 1, , Feng Pan 1,, , Yanyan Yin 2, and  Song Wang 3,4,

1. 

Key laboratory of Advanced Process Control for Light Industry (Ministry of Education), School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, China
2. 

School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Western Australia, 6102, Australia
3. 

School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO Box U1987, Perth, WA6845, Australia
4. 

Shenzhen Audencia Business School, WeBank Institute of Fintech, Guangdong Laboratory of Artificial Intelligence and Digital Economics (SZ), Shenzhen University, Shenzhen, 518060, China

* Corresponding author: Feng Pan

Received  May 2019 Revised  October 2019 Published  April 2020

We propose a novel method for constructing probabilistic robust disturbance rejection control for uncertain systems in which a scenario optimization method is used to deal with the nonlinear and unbounded uncertainties. For anti-disturbance, a reduced order disturbance observer is considered and a state-feedback controller is designed. Sufficient conditions are presented to ensure that the resulting closed-loop system is stable and a prescribed $ H_{\infty} $ performance index is satisfied. A numerical example is presented to illustrate the effectiveness of the techniques proposed and analyzed.

Keywords: Uncertain systems, discrete time systems, scenario optimization method, probabilistic control, disturbance rejection.
Mathematics Subject Classification: Primary: 93C55, 93D09; Secondary: 93B05.
Citation: Peng Cheng, Feng Pan, Yanyan Yin, Song Wang. Probabilistic robust anti-disturbance control of uncertain systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020076
References:
[1]

J. Ackermann, Robust Control: The Parameter Space Approach, Springer-Verlag, London, 2002. doi: 10.1007/978-1-4471-0207-6.  Google Scholar

[2]

T. Alamo, R. Tempo, A. Luque and D. R. Ramirez, Randomized methods for design of uncertain systems: Sample complexity and sequential algorithms, Automatica J. IFAC, 52 (2015), 160-172.  doi: 10.1016/j.automatica.2014.11.004.  Google Scholar

[3]

B. R. Barmish, New Tools for Robustness of Linear Systems, MacMillan, New York, 1994. Google Scholar

[4]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, 15, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[5]

M. C. Campi, S. Garatti and and M. Prandini, The scenario approach for systems and control design, Ann. Rev. Control, 33 (2009), 149-157.  doi: 10.1016/j.arcontrol.2009.07.001.  Google Scholar

[6]

G. C. Calafiore and M. C. Campi, The scenario approach to robust control design,, IEEE Trans. Automat. Control, 51 (2006), 742-753.  doi: 10.1109/TAC.2006.875041.  Google Scholar

[7]

G. Grimm, M. J. Messina, S. E. Tuna and A. R. Teel, Nominally robust model predictive control with state constraints, IEEE Trans. Automat. Control, 52 (2007), 1856-1870.  doi: 10.1109/TAC.2007.906187.  Google Scholar

[8]

P. Gahinet, Explicit controller formulas for LMI-based $H_{\infty}$ synthesis, Automatica J. IFAC, 32 (1996), 1007-1014.  doi: 10.1016/0005-1098(96)00033-7.  Google Scholar

[9]

L. Guo and W. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, Internat. J. Robust Nonlinear Control, 15 (2005), 109-125.  doi: 10.1002/rnc.978.  Google Scholar

[10]

X. Ji, M. Ren and H. Su, Comment on "Further enhancement on robust ${H_\infty }$ control design for discrete-time singular systems", IEEE Trans. Automat. Control, 60 (2015), 3119-3120.  doi: 10.1109/TAC.2015.2409951.  Google Scholar

[11]

L. Jin, Y. Yin, K. L. Teo and F. Liu, Event-triggered mixed $H\infty$ and passive control for Markov jump systems with bounded inputs, J. Ind. Manag. Optim.. doi: 10.3934/jimo.2020024.  Google Scholar

[12]

Y. Liu, Y. Yin, K. L. Teo, S. Wang and F. Liu, Probabilistic control of Markov jump systems by scenario optimization approach, J. Ind. Manag. Optim., 15 (2019), 1447-1453.  doi: 10.3934/jimo.2018103.  Google Scholar

[13]

H. Melkote, F. Khorrami, S. Jain and M. S. Mattice, Robust adaptive control of variable reluctance stepper motors, IEEE Trans. Control Systems Tech., 7 (1999), 212-221.  doi: 10.1109/87.748147.  Google Scholar

[14]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996.  doi: 10.1137/050622328.  Google Scholar

[15]

R. E. Skelton, T. Iwasaki and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, The Taylor & Francis Systems and Control Book Series, Taylor & Francis Group, London, 1998.  Google Scholar

[16]

H. Sun and L. Guo, Neural network-based DOBC for a class of nonlinear systems with unmatched disturbances, IEEE Trans. Neural Networks Learning Systems, 28 (2017), 482-489.  doi: 10.1109/TNNLS.2015.2511450.  Google Scholar

[17]

E. Tian, Z. Wang, L. Zou and D. Yue, Chance-constrained $H_{\infty}$ control for a class of time-varying systems with stochastic nonlinearities: The finite-horizon case, Automatica J. IFAC, 107 (2019), 296-305.  doi: 10.1016/j.automatica.2019.05.039.  Google Scholar

[18]

E. Tian, Z. Wang, L. Zou and D. Yue, Probabilistic constrained filtering for a class of nonlinear systems with improved static event-triggered communication, Internat. J. Robust Nonlinear Control, 29 (2019), 1484-1498.  doi: 10.1002/rnc.4447.  Google Scholar

[19]

B. Yao, M. Al-Majed and M. Tomizuka, High-performance robust motion control of machine tools: An adaptive robust control approach and comparative experiments, IEEE/ASME Trans. Mechatronics, 2 (1997), 63-76.  doi: 10.1109/ACC.1997.611956.  Google Scholar

[20]

Y. Yin, P. Shi, F. Liu, K. L. Teo and C. C. Lim, Robust filtering for nonlinear nonhomogeneous Markov jump systems by fuzzy approximation approach, IEEE Trans on Cybernetics, 45 (2015), 1706-1716.  doi: 10.1109/TCYB.2014.2358680.  Google Scholar

[21]

Y. Yin, Z. Lin, Y. Liu and K. L. Teo, Event-triggered constrained control of positive systems with input saturation, Internat. J. Robust Nonlinear Control, 28 (2018), 3532-3542.  doi: 10.1002/rnc.4097.  Google Scholar

[22]

Y. Yin and Z. Lin, Constrained control of uncertain nonhomogeneous Markovian jump systems, Internat. J. Robust Nonlinear Control, 27 (2017), 3937-3950.  doi: 10.1002/rnc.3774.  Google Scholar

[23]

J. Yang, B. Wu, S. Li and X. Yu, Design and qualitative robustness analysis of an DOBC approach for DC-DC buck converters with unmatched circuit parameter perturbations, IEEE Trans. Circuits Systems I: Regular Papers, 63 (2016), 551-560.  doi: 10.1109/TCSI.2016.2529238.  Google Scholar

show all references

References:
[1]

J. Ackermann, Robust Control: The Parameter Space Approach, Springer-Verlag, London, 2002. doi: 10.1007/978-1-4471-0207-6.  Google Scholar

[2]

T. Alamo, R. Tempo, A. Luque and D. R. Ramirez, Randomized methods for design of uncertain systems: Sample complexity and sequential algorithms, Automatica J. IFAC, 52 (2015), 160-172.  doi: 10.1016/j.automatica.2014.11.004.  Google Scholar

[3]

B. R. Barmish, New Tools for Robustness of Linear Systems, MacMillan, New York, 1994. Google Scholar

[4]

S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, 15, SIAM, Philadelphia, PA, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[5]

M. C. Campi, S. Garatti and and M. Prandini, The scenario approach for systems and control design, Ann. Rev. Control, 33 (2009), 149-157.  doi: 10.1016/j.arcontrol.2009.07.001.  Google Scholar

[6]

G. C. Calafiore and M. C. Campi, The scenario approach to robust control design,, IEEE Trans. Automat. Control, 51 (2006), 742-753.  doi: 10.1109/TAC.2006.875041.  Google Scholar

[7]

G. Grimm, M. J. Messina, S. E. Tuna and A. R. Teel, Nominally robust model predictive control with state constraints, IEEE Trans. Automat. Control, 52 (2007), 1856-1870.  doi: 10.1109/TAC.2007.906187.  Google Scholar

[8]

P. Gahinet, Explicit controller formulas for LMI-based $H_{\infty}$ synthesis, Automatica J. IFAC, 32 (1996), 1007-1014.  doi: 10.1016/0005-1098(96)00033-7.  Google Scholar

[9]

L. Guo and W. Chen, Disturbance attenuation and rejection for systems with nonlinearity via DOBC approach, Internat. J. Robust Nonlinear Control, 15 (2005), 109-125.  doi: 10.1002/rnc.978.  Google Scholar

[10]

X. Ji, M. Ren and H. Su, Comment on "Further enhancement on robust ${H_\infty }$ control design for discrete-time singular systems", IEEE Trans. Automat. Control, 60 (2015), 3119-3120.  doi: 10.1109/TAC.2015.2409951.  Google Scholar

[11]

L. Jin, Y. Yin, K. L. Teo and F. Liu, Event-triggered mixed $H\infty$ and passive control for Markov jump systems with bounded inputs, J. Ind. Manag. Optim.. doi: 10.3934/jimo.2020024.  Google Scholar

[12]

Y. Liu, Y. Yin, K. L. Teo, S. Wang and F. Liu, Probabilistic control of Markov jump systems by scenario optimization approach, J. Ind. Manag. Optim., 15 (2019), 1447-1453.  doi: 10.3934/jimo.2018103.  Google Scholar

[13]

H. Melkote, F. Khorrami, S. Jain and M. S. Mattice, Robust adaptive control of variable reluctance stepper motors, IEEE Trans. Control Systems Tech., 7 (1999), 212-221.  doi: 10.1109/87.748147.  Google Scholar

[14]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996.  doi: 10.1137/050622328.  Google Scholar

[15]

R. E. Skelton, T. Iwasaki and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, The Taylor & Francis Systems and Control Book Series, Taylor & Francis Group, London, 1998.  Google Scholar

[16]

H. Sun and L. Guo, Neural network-based DOBC for a class of nonlinear systems with unmatched disturbances, IEEE Trans. Neural Networks Learning Systems, 28 (2017), 482-489.  doi: 10.1109/TNNLS.2015.2511450.  Google Scholar

[17]

E. Tian, Z. Wang, L. Zou and D. Yue, Chance-constrained $H_{\infty}$ control for a class of time-varying systems with stochastic nonlinearities: The finite-horizon case, Automatica J. IFAC, 107 (2019), 296-305.  doi: 10.1016/j.automatica.2019.05.039.  Google Scholar

[18]

E. Tian, Z. Wang, L. Zou and D. Yue, Probabilistic constrained filtering for a class of nonlinear systems with improved static event-triggered communication, Internat. J. Robust Nonlinear Control, 29 (2019), 1484-1498.  doi: 10.1002/rnc.4447.  Google Scholar

[19]

B. Yao, M. Al-Majed and M. Tomizuka, High-performance robust motion control of machine tools: An adaptive robust control approach and comparative experiments, IEEE/ASME Trans. Mechatronics, 2 (1997), 63-76.  doi: 10.1109/ACC.1997.611956.  Google Scholar

[20]

Y. Yin, P. Shi, F. Liu, K. L. Teo and C. C. Lim, Robust filtering for nonlinear nonhomogeneous Markov jump systems by fuzzy approximation approach, IEEE Trans on Cybernetics, 45 (2015), 1706-1716.  doi: 10.1109/TCYB.2014.2358680.  Google Scholar

[21]

Y. Yin, Z. Lin, Y. Liu and K. L. Teo, Event-triggered constrained control of positive systems with input saturation, Internat. J. Robust Nonlinear Control, 28 (2018), 3532-3542.  doi: 10.1002/rnc.4097.  Google Scholar

[22]

Y. Yin and Z. Lin, Constrained control of uncertain nonhomogeneous Markovian jump systems, Internat. J. Robust Nonlinear Control, 27 (2017), 3937-3950.  doi: 10.1002/rnc.3774.  Google Scholar

[23]

J. Yang, B. Wu, S. Li and X. Yu, Design and qualitative robustness analysis of an DOBC approach for DC-DC buck converters with unmatched circuit parameter perturbations, IEEE Trans. Circuits Systems I: Regular Papers, 63 (2016), 551-560.  doi: 10.1109/TCSI.2016.2529238.  Google Scholar

Figure 1.  State trajectory of a-posteriori Monte-Carlo analysis
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Figure 2.  Estimation of disturbance
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Figure 3.  Trajectory of controlled output
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