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September  2021, 17(5): 2441-2450. doi: 10.3934/jimo.2020076

Probabilistic robust anti-disturbance control of uncertain systems

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  September 2021 Early access  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.

Citation: Peng Cheng, Feng Pan, Yanyan Yin, Song Wang. Probabilistic robust anti-disturbance control of uncertain systems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2441-2450. 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.

[2]

T. AlamoR. TempoA. 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.

[3]

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

[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.

[5]

M. C. CampiS. 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.

[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.

[7]

G. GrimmM. J. MessinaS. 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.

[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.

[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.

[10]

X. JiM. 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.

[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.

[12]

Y. LiuY. YinK. L. TeoS. 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.

[13]

H. MelkoteF. KhorramiS. 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.

[14]

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

[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.

[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.

[17]

E. TianZ. WangL. 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.

[18]

E. TianZ. WangL. 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.

[19]

B. YaoM. 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.

[20]

Y. YinP. ShiF. LiuK. 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.

[21]

Y. YinZ. LinY. 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.

[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.

[23]

J. YangB. WuS. 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.

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.

[2]

T. AlamoR. TempoA. 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.

[3]

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

[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.

[5]

M. C. CampiS. 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.

[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.

[7]

G. GrimmM. J. MessinaS. 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.

[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.

[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.

[10]

X. JiM. 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.

[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.

[12]

Y. LiuY. YinK. L. TeoS. 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.

[13]

H. MelkoteF. KhorramiS. 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.

[14]

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

[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.

[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.

[17]

E. TianZ. WangL. 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.

[18]

E. TianZ. WangL. 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.

[19]

B. YaoM. 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.

[20]

Y. YinP. ShiF. LiuK. 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.

[21]

Y. YinZ. LinY. 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.

[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.

[23]

J. YangB. WuS. 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.

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