American Institute of Mathematical Sciences

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January  2021, 17(1): 67-79. doi: 10.3934/jimo.2019099

A chance-constrained stochastic model predictive control problem with disturbance feedback

Yuan Tan 1, , Qingyuan Cao 2, , Lan Li 3, , Tianshi Hu 4,, and  Min Su 5,

1. 

College of Electrical and Information Technology, Sichuan University, Chengdu, China
2. 

Business School, The University of Edinburgh, Edinburgh, UK
3. 

School of Electronic Engineering, Chengdu University of Information Technology, Chengdu, China
4. 

Inovation and Entrepreneurship College, Xihua University, Chengdu, China
5. 

College of Electrical and Information Technology, Sichuan University, Chengdu, China

* Corresponding author: Tianshi Hu

Received  February 2019 Revised  March 2019 Published  January 2021 Early access  July 2019

Fund Project: This work was partially supported by a grant from National Natural Science Foundation of China under number 61701124, a grant from Science and Technology on Space Intelligent Control Laboratory, No. KGJZDSYS-2018-03, a grant from Sichuan Province Government under the application number 2019YJ0105, and a grant from Fundamental Research Funds for the Central Universities (China)

In this paper, we develop two algorithms for stochastic model predictive control (SMPC) problems with discrete linear systems. Participially, chance constraints on the state and control are considered. Different from the state-of-the-art robust model predictive control (RMPC) algorithm, the proposed is less conservative. Meanwhile, the proposed algorithms do not assume the full knowledge of the disturbance distribution. It only requires the mean and variance of the disturbance. Rigorous computational analysis is carried out for the proposed algorithms. Numerical results are provided to demonstrate the effectiveness and the superior of the proposed SMPC algorithms.

Keywords: Stochastic model predictive control, chance constraint, robust optimization.
Mathematics Subject Classification: 91B70.
Citation: Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial and Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
References:
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A. Bental and M. Teboulle, Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming. Mgmt. Science, (2011).

[2]

D. Bernardini and A. Bemporad, Scenario-based model predictive control of stochastic constrained linear systems, IEEE Conference on Decision & Control, IEEE, (2009). doi: 10.1109/CDC.2009.5399917.

[3]

G. C. Calafiore and L. Fagiano, Robust model predictive control via scenario optimization, IEEE Transactions on Automatic Control, 58 (2013), 219-224. doi: 10.1109/TAC.2012.2203054.

[4]

M. Cannon, B. Kouvaritakis and D. Ng, Probabilistic tubes in linear stochastic model predictive control, Systems & Control Letters, 58 (2009), 747-753.  doi: 10.1016/j.sysconle.2009.08.004.

[5]

W. Chen, M. Sim, J. Sun and C.-P. Teo, From CVaR to uncertainty set: Implications in joint chance-constrained optimization, Ops. Research, 58 (2010), 470-485.  doi: 10.1287/opre.1090.0712.

[6]

E. Cinquemani, M. Agarwal, D. Chatterjee and J. Lygeros, Convexity and convex approximations of discrete-time stochastic control problems with constraints, Automatica, 47 (2011), 2082-2087.  doi: 10.1016/j.automatica.2011.01.023.

[7]

M. Farina, L. Giulioni and L. Magni, A probabilistic approach to model predictive control, in 52nd IEEE Conference on Decision and Control, IEEE, (2013). doi: 10.1109/CDC.2013.6761117.

[8]

M. Farina, L. Giulioni, L. Magni and R. Scattolini, An approach to output-feedback MPC of stochastic linear discrete-time systems, Automatica, 55 (2015), 140-149.  doi: 10.1016/j.automatica.2015.02.039.

[9]

M. Farina, L. Giulioni and R. Scattolini, Stochastic linear Model Predictive Control with chance constraints - A review, J. of Process Control, 44 (2016), 53-67.  doi: 10.1016/j.jprocont.2016.03.005.

[10]

M. Farina and R. Scattolini, Model predictive control of linear systems with multiplicative unbounded uncertainty and chance constraints, Automatica, 70 (2016), 258-265.  doi: 10.1016/j.automatica.2016.04.008.

[11]

Z. H. Gong, C. Y. Liu, K. L. Teo and J. Sun, Distributionally robust parameter identification of a time-delay dynamical system with stochastic measurements, Appl. Math. Modelling, 69 (2019), 685-695.  doi: 10.1016/j.apm.2018.09.040.

[12]

Z. H. Gong, C. Y. Liu, J. Sun and K. L. Teo, Distributional robust $L_1$-estimation in multiple linear regression, Optim. Letters, 13 (2019), 935-947.  doi: 10.1007/s11590-018-1299-x.

[13]

M. Grantand and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, (2014). Retrieved from: http://cvxr.com/cvx.

[14]

P. Hokayem, D. Chatterjee and J. Lygeros, On stochastic receding horizon control with bounded control inputs: A vector space approach, IEE Trans. on Automat. Control, 56 (2011), 2704-2710.  doi: 10.1109/TAC.2011.2159422.

[15]

P. Hokayem, E. Cinquemani, D. Chatterjee, F. Ramponi and J. Lygeros, Stochastic receding horizon control with output feedback and bounded controls, Automatica, 48 (2012), 77-88.  doi: 10.1016/j.automatica.2011.09.048.

[16]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Trans. on Wireless Comms., 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.

[17]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109.  doi: 10.1109/LSP.2017.2773601.

[18]

B. Li, J. Sun, K. L. Teo, C. J. Yu and M. Zhang, A distributionally robust approach to a class of three-stage stochastic linear programs. Pacific J. of Optim., 15 (2019), 219-236.

[19]

B. Li, J. Sun, H. L. Xu and M. Zhang, A class of two-stage distributionally robust stochastic games, J. of Indust. and Mgmt. Optim., 15 (2019), 387-400. 

[20]

B. Li, Q. Xun, J. Sun, K. L. Teo and C. J. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Appl. Math. Modelling, 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039.

[21]

M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Algebra and its Appl., 284 (1998), 193-228.  doi: 10.1016/S0024-3795(98)10032-0.

[22]

L. Magni, G. D. Nicolao and R. Scattolini, Robust model predictive control for nonlinear discrete-time systems, Int. J. of Robust & Nonlinear Control, 13 (2003), 229-246.  doi: 10.1002/rnc.815.

[23]

D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814. doi: 10.1016/S0005-1098(99)00214-9.

[24]

D. Q. Mayne, M. M. Seron and S. V. Raković, Robust model predictive control of constrained linear systems with bounded disturbances, Automatica, 41 (2005), 219-224.  doi: 10.1016/j.automatica.2004.08.019.

[25]

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

[26]

J. A. Paulson, E. A. Buehler, R. D. Braatz and A. Mesbah, Stochastic model predictive control with joint chance constraints, Int. J. of Control, (2017), 1–14. doi: 10.1080/00207179.2017.1323351.

[27]

S. Qu, Y. Zhou, Y. Zhang, M. I. M. Wahab, G. Zhang and Y. Ye, Optimal strategy for a green supply chain considering shipping policy and default risk, Comp. & Indust. Engineering, 131 (2019), 172-186.  doi: 10.1016/j.cie.2019.03.042.

[28]

D. M. Raimondo, D. Limon and M. Lazar, Min-max model predictive control of nonlinear systems: A unifying overview on stability, European J. of Control, 15 (2009), 5-21. doi: 10.3166/ejc.15.5-21.

[29]

D. R. Ramírez, T. Alamo and E. F. Camacho, Min-Max MPC based on a computationally efficient upper bound of the worst case cost, J. of Process Control, 16 (2006), 511-519.  doi: 10.1016/j.jprocont.2005.07.005.

[30]

G. Schildbach, P. Goulart and M. Morari, Linear controller design for chance constrained systems, Automatica, 51 (2015), 278-284. doi: 10.1016/j.automatica.2014.10.096.

[31]

M. Y. Shin, Compution in constrained stochanstic model perdictive control of linear systems, Ph.D dissertation, Stanford University in California, 2011.

[32]

Y. F. Sun, G. Aw, B. Li, K. L. Teo and J. Sun., CVaR-based robust models for portfolio selection. Journal of Industrial and Management Optimization, 2018. doi: 10.3934/jimo.2019032.

[33]

D. P. Tesi, MS Thesis, Ph.D thesis, University of Pavia in Italy, 2009.

show all references

References:
[1]

A. Bental and M. Teboulle, Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming. Mgmt. Science, (2011).

[2]

D. Bernardini and A. Bemporad, Scenario-based model predictive control of stochastic constrained linear systems, IEEE Conference on Decision & Control, IEEE, (2009). doi: 10.1109/CDC.2009.5399917.

[3]

G. C. Calafiore and L. Fagiano, Robust model predictive control via scenario optimization, IEEE Transactions on Automatic Control, 58 (2013), 219-224. doi: 10.1109/TAC.2012.2203054.

[4]

M. Cannon, B. Kouvaritakis and D. Ng, Probabilistic tubes in linear stochastic model predictive control, Systems & Control Letters, 58 (2009), 747-753.  doi: 10.1016/j.sysconle.2009.08.004.

[5]

W. Chen, M. Sim, J. Sun and C.-P. Teo, From CVaR to uncertainty set: Implications in joint chance-constrained optimization, Ops. Research, 58 (2010), 470-485.  doi: 10.1287/opre.1090.0712.

[6]

E. Cinquemani, M. Agarwal, D. Chatterjee and J. Lygeros, Convexity and convex approximations of discrete-time stochastic control problems with constraints, Automatica, 47 (2011), 2082-2087.  doi: 10.1016/j.automatica.2011.01.023.

[7]

M. Farina, L. Giulioni and L. Magni, A probabilistic approach to model predictive control, in 52nd IEEE Conference on Decision and Control, IEEE, (2013). doi: 10.1109/CDC.2013.6761117.

[8]

M. Farina, L. Giulioni, L. Magni and R. Scattolini, An approach to output-feedback MPC of stochastic linear discrete-time systems, Automatica, 55 (2015), 140-149.  doi: 10.1016/j.automatica.2015.02.039.

[9]

M. Farina, L. Giulioni and R. Scattolini, Stochastic linear Model Predictive Control with chance constraints - A review, J. of Process Control, 44 (2016), 53-67.  doi: 10.1016/j.jprocont.2016.03.005.

[10]

M. Farina and R. Scattolini, Model predictive control of linear systems with multiplicative unbounded uncertainty and chance constraints, Automatica, 70 (2016), 258-265.  doi: 10.1016/j.automatica.2016.04.008.

[11]

Z. H. Gong, C. Y. Liu, K. L. Teo and J. Sun, Distributionally robust parameter identification of a time-delay dynamical system with stochastic measurements, Appl. Math. Modelling, 69 (2019), 685-695.  doi: 10.1016/j.apm.2018.09.040.

[12]

Z. H. Gong, C. Y. Liu, J. Sun and K. L. Teo, Distributional robust $L_1$-estimation in multiple linear regression, Optim. Letters, 13 (2019), 935-947.  doi: 10.1007/s11590-018-1299-x.

[13]

M. Grantand and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, (2014). Retrieved from: http://cvxr.com/cvx.

[14]

P. Hokayem, D. Chatterjee and J. Lygeros, On stochastic receding horizon control with bounded control inputs: A vector space approach, IEE Trans. on Automat. Control, 56 (2011), 2704-2710.  doi: 10.1109/TAC.2011.2159422.

[15]

P. Hokayem, E. Cinquemani, D. Chatterjee, F. Ramponi and J. Lygeros, Stochastic receding horizon control with output feedback and bounded controls, Automatica, 48 (2012), 77-88.  doi: 10.1016/j.automatica.2011.09.048.

[16]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust linear receiver design for multi-access space-time block coded MIMO systems, IEEE Trans. on Wireless Comms., 16 (2017), 464-474.  doi: 10.1109/TWC.2016.2625246.

[17]

B. Li, Y. Rong, J. Sun and K. L. Teo, A distributionally robust minimum variance beamformer design, IEEE Signal Processing Letters, 25 (2018), 105-109.  doi: 10.1109/LSP.2017.2773601.

[18]

B. Li, J. Sun, K. L. Teo, C. J. Yu and M. Zhang, A distributionally robust approach to a class of three-stage stochastic linear programs. Pacific J. of Optim., 15 (2019), 219-236.

[19]

B. Li, J. Sun, H. L. Xu and M. Zhang, A class of two-stage distributionally robust stochastic games, J. of Indust. and Mgmt. Optim., 15 (2019), 387-400. 

[20]

B. Li, Q. Xun, J. Sun, K. L. Teo and C. J. Yu, A model of distributionally robust two-stage stochastic convex programming with linear recourse, Appl. Math. Modelling, 58 (2018), 86-97.  doi: 10.1016/j.apm.2017.11.039.

[21]

M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Algebra and its Appl., 284 (1998), 193-228.  doi: 10.1016/S0024-3795(98)10032-0.

[22]

L. Magni, G. D. Nicolao and R. Scattolini, Robust model predictive control for nonlinear discrete-time systems, Int. J. of Robust & Nonlinear Control, 13 (2003), 229-246.  doi: 10.1002/rnc.815.

[23]

D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert, Constrained model predictive control: Stability and optimality, Automatica, 36 (2000), 789-814. doi: 10.1016/S0005-1098(99)00214-9.

[24]

D. Q. Mayne, M. M. Seron and S. V. Raković, Robust model predictive control of constrained linear systems with bounded disturbances, Automatica, 41 (2005), 219-224.  doi: 10.1016/j.automatica.2004.08.019.

[25]

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

[26]

J. A. Paulson, E. A. Buehler, R. D. Braatz and A. Mesbah, Stochastic model predictive control with joint chance constraints, Int. J. of Control, (2017), 1–14. doi: 10.1080/00207179.2017.1323351.

[27]

S. Qu, Y. Zhou, Y. Zhang, M. I. M. Wahab, G. Zhang and Y. Ye, Optimal strategy for a green supply chain considering shipping policy and default risk, Comp. & Indust. Engineering, 131 (2019), 172-186.  doi: 10.1016/j.cie.2019.03.042.

[28]

D. M. Raimondo, D. Limon and M. Lazar, Min-max model predictive control of nonlinear systems: A unifying overview on stability, European J. of Control, 15 (2009), 5-21. doi: 10.3166/ejc.15.5-21.

[29]

D. R. Ramírez, T. Alamo and E. F. Camacho, Min-Max MPC based on a computationally efficient upper bound of the worst case cost, J. of Process Control, 16 (2006), 511-519.  doi: 10.1016/j.jprocont.2005.07.005.

[30]

G. Schildbach, P. Goulart and M. Morari, Linear controller design for chance constrained systems, Automatica, 51 (2015), 278-284. doi: 10.1016/j.automatica.2014.10.096.

[31]

M. Y. Shin, Compution in constrained stochanstic model perdictive control of linear systems, Ph.D dissertation, Stanford University in California, 2011.

[32]

Y. F. Sun, G. Aw, B. Li, K. L. Teo and J. Sun., CVaR-based robust models for portfolio selection. Journal of Industrial and Management Optimization, 2018. doi: 10.3934/jimo.2019032.

[33]

D. P. Tesi, MS Thesis, Ph.D thesis, University of Pavia in Italy, 2009.

Figure 1.  Optimal trajectories from (-5, 3) to (0, 0)
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Figure 2.  Violated results of 100,000 sample trajectories
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Figure 3.  100 sample trajectories of SMPC implementation
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Figure 4.  Simulation results of the proposed algorithm 2 with different violation probabilities
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Figure 5.  Simulation results of the proposed algorithm and RMPC
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