# American Institute of Mathematical Sciences

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

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

 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.

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 & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
##### References:
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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] M. Grantand and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, (2014). Retrieved from: http://cvxr.com/cvx. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. on Optim., 17 (2006), 969-996.  doi: 10.1137/050622328.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] M. Y. Shin, Compution in constrained stochanstic model perdictive control of linear systems, Ph.D dissertation, Stanford University in California, 2011. Google Scholar [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.  Google Scholar [33] D. P. Tesi, MS Thesis, Ph.D thesis, University of Pavia in Italy, 2009. Google Scholar

show all references

##### References:
 [1] A. Bental and M. Teboulle, Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming. Mgmt. Science, (2011). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] M. Grantand and S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1, (2014). Retrieved from: http://cvxr.com/cvx. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. on Optim., 17 (2006), 969-996.  doi: 10.1137/050622328.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] M. Y. Shin, Compution in constrained stochanstic model perdictive control of linear systems, Ph.D dissertation, Stanford University in California, 2011. Google Scholar [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.  Google Scholar [33] D. P. Tesi, MS Thesis, Ph.D thesis, University of Pavia in Italy, 2009. Google Scholar
Optimal trajectories from (-5, 3) to (0, 0)
Violated results of 100,000 sample trajectories
100 sample trajectories of SMPC implementation
Simulation results of the proposed algorithm 2 with different violation probabilities
Simulation results of the proposed algorithm and RMPC
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