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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 |
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.
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. |
[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. 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. |
[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. 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.
|
[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. 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. |
[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. |
[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. 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. |
[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. 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.
|
[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. 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. |
[33] |
D. P. Tesi, MS Thesis, Ph.D thesis, University of Pavia in Italy, 2009. Google Scholar |





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