Conservativeness of Probabilistic Constrained Optimal Control Method for Unknown Probability Distribution
In recent decades, probabilistic constrained optimal
control problems have attracted much attention in many research
fields. Although probabilistic constraints are generally intractable
in an optimization problem, several tractable methods haven been
proposed to handle probabilistic constraints. In most methods,
probabilistic constraints are reduced to deterministic constraints
that are tractable in an optimization problem. However, there is a
gap between the transformed deterministic constraints in case of
known and unknown probability distribution. This paper examines
the conservativeness of probabilistic constrained optimization method
for unknown probability distribution. The objective of this paper is
to provide a quantitative assessment of the conservatism for tractable
constraints in probabilistic constrained optimization with unknown
probability distribution.
[1] D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert,
Constrained Model Predictive Control: Stability and Optimality,
Automatica, Vol. 36, 2000, pp.789-814.
[2] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with
Numerical Solution for Thermal Fluid Systems, Proceedings of SICE
Annual Conference, 2012, pp.1298-1303.
[3] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with
Numerical Solution for Spatiotemporal Dynamic Systems, Proceedings
of IEEE Conference on Decision and Control, 2013, pp.2920-2925.
[4] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control
for Hot Strip Mill Cooling Systems, IEEE/ASME Transactions on
Mechatronics, vol. 18, 2013, pp. 998-1005.
[5] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control
With Numerical Solution for Nonlinear Parabolic Partial Differential
Equations, IEEE Transactions on Automatic Control, Vol. 58, 2013, pp.
725-730.
[6] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Receding Horizon Control
for High-Dimensional Burgers ’ Equations with Boundary Control
Inputs, Transactions of the Japan Society for Aeronautical and Space
Sciences, Vol. 56, 2013, pp. 137-144.
[7] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Output Feedback Receding
Horizon Control for Spatiotemporal Dynamic Systems, Proceedings of
Asian Control Conference, 2013.
[8] R. Satoh, T. Hashimoto and T. Ohtsuka, Receding Horizon Control for
Mass Transport Phenomena in Thermal Fluid Systems, Proceedings of
Australian Control Conference, 2014, pp.273-278.
[9] T. Hashimoto, Receding Horizon Control for a Class of Discrete-time
Nonlinear Implicit Systems, Proceedings of IEEE Conference on
Decision and Control, 2014, pp.5089-5094.
[10] M. V. Kothare, V. Balakrishnan and M. Morari, Robust Constrained
Model Predictive Control Using Linear Matrix Inequalities, Automatica,
Vol. 32, 1996, pp.1361-1379.
[11] P. Scokaert and D. Mayne, Min-max Feedback Model Predictive Control
for Constrained Linear Systems, IEEE Trans. Automat. Contr., Vol. 43,
1998, pp.1136-1142.
[12] A. Bemporad, F. Borrelli and M. Morari, Min-max Control of
Constrained Uncertain Discrete-time Linear Systems, IEEE Trans.
Automat. Contr., Vol. 48, 2003, pp.1600-1606.
[13] T. Alamo, D. Pe˜na, D. Limon and E. Camacho, Constrained Min-max
Predictive Control: Modifications of the Objective Function Leading to
Polynomial Complexity, IEEE Trans. Automat. Contr., Vol. 50, 2005,
pp.710-714.
[14] D. Pe˜na, T. Alamo, A. Bemporad and E. Camacho, A Decomposition
Algorithm for Feedback Min-max Model Predictive Control, IEEE
Trans. Automat. Contr., Vol. 51, 2006, pp.1688-1692.
[15] D. Bertsimas and D. B. Brown, Constrained Stochastic LQC: A
Tractable Approach, IEEE Trans. Automat. Contr., Vol. 52, 2007,
pp.1826-1841.
[16] P. Hokayema, E. Cinquemani, D. Chatterjee, F Ramponid and J.
Lygeros, Stochastic Receding Horizon Control with Output Feedback
and Bounded Controls, Automatica, Vol. 48, 2012, pp.77-88.
[17] M. Cannon, B. Kouvaritakis and D. Ng, Probabilistic Tubes in Linear
Stochastic Model Predictive Control, Systems & Control Letters, Vol.
58, 2009, pp.747-753.
[18] M. Cannon, B. Kouvaritakis and X. Wu, Model Predictive Control for
Systems with Stochastic Multiplicative Uncertainty and Probabilistic
Constraints, Automatica, Vol. 45, 2009, pp.167-172.
[19] M. Cannon, B. Kouvaritakis and X. Wu, Probabilistic Constrained MPC
for Multiplicative and Additive Stochastic Uncertainty, IEEE Trans.
Automat. Contr., Vol. 54, 2009, pp.1626-1632.
[20] E. Cinquemani, M. Agarwal, D. Chatterjee and J. Lygeros, Convexity
and Convex Approximations of Discrete-time Stochastic Control
Problems with Constraints, Automatica, Vol. 47, 2011, pp.2082-2087.
[21] M. Ono, L. Blackmore and B. C. Williams, Chance Constrained Finite
Horizon Optimal Control with Nonconvex Constraints, Proceedings of
American Control Conference, 2010, pp.1145-1152.
[22] G. C. Calafiore and M. C. Campi, The Scenario Approach to Robust
Control Design, IEEE Trans. Automat. Contr., Vol. 51, 2006, pp.742-753.
[23] J. Matuˇsko and F. Borrelli, Scenario-Based Approach to Stochastic
Linear Predictive Control, Proceedings of the 51st IEEE Conference
on Decision and Control, 2012, pp.5194-5199.
[24] A. Nemirovski and A. Shapiro, Convex Approximations of Chance
Constrained Programs, SIAM J. Control Optim., Vol. 17, 2006,
pp.969-996.
[25] Z. Zhou and R. Cogill, An Algorithm for State Constrained Stochastic
Linear-Quadratic Control, Proceedings of American Control Conference,
2011, pp.1476-1481.
[26] T. Hashimoto, I. Yoshimoto, T. Ohtsuka, Probabilistic Constrained
Model Predictive Control for Schr¨odinger Equation with Finite
Approximation, Proceedings of SICE Annual Conference, 2012,
pp.1613-1618.
[27] T. Hashimoto, Probabilistic Constrained Model Predictive Control for
Linear Discrete-time Systems with Additive Stochastic Disturbances,
Proceedings of IEEE Conference on Decision and Control, 2013,
pp.6434-6439.
[28] T. Hashimoto, Computational Simulations on Stability of Model
Predictive Control for Linear Discrete-time Stochastic Systems,
International Journal of Computer, Electrical, Automation, Control and
Information Engineering, 2015, pp.1385-1390.
[29] M. Farina, L. Giulioni, L. Magni and R. Scattolini, A Probabilistic
Approach to Model Predictive Control, Proceedings of the 52nd IEEE
Conference on Decision and Control, 2013, pp.7734-7739.
[30] B. Øksendal, Stochastic Differential Equations: An Introduction with
Applications, Springer, 6th edition, 2010.
[31] S. Boucheron, G. Lugosi and P. Massart Concentration Inequalities: A
Nonasymptotic Thepry of Independence, Oxford University Press, 2013.
[32] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series
in Operation Research and Financial Engineering, Springer, 2006.
[1] D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert,
Constrained Model Predictive Control: Stability and Optimality,
Automatica, Vol. 36, 2000, pp.789-814.
[2] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with
Numerical Solution for Thermal Fluid Systems, Proceedings of SICE
Annual Conference, 2012, pp.1298-1303.
[3] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with
Numerical Solution for Spatiotemporal Dynamic Systems, Proceedings
of IEEE Conference on Decision and Control, 2013, pp.2920-2925.
[4] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control
for Hot Strip Mill Cooling Systems, IEEE/ASME Transactions on
Mechatronics, vol. 18, 2013, pp. 998-1005.
[5] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control
With Numerical Solution for Nonlinear Parabolic Partial Differential
Equations, IEEE Transactions on Automatic Control, Vol. 58, 2013, pp.
725-730.
[6] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Receding Horizon Control
for High-Dimensional Burgers ’ Equations with Boundary Control
Inputs, Transactions of the Japan Society for Aeronautical and Space
Sciences, Vol. 56, 2013, pp. 137-144.
[7] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Output Feedback Receding
Horizon Control for Spatiotemporal Dynamic Systems, Proceedings of
Asian Control Conference, 2013.
[8] R. Satoh, T. Hashimoto and T. Ohtsuka, Receding Horizon Control for
Mass Transport Phenomena in Thermal Fluid Systems, Proceedings of
Australian Control Conference, 2014, pp.273-278.
[9] T. Hashimoto, Receding Horizon Control for a Class of Discrete-time
Nonlinear Implicit Systems, Proceedings of IEEE Conference on
Decision and Control, 2014, pp.5089-5094.
[10] M. V. Kothare, V. Balakrishnan and M. Morari, Robust Constrained
Model Predictive Control Using Linear Matrix Inequalities, Automatica,
Vol. 32, 1996, pp.1361-1379.
[11] P. Scokaert and D. Mayne, Min-max Feedback Model Predictive Control
for Constrained Linear Systems, IEEE Trans. Automat. Contr., Vol. 43,
1998, pp.1136-1142.
[12] A. Bemporad, F. Borrelli and M. Morari, Min-max Control of
Constrained Uncertain Discrete-time Linear Systems, IEEE Trans.
Automat. Contr., Vol. 48, 2003, pp.1600-1606.
[13] T. Alamo, D. Pe˜na, D. Limon and E. Camacho, Constrained Min-max
Predictive Control: Modifications of the Objective Function Leading to
Polynomial Complexity, IEEE Trans. Automat. Contr., Vol. 50, 2005,
pp.710-714.
[14] D. Pe˜na, T. Alamo, A. Bemporad and E. Camacho, A Decomposition
Algorithm for Feedback Min-max Model Predictive Control, IEEE
Trans. Automat. Contr., Vol. 51, 2006, pp.1688-1692.
[15] D. Bertsimas and D. B. Brown, Constrained Stochastic LQC: A
Tractable Approach, IEEE Trans. Automat. Contr., Vol. 52, 2007,
pp.1826-1841.
[16] P. Hokayema, E. Cinquemani, D. Chatterjee, F Ramponid and J.
Lygeros, Stochastic Receding Horizon Control with Output Feedback
and Bounded Controls, Automatica, Vol. 48, 2012, pp.77-88.
[17] M. Cannon, B. Kouvaritakis and D. Ng, Probabilistic Tubes in Linear
Stochastic Model Predictive Control, Systems & Control Letters, Vol.
58, 2009, pp.747-753.
[18] M. Cannon, B. Kouvaritakis and X. Wu, Model Predictive Control for
Systems with Stochastic Multiplicative Uncertainty and Probabilistic
Constraints, Automatica, Vol. 45, 2009, pp.167-172.
[19] M. Cannon, B. Kouvaritakis and X. Wu, Probabilistic Constrained MPC
for Multiplicative and Additive Stochastic Uncertainty, IEEE Trans.
Automat. Contr., Vol. 54, 2009, pp.1626-1632.
[20] E. Cinquemani, M. Agarwal, D. Chatterjee and J. Lygeros, Convexity
and Convex Approximations of Discrete-time Stochastic Control
Problems with Constraints, Automatica, Vol. 47, 2011, pp.2082-2087.
[21] M. Ono, L. Blackmore and B. C. Williams, Chance Constrained Finite
Horizon Optimal Control with Nonconvex Constraints, Proceedings of
American Control Conference, 2010, pp.1145-1152.
[22] G. C. Calafiore and M. C. Campi, The Scenario Approach to Robust
Control Design, IEEE Trans. Automat. Contr., Vol. 51, 2006, pp.742-753.
[23] J. Matuˇsko and F. Borrelli, Scenario-Based Approach to Stochastic
Linear Predictive Control, Proceedings of the 51st IEEE Conference
on Decision and Control, 2012, pp.5194-5199.
[24] A. Nemirovski and A. Shapiro, Convex Approximations of Chance
Constrained Programs, SIAM J. Control Optim., Vol. 17, 2006,
pp.969-996.
[25] Z. Zhou and R. Cogill, An Algorithm for State Constrained Stochastic
Linear-Quadratic Control, Proceedings of American Control Conference,
2011, pp.1476-1481.
[26] T. Hashimoto, I. Yoshimoto, T. Ohtsuka, Probabilistic Constrained
Model Predictive Control for Schr¨odinger Equation with Finite
Approximation, Proceedings of SICE Annual Conference, 2012,
pp.1613-1618.
[27] T. Hashimoto, Probabilistic Constrained Model Predictive Control for
Linear Discrete-time Systems with Additive Stochastic Disturbances,
Proceedings of IEEE Conference on Decision and Control, 2013,
pp.6434-6439.
[28] T. Hashimoto, Computational Simulations on Stability of Model
Predictive Control for Linear Discrete-time Stochastic Systems,
International Journal of Computer, Electrical, Automation, Control and
Information Engineering, 2015, pp.1385-1390.
[29] M. Farina, L. Giulioni, L. Magni and R. Scattolini, A Probabilistic
Approach to Model Predictive Control, Proceedings of the 52nd IEEE
Conference on Decision and Control, 2013, pp.7734-7739.
[30] B. Øksendal, Stochastic Differential Equations: An Introduction with
Applications, Springer, 6th edition, 2010.
[31] S. Boucheron, G. Lugosi and P. Massart Concentration Inequalities: A
Nonasymptotic Thepry of Independence, Oxford University Press, 2013.
[32] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series
in Operation Research and Financial Engineering, Springer, 2006.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:70874", author = "Tomoaki Hashimoto", title = "Conservativeness of Probabilistic Constrained Optimal Control Method for Unknown Probability Distribution", abstract = "In recent decades, probabilistic constrained optimal
control problems have attracted much attention in many research
fields. Although probabilistic constraints are generally intractable
in an optimization problem, several tractable methods haven been
proposed to handle probabilistic constraints. In most methods,
probabilistic constraints are reduced to deterministic constraints
that are tractable in an optimization problem. However, there is a
gap between the transformed deterministic constraints in case of
known and unknown probability distribution. This paper examines
the conservativeness of probabilistic constrained optimization method
for unknown probability distribution. The objective of this paper is
to provide a quantitative assessment of the conservatism for tractable
constraints in probabilistic constrained optimization with unknown
probability distribution.", keywords = "Optimal control, stochastic systems, discrete-time
systems, probabilistic constraints.", volume = "9", number = "9", pages = "523-5", }
