Computational Simulations on Stability of Model Predictive Control for Linear Discrete-time Stochastic Systems

Model predictive control is a kind of optimal feedback control in which control performance over a finite future is optimized with a performance index that has a moving initial time and a moving terminal time. This paper examines the stability of model predictive control for linear discrete-time systems with additive stochastic disturbances. A sufficient condition for the stability of the closed-loop system with model predictive control is derived by means of a linear matrix inequality. The objective of this paper is to show the results of computational simulations in order to verify the effectiveness of the obtained stability condition.

Authors:

• Tomoaki Hashimoto

Keywords:

• Computational simulations
• optimal control
• predictive control
• stochastic systems
• discrete-time systems.

References:

[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 X. Wu, Probabilistic Constrained MPC
for Multiplicative and Additive Stochastic Uncertainty, IEEE Trans.
Automat. Contr., Vol. 54, 2009, pp.1626-1632.
[18] 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.
[19] 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.
[20] Z. Zhou and R. Cogill, An Algorithm for State Constrained Stochastic
Linear-Quadratic Control, Proceedings of American Control Conference,
2011, pp.1476-1481.
[21] 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.
[22] 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.
[23] B. Øksendal, Stochastic Differential Equations: An Introduction with
Applications, Springer, 6th edition, 2010.
[24] J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and
Design, Nob Hill Publishing, 2009.
[25] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series
in Operation Research and Financial Engineering, Springer, 2006.
[26] L. E. Ghaoui and S. I. Niculescu, Advances in Linear Matrix Inequality
Methods in Control, Society for Industrial and Applied Mathematics,
1987.