Model Predictive Control of Stochastic Linear Systems with Probability Constraints

Abstract

This paper presents a strategy for computing model predictive control of linear Gaussian noise systems with probability constraints. As usual, constraints are taken on the system state and control input. The novelty relies on setting bounds on the underlying cumulative probability distribution, and showing that the model predictive control can be computed in an efficient manner through these novel bounds— an application confirms this assertion. Indeed real-time experiments were carried out to control a direct current (DC) motor. The corresponding data show the effectiveness and usefulness of the approach.

References

[1] Bazaraa, M.S.; Sherali, H.D.; Shetty, C.M. (2006). Nonlinear programming: theory and algorithms, 3rd edn., Wiley-Interscience, New Jersey, 2006.
https://doi.org/10.1002/0471787779

[2] Bernardini, D.; Bemporad, A. (2012). Stabilizing model predictive control of stochastic constrained linear systems, IEEE Trans. Autom. Control, 57, 1468–1480, 2012.
https://doi.org/10.1109/TAC.2011.2176429

[3] Blackmore, L.; Ono, M. (2009). Convex chance constrained predictive control without sampling, In: AIAA Guidance, Navigation and Control Conference, Chicago, Illinois, USA, 1-17, 2009.
https://doi.org/10.2514/6.2009-5876

[4] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory, SIAM, Philadelphia, 1994.
https://doi.org/10.1137/1.9781611970777

[5] Cannon, M.; Kouvaritakis, B.; Rakovic, S.; Cheng Q. (2011). Stochastic tubes in model predictive control with probabilistic constraints, IEEE Trans. Autom. Control, 56, 194–200, 2011.
https://doi.org/10.1109/TAC.2010.2086553

[6] Cao, G.; Lai, E.M.-K.; Alam, F. (2017). Gaussian process model predictive control of unknown non-linear systems, IET Control Theory Appl., 11, 703–713, 2017.
https://doi.org/10.1049/iet-cta.2016.1061

[7] Caruntu, C.F.; Balau, A.E.; Lazar, M.; van den Bosch, P.P.J.; Di Cairano, S. (2016). Driveline oscillations damping: A tractable predictive control solution based on a piecewise affine model, Nonlinear Analysis: Hybrid Systems, 19, 168–185, 2016.
https://doi.org/10.1016/j.nahs.2015.10.001

[8] Costa Junior, A.G.; Riul J.A.; Montenegro, P.H.M. (2016). Application of the subspace identification method using the N4SID technique for a robotic manipulator, IEEE Latin America Transactions, 14, 1588–1993, 2016.
https://doi.org/10.1109/TLA.2016.7483487

[9] Farina, M.; Giulioni, L.; Scattolini, R. (2016). Stochastic linear model predictive control with chance constraints-A review, Journal of Process Control, 44, 53–67, 2016.
https://doi.org/10.1016/j.jprocont.2016.03.005

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

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

[12] Hashorva, E.; Hüsler, J. (2003). On multivariate Gaussian tails, Annals of the Institute of Statistical Mathematics, 55, 507–522, 2003.
https://doi.org/10.1007/BF02517804

[13] Katayama, T. (2005). Subspace methods for system identification, communications and control engineering, Springer-Verlag, London, 2005.
https://doi.org/10.1007/1-84628-158-X

[14] Kwon, W.H., Han, S.H. (2005). Receding horizon control: model predictive control for state models, Springer-Verlag, New York, 2005.

[15] Li, P.; Wendt, M.; Wozny, G. (2002). A probabilistically constrained model predictive controller, Automatica, 38, 1171–1176, 2002.
https://doi.org/10.1016/S0005-1098(02)00002-X

[16] Li, J.W.; Li, D.W.; Xi, Y.G. (2017). H1 predictive control with probability constraints for linear stochastic systems, IET Control Theory Appl., 11, 557–566, 2017.
https://doi.org/10.1049/iet-cta.2016.0915

[17] Lu, D.; Li, W.V. (2009). A note on multivariate Gaussian estimates, Journal of Mathematical Analysis and Applications,354, 704–707, 2009.
https://doi.org/10.1016/j.jmaa.2009.01.046

[18] Oliveira, R.C.L.; Vargas, A.N.; do Val, J.B.R.; Peres, P.L.D. (2014). Mode-independent H2-control of a DC motor modeled as a Markov jump linear system, IEEE Transactions on Control Systems Technology, 22, 1915–1919, 2014.
https://doi.org/10.1109/TCST.2013.2293627

[19] Rubagotti, M.; Patrinos, P.; Guiggiani, A.; Bemporad, A. (2016). Real-time model predictive control based on dual gradient projection: theory and fixed-point FPGA implementation, Int. J. Robust Nonlinear Control, 26, 3292–3310, 2016.
https://doi.org/10.1002/rnc.3507

[20] Schwarm, A.T.; Nikolaou, M. (1999). Chance-constrained model predictive control, AIChE Journal, 45, 1743–1752, 1999.
https://doi.org/10.1002/aic.690450811

[21] Vargas, A.N.; Costa, E.F.; do Val, J.B.R. (2013). On the control of Markov jump linear systems with no mode observation: application to a DC motor device, Int. J. Robust Nonlinear Control, 23, 1136–1950, 2013.
https://doi.org/10.1002/rnc.2911

[22] Vargas, A. N.; do Val, J.B.R. (2010). Average cost and stability of time-varying linear systems, IEEE Trans. Autom. Control, 55, 714–720, 2010.
https://doi.org/10.1109/TAC.2010.2040423

[23] Wang, S.; Yu, M.; Sun, X. (2015). Robust H1 control for time-delay networked control systems with probability constraints, IET Control Theory Appl., 9, 482–2489, 2015.
https://doi.org/10.1049/iet-cta.2015.0143

[24] Yang, H.; Guo, M.C.; Xia, Y.; Cheng, L. (2018). Trajectory tracking for wheeled mobile robots via model predictive control with softening constraints, IET Control Theory Appl., 12, 206–214, 2018.
https://doi.org/10.1049/iet-cta.2017.0395

[25] Yan, J.; Bitmead, R.R. (2005). Incorporating state estimation into model predictive control and its application to network traffic control, Automatica, 41, 595–604, 2005.
https://doi.org/10.1016/j.automatica.2004.11.022

[26] Zeilinger, M.N.; Raimondo, D.M.; Domahidi, A.; Morari, M.; Jones, C.N. (2014). Flocking of multi-agents with a virtual leader, Automatica, 50, 683–694, 2014.
https://doi.org/10.1016/j.automatica.2013.11.019
Published
2018-11-29
How to Cite
CARUNTU, Constantin F. et al. Model Predictive Control of Stochastic Linear Systems with Probability Constraints. INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL, [S.l.], v. 13, n. 6, p. 927-937, nov. 2018. ISSN 1841-9844. Available at: <http://univagora.ro/jour/index.php/ijccc/article/view/3383>. Date accessed: 16 july 2020. doi: https://doi.org/10.15837/ijccc.2018.6.3383.

Keywords

probability constraints, stochastic systems, linear systems, control