# EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities

### Sorin Olaru, Didier Dumur, Simona Dobre

#### Abstract

This paper proposes a geometrical analysis of the polyhedral feasible domains for the predictive control laws under constraints. The fact that the system dynamics influence the topology of such polyhedral domains is well known from the studies dedicated to the feasibility of the control laws. Formally the system state acts as a vector of parameters for the optimization problem to be solved on-line and its influence can be fully described by the use of parameterized polyhedra and their dual constraints/generators representation. Problems like the constraints redundancy or the construction of the associated explicit control laws at least for linear or quadratic cost functions can thus receive fully geometrical solutions. Convex nonlinear constraints can be approximated using a description based on the parameterized vertices. In the case of nonconvex regions the explicit solutions can be obtain by constructing Voronoi partitions based on a collection of points distributed over the borders of the feasible domain.

#### References

- Bemporad, A., Borrelli, F., and Morari, M. (2001). Robust Model Predictive Control: Piecewise Linear Explicit Solution. In European Control Conference, pages 939-944, Porto, Portugal.
- Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E. (2002). The Explicit Linear Quadratic Regulator for Constrained Systems. Automatica, 38(1):3-20.
- Borelli, F. (2003). Constrained Optimal Control of Linear and Hybrid Systems. Springer-Verlag, Berlin.
- Goodwin, G., Seron, M., and Dona, J. D. (2004). Constrained Control and Estimation. Springer, Berlin.
- Kerrigan, E. and Maciejowski, J. (2004). Feedback minmax model predictive control using a single linear program: Robust stability and the explicit solution. International Journal of Robust and Nonlinear Control, 14(4):395-413.
- Leverge, H. (1994). A note on chernikova's algorithm. In Technical Report 635. IRISA, France.
- Loechner, V. and Wilde, D. K. (1997). Parameterized polyhedra and their vertices. International Journal of Parallel Programming, V25(6):525-549.
- Papers, (1983). Birkhauser.
- Olaru, S. and Dumur, D. (2005). Compact explicit mpc with guarantee of feasibility for tracking. In 44th IEEE Conference on Decision and Control, and European Control Conference. CDC-ECC 7805., pages 969-974.
- Olaru, S. B. and Dumur, D. A parameterized polyhedra approach for explicit constrained predictive control. In 43rd IEEE Conference on Decision and Control, 2004. CDC., pages 1580-1585 Vol.2.
- Schrijver, A. (1986). Theory of Linear and Integer Programming. John Wiley and Sons, NY.
- Scokaert, P. O., Mayne, D. Q., and Rawlings, J. B. (1999). Suboptimal model predictive control (feasibility implies stability). In IEEE Transactions on Automatic Control, volume 44, pages 648-654.
- Seron, M., Goodwin, G., and Dona, J. D. (2003). Characterisation of receding horizon control for constrained linear systems. In Asian Journal of Control, volume 5, pages 271-286.
- Tondel, P., Johansen, T., and Bemporad, A. (2003). Evaluation of piecewise affine control via binary search tree. Automatica, 39:945-950.
- Wilde, D. (1993). A library for doing polyhedral operations. In Technical report 785. IRISA, France.

#### Paper Citation

#### in Harvard Style

Olaru S., Dumur D. and Dobre S. (2007). **EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities** . In *Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO,* ISBN 978-972-8865-84-9, pages 70-77. DOI: 10.5220/0001633200700077

#### in Bibtex Style

@conference{icinco07,

author={Sorin Olaru and Didier Dumur and Simona Dobre},

title={EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities},

booktitle={Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO,},

year={2007},

pages={70-77},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001633200700077},

isbn={978-972-8865-84-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics - Volume 3: ICINCO,

TI - EXPLICIT PREDICTIVE CONTROL LAWS - On the Geometry of Feasible Domains and the Presence of Nonlinearities

SN - 978-972-8865-84-9

AU - Olaru S.

AU - Dumur D.

AU - Dobre S.

PY - 2007

SP - 70

EP - 77

DO - 10.5220/0001633200700077