L. Explicit-robustness-margin, 40 3.3.2 Construction based on the vertex representation, p.45

B. Barmish and G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching assumptions. Automatic Control, IEEE Transactions on, vol.27, issue.1, pp.153-158, 1982.

A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems, Automatica, vol.38, issue.1, pp.3-20, 2002.
DOI : 10.1016/S0005-1098(01)00174-1

A. Bemporad, F. Borrelli, and M. Morari, Min-max control of constrained uncertain discrete-time linear systems, IEEE Transactions on Automatic Control, vol.48, issue.9, pp.1600-1606, 2003.
DOI : 10.1109/TAC.2003.816984

H. Benlaoukli and S. Olaru, Model Predictive Control ??? Numerical Methods for the Invariant Sets Approximation, Numerical Analysis and Its Applications, pp.192-199, 2009.
DOI : 10.1109/TAC.2005.843854

URL : https://hal.archives-ouvertes.fr/hal-00431013

H. Benlaoukli, M. Hovd, S. Olaru, and P. Boucher, On the construction of invariant sets for piecewise affine systems using the transition graph, 2009 IEEE International Conference on Control and Automation, pp.122-127, 2009.
DOI : 10.1109/ICCA.2009.5410557

URL : https://hal.archives-ouvertes.fr/hal-00418327

G. Bitsoris, On the positive invariance of polyhedral sets for discrete-time systems, Systems & Control Letters, vol.11, issue.3, pp.243-248, 1988.
DOI : 10.1016/0167-6911(88)90065-5

G. Bitsoris, Positively invariant polyhedral sets of discrete-time linear systems, International Journal of Control, vol.82, issue.6, pp.1713-1726, 1988.
DOI : 10.1080/00207178808906132

G. Bitsoris and M. Vassilaki, Constrained regulation of linear systems, Automatica, vol.31, issue.2, pp.223-227, 1995.
DOI : 10.1016/0005-1098(94)E0053-K

F. Blanchini, Ultimate boundedness control for uncertain discrete-time systems via set-induced lyapunov functions. Automatic Control, IEEE Transactions on, vol.39, issue.2, pp.428-433, 1994.

F. Blanchini, Set invariance in control, Automatica, vol.35, issue.11, pp.1747-1767, 1999.
DOI : 10.1016/S0005-1098(99)00113-2

F. Blanchini and S. Miani, Set-theoretic methods in control, 2008.
DOI : 10.1007/978-3-319-17933-9

S. Boyd and L. Vandenberghe, Convex optimization, 2004.

S. P. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory, SIAM, vol.15, 1994.
DOI : 10.1137/1.9781611970777

T. F. Chan, Rank revealing QR factorizations, Linear Algebra and its Applications, vol.88, issue.89, pp.67-82, 1987.
DOI : 10.1016/0024-3795(87)90103-0

V. Chmielewski and . Manousiouthakis, On constrained infinite-time linear quadratic optimal control, Decision and Control Proceedings of the 35th IEEE Conference on, pp.1319-1324, 1996.

H. Crapo and W. Whiteley, Plane self stresses and projected polyhedra 1: the basic pattern, Structural Topology, vol.19, pp.55-73, 1993.

H. Crapo and W. Whiteley, Spaces of stresses, projections and parallel drawings for spherical polyhedra. Contributions to Algebra and Geometry, pp.259-281, 1994.

J. Daafouz and J. Bernussou, Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties, Systems & Control Letters, vol.43, issue.5, pp.355-359, 2001.
DOI : 10.1016/S0167-6911(01)00118-9

J. R. De-loera, S. Francisco, and A. Jesús, Triangulations. structures for algorithms and applications, Algorithms and Computation in Mathematics, vol.25, 2010.

P. Dorato, Non-fragile controller design: an overview, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), pp.2829-2831, 1998.
DOI : 10.1109/ACC.1998.688371

H. Edelsbrunner and R. Seidel, Voronoi diagrams and arrangements, Discrete Comput. Geom, 1986.

C. Feller, T. A. Johansen, and S. Olaru, An improved algorithm for combinatorial multi-parametric quadratic programming, Automatica, vol.49, issue.5, pp.1370-1376, 2013.
DOI : 10.1016/j.automatica.2013.02.022

URL : https://hal.archives-ouvertes.fr/hal-00827224

R. A. Freeman and P. V. Kokotovic, Inverse Optimality in Robust Stabilization, SIAM Journal on Control and Optimization, vol.34, issue.4, pp.1365-1391, 1996.
DOI : 10.1137/S0363012993258732

K. Fukuda, cdd/cdd+ reference manual. Institute for Operations Research, 1997.

T. Gal, Postoptimal analyses, parametric programming and related topics, 1995.
DOI : 10.1515/9783110871203

B. D. George and B. C. Eaves, Fourier-motzkin elimination and its dual, Journal of Combinatorial Theory, Series A, vol.14, issue.3, pp.288-297, 1973.

E. G. Gilbert and K. T. Tan, Linear systems with state and control constraints: the theory and application of maximal output admissible sets, IEEE Transactions on Automatic Control, vol.36, issue.9, pp.1008-1020, 1991.
DOI : 10.1109/9.83532

A. Grancharova and T. A. Johansen, Explicit Nonlinear Model Predictive Control, 2012.
DOI : 10.1007/978-3-642-28780-0

B. Grünbaum, Convex polytopes, 1967.

M. Gulan, N. A. Nguyen, S. Olaru, P. Rodriguez-ayerbe, and B. Rohal-'-ilkiv, Implications of Inverse Parametric Optimization in Model Predictive Control, Developments in Model- Based Optimization and Control, 2015.
DOI : 10.1007/978-3-319-26687-9_3

URL : https://hal.archives-ouvertes.fr/hal-01259959

P. Gutman and M. Cwikel, Admissible sets and feedback control for discretetime linear dynamical systems with bounded controls and states. Automatic Control, IEEE Transactions on, vol.31, issue.4, pp.373-376, 1986.

P. Gutman and M. Cwikel, An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states. Automatic Control, IEEE Transactions on, vol.32, issue.3, pp.251-254, 1987.

D. Hartvigsen, Recognizing Voronoi Diagrams with Linear Programming, ORSA Journal on Computing, vol.4, issue.4, 1992.
DOI : 10.1287/ijoc.4.4.369

W. P. Heemels, B. D. Schutter, and A. Bemporad, Equivalence of hybrid dynamical models, Automatica, vol.37, issue.7, pp.1085-1091, 2001.
DOI : 10.1016/S0005-1098(01)00059-0

A. Hempel, P. Goulart, and J. Lygeros, Inverse Parametric Optimization With an Application to Hybrid System Control, IEEE Transactions on Automatic Control, vol.60, issue.4, pp.1064-1069, 2015.
DOI : 10.1109/TAC.2014.2336992

J. C. Hennet, Discrete Time Constrained Linear Systems, Control and Dynamic Systems, vol.71, pp.157-213, 1995.
DOI : 10.1016/S0090-5267(06)80018-6

M. Herceg, M. Kvasnica, C. Jones, and M. Morari, Multi-Parametric Toolbox 3.0, Proc. of the European Control Conference, pp.502-510, 2013.

M. Hovd and S. Olaru, Relaxing PWQ Lyapunov stability criteria for PWA systems, Automatica, vol.49, issue.2, 2012.
DOI : 10.1016/j.automatica.2012.10.013

URL : https://hal.archives-ouvertes.fr/hal-00794374

M. Hovd and S. Olaru, Piecewise quadratic lyapunov functions for stability verification of approximate explicit mpc. Modeling, Identification and Control, pp.45-53, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00524537

T. A. Johansen, On multi-parametric nonlinear programming and explicit nonlinear model predictive control, Proceedings of the 41st IEEE Conference on Decision and Control, 2002., pp.2768-2773, 2002.
DOI : 10.1109/CDC.2002.1184260

T. A. Johansen, Approximate explicit receding horizon control of constrained nonlinear systems, Automatica, vol.40, issue.2, pp.293-300, 2004.
DOI : 10.1016/j.automatica.2003.09.021

R. E. Kalman, When Is a Linear Control System Optimal?, Journal of Basic Engineering, vol.86, issue.1, pp.51-60, 1964.
DOI : 10.1115/1.3653115

L. H. Keel and S. P. Bhattacharyya, Robust, fragile, or optimal? Transactions on Automatic Control, pp.1098-1105, 1997.

E. C. Kerrigan, Robust constraint satisfaction: Invariant sets and predictive control, 2001.

H. K. Khalil, Nonlinear systems, p.323, 2002.

E. Kofman, H. Haimovich, and M. M. Seron, A systematic method to obtain ultimate bounds for perturbed systems, International Journal of Control, vol.18, issue.2, pp.167-178, 2007.
DOI : 10.1109/9.763226

I. Kolmanovsky and E. G. Gilbert, Theory and computation of disturbance invariant sets for discrete-time linear systems, Mathematical Problems in Engineering, vol.4, issue.4, pp.317-367, 1998.
DOI : 10.1155/S1024123X98000866

H. Kong, G. Goodwin, and M. M. Seron, A revisit to inverse optimality of linear systems, International Journal of Control, vol.5, issue.10, pp.1506-1514, 2012.
DOI : 10.1049/iet-cta.2007.0371

M. V. Kothare, V. Balakrishnan, and M. Morari, Robust constrained model predictive control using linear matrix inequalities, Automatica, vol.32, issue.10, pp.1361-1379, 1996.
DOI : 10.1016/0005-1098(96)00063-5

M. Krstic and Z. H. Li, Inverse optimal design of input-to-state stabilizing nonlinear controllers, IEEE Transactions on Automatic Control, vol.43, issue.3, pp.336-350, 1998.
DOI : 10.1109/9.661589

M. Krstic and P. Tsiotras, Inverse optimal stabilization of a rigid spacecraft Automatic Control, IEEE Transactions on, vol.44, issue.5, pp.1042-1049, 1999.

M. Kvasnica, Real-time model predictive control via multi-parametric programming: theory and tools, 2009.

M. Kvasnica and M. Fikar, Clipping-based complexity reduction in explicit mpc Automatic Control, IEEE Transactions on, vol.57, issue.7, pp.1878-1883, 2012.

M. Kvasnica, J. Hledík, I. Rauová, and M. Fikar, Complexity reduction of explicit model predictive control via separation, Automatica, vol.49, issue.6, pp.1776-1781, 2013.
DOI : 10.1016/j.automatica.2013.02.018

V. Larin, About the inverse problem of optimal control, 2003.

C. W. Lee, P.L.-Spheres, convex polytopes, and stress, Discrete & Computational Geometry, vol.16, issue.4, 1996.
DOI : 10.1007/BF02711516

URL : https://hal.archives-ouvertes.fr/jpa-00224550

J. Lofberg, YALMIP : a toolbox for modeling and optimization in MATLAB, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), pp.284-289, 2004.
DOI : 10.1109/CACSD.2004.1393890

C. Løvaas, M. M. Seron, and G. C. Goodwin, Inverse minimax optimality of model predictive control policies, Systems & Control Letters, vol.58, issue.1, pp.31-38, 2009.
DOI : 10.1016/j.sysconle.2008.07.005

J. C. Maxwell, On reciprocal diagrams and diagrams of forces. Philosophical Magazine, ser. 4, pp.250-261

H. N. Nguyen, Constrained Control of Uncertain, Time-Varying, Discrete-Time Systems: An Interpolation-Based Approach, Lecture Notes in Control and Information Sciences, vol.451, 2014.
DOI : 10.1007/978-3-319-02827-9

H. N. Nguyen, P. O. Gutman, S. Olaru, and M. Hovd, Explicit constraint control based on interpolation techniques for time-varying and uncertain linear discrete-time systems, World Congress, pp.5741-5746, 2011.
DOI : 10.3182/20110828-6-IT-1002.03517

URL : https://hal.archives-ouvertes.fr/hal-00592340

N. A. Nguyen, S. Olaru, P. Rodriguez-ayerbe, M. Hovd, and I. Necoara, Constructive solution of inverse parametric linear/quadratic programming problems . a. URL https
URL : https://hal.archives-ouvertes.fr/hal-01207234

N. A. Nguyen, S. Olaru, P. Rodriguez-ayerbe, and M. Kvasnica, Convex liftingsbased robust control design
URL : https://hal.archives-ouvertes.fr/hal-01509316

N. A. Nguyen, S. Olaru, G. Bitsoris, P. Rodriguez-ayerbe, and M. Hovd, Explicit fragility margins for PWA control laws of discrete-time linear systems, 2014 European Control Conference (ECC), 2014.
DOI : 10.1109/ECC.2014.6862538

URL : https://hal.archives-ouvertes.fr/hal-01087256

N. A. Nguyen, S. Olaru, P. Rodriguez-ayerbe, and M. Hovd, An inverse optimality argument to improve robustness in constrained control, IFAC World Congress, pp.1631-1636, 2014.
DOI : 10.3182/20140824-6-ZA-1003.02586

URL : https://hal.archives-ouvertes.fr/hal-01086391

N. A. Nguyen, S. Olaru, P. Rodriguez-ayerbe, M. Hovd, and I. Necoara, On the lifting problems and their connections with piecewise affine control law design, 2014 European Control Conference (ECC), 2014.
DOI : 10.1109/ECC.2014.6862605

URL : https://hal.archives-ouvertes.fr/hal-01087243

N. A. Nguyen, S. Olaru, P. Rodriguez-ayerbe, M. Hovd, and I. Necoara, Inverse parametric convex programming problems via convex liftings, 19th IFAC World Congress, 2014.
DOI : 10.3182/20140824-6-ZA-1003.02364

URL : https://hal.archives-ouvertes.fr/hal-01086492

A. Nguyen, S. Olaru, G. Rodriguez-ayerbe, P. Bitsoris, and M. Hovd, Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law, Automatica, vol.68, 2015.
DOI : 10.1016/j.automatica.2015.10.048

URL : https://hal.archives-ouvertes.fr/hal-01408554

N. A. Nguyen, S. Olaru, and P. Rodriguez-ayerbe, Any discontinuous PWA function is optimal solution to a parametric linear programming problem, 2015 54th IEEE Conference on Decision and Control (CDC), 2015.
DOI : 10.1109/CDC.2015.7403151

URL : https://hal.archives-ouvertes.fr/hal-01259879

N. A. Nguyen, S. Olaru, and P. Rodriguez-ayerbe, Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra, 2015 54th IEEE Conference on Decision and Control (CDC), 2015.
DOI : 10.1109/CDC.2015.7403150

URL : https://hal.archives-ouvertes.fr/hal-01259895

N. A. Nguyen, S. Olaru, and P. Rodriguez-ayerbe, Recognition of additively weighted Voronoi diagrams and weighted Delaunay decompositions, 2015 European Control Conference (ECC), 2015.
DOI : 10.1109/ECC.2015.7330565

URL : https://hal.archives-ouvertes.fr/hal-01259935

N. A. Nguyen, S. Olaru, and P. Rodriguez-ayerbe, On the complexity of the convex liftings-based solution to inverse parametric convex programming problems, 2015 European Control Conference (ECC), 2015.
DOI : 10.1109/ECC.2015.7331064

URL : https://hal.archives-ouvertes.fr/hal-01140609

N. A. Nguyen, S. Olaru, and P. Rodriguez-ayerbe, Robust control design based on convex liftings, the 8th IFAC Symposium on Robust Control Design, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01159892

N. A. Nguyen, S. Olaru, P. Rodriguez-ayerbe, and M. , Hovd, and I. Necoara. Fully inverse parametric linear/quadratic programming problems via convex liftings

S. Olaru and D. Dumur, A parameterized polyhedra approach for explicit constrained predictive control, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), pp.1580-1585, 2004.
DOI : 10.1109/CDC.2004.1430269

S. Olaru and D. Dumur, Avoiding constraints redundancy in predictive control optimization routines. Automatic Control, IEEE Transactions on, vol.50, issue.9, pp.1459-1465, 2005.

S. Olaru and D. Dumur, On the continuity and complexity of control laws based on multiparametric linear programs, Proceedings of the 45th IEEE Conference on Decision and Control, pp.5465-5470, 2006.
DOI : 10.1109/CDC.2006.377330

URL : https://hal.archives-ouvertes.fr/hal-00261371

S. Olaru, J. A. De-doná, M. M. Seron, and F. Stoican, Positive invariant sets for fault tolerant multisensor control schemes, International Journal of Control, vol.40, issue.12, pp.2622-2640, 2010.
DOI : 10.1016/j.automatica.2004.01.014

URL : https://hal.archives-ouvertes.fr/hal-00293927

S. Olaru, N. A. Nguyen, G. Bitsoris, P. Rodriguez-ayerbe, and M. Hovd, Explicit robustness margins for discrete-time linear systems with PWA control, 2013 17th International Conference on System Theory, Control and Computing (ICSTCC), 2013.
DOI : 10.1109/ICSTCC.2013.6688989

URL : https://hal.archives-ouvertes.fr/hal-00917917

E. Ostertag, Mono-and Multivariable Control and Estimation: Linear, Quadratic and LMI Methods, 2011.
DOI : 10.1007/978-3-642-13734-1

E. N. Pistikopoulos, M. C. Georgiadis, and V. Dua, Multi-parametric programming, 2007.
DOI : 10.1002/9783527631216

S. V. Rakovi´crakovi´c and M. Lazar, The Minkowski???Lyapunov equation for linear dynamics: Theoretical foundations, Automatica, vol.50, issue.8, pp.2015-2024, 2014.
DOI : 10.1016/j.automatica.2014.05.023

S. V. Rakovic, P. Grieder, M. Kvasnica, D. Q. Mayne, and M. Morari, Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), pp.1418-1423, 2004.
DOI : 10.1109/CDC.2004.1430242

S. V. Rakovic, E. C. Kerrigan, K. I. Kouramas, and D. Q. Mayne, Invariant approximations of the minimal robust positively invariant set. Automatic Control, IEEE Transactions on, vol.50, issue.3, pp.406-410, 2005.

S. V. Rakovic, E. C. Kerrigan, D. Q. Mayne, and J. Lygeros, Reachability analysis of discrete-time systems with disturbances. Automatic Control, IEEE Transactions on, vol.51, issue.4, pp.546-561, 2006.

J. Richalet, A. Rault, J. L. Testud, and J. Papon, Model predictive heuristic control, Automatica, vol.14, issue.5, pp.413-428, 1978.
DOI : 10.1016/0005-1098(78)90001-8

R. T. Rockafellar, Convex analysis. Number 28, 1970.

C. Rowe and J. Maciejowski, Tuning mpc using h ? loop shaping, American Control Conference Proceedings of the 2000, pp.1332-1336, 2000.

W. J. Rugh and J. S. Shamma, Research on gain scheduling, Automatica, vol.36, issue.10, pp.1401-1425, 2000.
DOI : 10.1016/S0005-1098(00)00058-3

K. Rybnikov, Polyhedral Partitions and Stresses, 1999.

R. Schneider, Convex bodies: the Brunn?Minkowski theory, 2013.
DOI : 10.1017/CBO9780511526282

A. Schrijver, Theory of linear and integer programming, 1998.

A. Schulz, Lifting Planar Graphs to Realize Integral 3-Polytopes and Topics in Pseudo-Triangulations, 2008.

F. Scibilia, S. Olaru, and M. Hovd, Approximate explicit linear mpc via delaunay tessellation, Control Conference (ECC), pp.2833-2838, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00418337

M. M. Seron, G. C. Goodwin, and J. A. Doná, Characterisation Of Receding Horizon Control For Constrained Linear Systems, Asian Journal of Control, vol.6, issue.6, pp.271-286, 2003.
DOI : 10.1111/j.1934-6093.2003.tb00118.x

J. Spjøtvold, E. C. Kerrigan, C. N. Jones, P. Tøndel, and T. A. Johansen, On the facet-to-facet property of solutions to convex parametric quadratic programs, Automatica, vol.42, issue.12, pp.422209-2214, 2006.
DOI : 10.1016/j.automatica.2006.06.026

F. Stoican and S. Olaru, Set-theoretic Fault-tolerant Control in Multisensor Systems, 2013.
DOI : 10.1002/9781118649428

URL : https://hal.archives-ouvertes.fr/hal-00826961

A. Suardi, S. Longo, E. C. Kerrigan, and G. A. Constantinides, Robust explicit MPC design under finite precision arithmetic, Proceedings of the 19th World Congress The International Federation of Automatic Control, 2014.
DOI : 10.3182/20140824-6-ZA-1003.01033

M. Sznaier and M. J. Damborg, Suboptimal control of linear systems with state and control inequality constraints, 26th IEEE Conference on Decision and Control, 1987.
DOI : 10.1109/CDC.1987.272491

F. Tahir and I. M. Jaimoukha, Robust Positively Invariant Sets for Linear Systems subject to model-uncertainty and disturbances, Nonlinear Model Predictive Control, pp.213-217, 2012.
DOI : 10.3182/20120823-5-NL-3013.00032

P. Tøndel, T. A. Johansen, and A. Bemporad, An algorithm for multi-parametric quadratic programming and explicit MPC solutions, Automatica, vol.39, issue.3, pp.489-497, 2003.
DOI : 10.1016/S0005-1098(02)00250-9

M. Vassilaki and G. Bitsoris, Constrained regulation of linear continuous-time dynamical systems, Systems & Control Letters, vol.13, issue.3, pp.247-252, 1989.
DOI : 10.1016/0167-6911(89)90071-6

M. Vassilaki, J. C. Hennet, and G. Bitsoris, Feedback control of linear discrete-time systems under state and control constraints, International Journal of Control, vol.13, issue.6, pp.1727-1735, 1988.
DOI : 10.1109/TAC.1985.1103785

Y. Wang and S. Boyd, Fast Model Predictive Control Using Online Optimization, IEEE Transactions on Control Systems Technology, vol.18, issue.2, pp.267-278, 2010.
DOI : 10.1109/TCST.2009.2017934

G. M. Ziegler, Lectures on polytopes, 1995.
DOI : 10.1007/978-1-4613-8431-1