Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law

In this paper, we focus on the robustness and fragility problem for piecewise affine (PWA) control laws for discrete-time linear system dynamics in the presence of parametric uncertainty of the state space model. A generic geometrical approach will be used to obtain robustness/fragility margins with respect to the positive invariance properties. For PWA control laws defined over a bounded region in the state space, it is shown that these margins can be described in terms of polyhedral sets in parameter space. The methodology is further extended to the fragility problem with respect to the partition defining the controller. Finally, several computational aspects are presented regarding the transformation from the theoretical formulations to explicit representations (vertex/halfspace representation of polytopes) of these sets.

[1]  P. Boucher,et al.  On the construction of invariant sets for piecewise affine systems using the transition graph , 2009, 2009 IEEE International Conference on Control and Automation.

[2]  Sorin Olaru,et al.  Explicit fragility margins for PWA control laws of discrete-time linear systems , 2014, 2014 European Control Conference (ECC).

[3]  J. Hennet,et al.  Feedback control of linear discrete-time systems under state and control constraints , 1988 .

[4]  David Avis,et al.  A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra , 1991, SCG '91.

[5]  Morten Hovd,et al.  Implicit improved vertex control for uncertain, time-varying linear discrete-time systems with state and control constraints , 2013, Autom..

[6]  Efstratios N. Pistikopoulos,et al.  An algorithm for robust explicit/multi-parametric model predictive control , 2013, Autom..

[7]  Nikolaos Athanasopoulos,et al.  Construction of invariant polytopic sets with specified complexity , 2014, Int. J. Control.

[8]  M. Hovd,et al.  Explicit robustness margins for discrete-time linear systems with PWA control , 2013, 2013 17th International Conference on System Theory, Control and Computing (ICSTCC).

[9]  Miroslav Fikar,et al.  Clipping-Based Complexity Reduction in Explicit MPC , 2012, IEEE Transactions on Automatic Control.

[10]  Manfred Morari,et al.  Multi-Parametric Toolbox 3.0 , 2013, 2013 European Control Conference (ECC).

[11]  Sorin Olaru,et al.  Robust control design based on convex liftings , 2015 .

[12]  Didier Dumur,et al.  A parameterized polyhedra approach for explicit constrained predictive control , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[13]  Sorin Olaru,et al.  Model Predictive Control - Numerical Methods for the Invariant Sets Approximation , 2009, NAA.

[14]  Tor Arne Johansen,et al.  Approximate explicit constrained linear model predictive control via orthogonal search tree , 2003, IEEE Trans. Autom. Control..

[15]  P. Dorato,et al.  Non-fragile controller design: an overview , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[16]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[17]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[18]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[19]  George B. Dantzig,et al.  Fourier-Motzkin Elimination and Its Dual , 1973, J. Comb. Theory, Ser. A.

[20]  Shankar P. Bhattacharyya,et al.  Robust, fragile or optimal? , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[21]  Alberto Bemporad,et al.  An algorithm for multi-parametric quadratic programming and explicit MPC solutions , 2003, Autom..

[22]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[23]  Bart De Schutter,et al.  Equivalence of hybrid dynamical models , 2001, Autom..

[24]  Graham C. Goodwin,et al.  Characterisation Of Receding Horizon Control For Constrained Linear Systems , 2003 .

[25]  Mi-Ching Tsai,et al.  Robust and Optimal Control , 2014 .

[26]  K. T. Tan,et al.  Linear systems with state and control constraints: the theory and application of maximal output admissible sets , 1991 .

[27]  Jean-Claude Hennet,et al.  Discrete Time Constrained Linear Systems , 1995 .

[28]  M. Kothare,et al.  Robust constrained model predictive control using linear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[29]  Imad M. Jaimoukha,et al.  Robust Positively Invariant Sets for Linear Systems subject to model-uncertainty and disturbances , 2012 .

[30]  Eric C. Kerrigan,et al.  Robust explicit MPC design under finite precision arithmetic , 2014 .

[31]  G. Bitsoris.  Positively invariant polyhedral sets of discrete-time linear systems , 1988 .

[32]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[33]  J. Maciejowski,et al.  Feedback min‐max model predictive control using a single linear program: robust stability and the explicit solution , 2004 .