Explicit robust constrained control for linear systems : analysis, implementation and design based on optimization

Piecewise affine (PWA) feedback control laws have received significant attention due to their relevance for the control of constrained systems, hybrid systems; equally for the approximation of nonlinear control. However, they are associated with serious implementation issues. Motivated from the interest in this class of particular controllers, this thesis is mostly related to their analysis and design.The first part of this thesis aims to compute the robustness and fragility margins for a given PWA control law and a linear discrete-time system. More precisely, the robustness margin is defined as the set of linear time-varying systems such that the given PWA control law keeps the trajectories inside a given feasible set. On a different perspective, the fragility margin contains all the admissible variations of the control law coefficients such that the positive invariance of the given feasible set is still guaranteed. It will be shown that if the given feasible set is a polytope, then so are these robustness/fragility margins.The second part of this thesis focuses on inverse optimality problem for the class of PWA controllers. Namely, the goal is to construct an optimization problem whose optimal solution is equivalent to the given PWA function. The methodology is based on emph convex lifting: an auxiliary 1− dimensional variable which enhances the convexity characterization into recovered optimization problem. Accordingly, if the given PWA function is continuous, the optimal solution to this reconstructed optimization problem will be shown to be unique. Otherwise, if the continuity of this given PWA function is not fulfilled, this function will be shown to be one optimal solution to the recovered problem.In view of applications in linear model predictive control (MPC), it will be shown that any continuous PWA control law can be obtained by a linear MPC problem with the prediction horizon at most equal to 2 prediction steps. Aside from the theoretical meaning, this result can also be of help to facilitate implementation of PWA control laws by avoiding storing state space partition. Another utility of convex liftings will be shown in the last part of this thesis to be a control Lyapunov function. Accordingly, this convex lifting will be deployed in the so-called robust control design based on convex liftings for linear system affected by bounded additive disturbances and polytopic uncertainties. Both implicit and explicit controllers can be obtained. This method can also guarantee the recursive feasibility and robust stability. However, this control Lyapunov function is only defined over the maximal λ −contractive set for a given 0 ≤ λ < 1 which is known to be smaller than the maximal controllable set. Therefore, an extension of the above method to the N-steps controllable set will be presented. This method is based on a cascade of convex liftings where an auxiliary variable will be used to emulate a Lyapunov function. Namely, this variable will be shown to be non-negative, to strictly decrease for N first steps and to stay at 0 afterwards. Accordingly, robust stability is sought.

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

[2]  Carl W. Lee,et al.  P.L.-Spheres, convex polytopes, and stress , 1996, Discret. Comput. Geom..

[3]  R. E. Kalman,et al.  When Is a Linear Control System Optimal , 1964 .

[4]  Didier Dumur,et al.  Avoiding constraints redundancy in predictive control optimization routines , 2005, IEEE Transactions on Automatic Control.

[5]  Per-Olof Gutman,et al.  Explicit constraint control based on interpolation techniques for time-varying and uncertain linear discrete-time systems , 2011 .

[6]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[7]  Ion Necoara,et al.  Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems , 2017, J. Optim. Theory Appl..

[8]  Z. Rekasius,et al.  On an inverse problem in optimal control , 1964 .

[9]  David Q. Mayne,et al.  Reachability analysis of discrete-time systems with disturbances , 2006, IEEE Transactions on Automatic Control.

[10]  Morten Hovd,et al.  Approximate explicit linear MPC via Delaunay tessellation , 2009, 2009 European Control Conference (ECC).

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

[12]  Ion Necoara,et al.  Inverse Parametric Convex Programming Problems Via Convex Liftings , 2014 .

[13]  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).

[14]  Hoai-Nam Nguyen,et al.  Constrained Control of Uncertain, Time-Varying, Discrete-Time Systems: An Interpolation-Based Approach , 2013 .

[15]  David Q. Mayne,et al.  Invariant approximations of the minimal robust positively Invariant set , 2005, IEEE Transactions on Automatic Control.

[16]  Sorin Olaru,et al.  Any discontinuous PWA function is optimal solution to a parametric linear programming problem , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[17]  Miroslav Fikar,et al.  Complexity reduction of explicit model predictive control via separation , 2013, Autom..

[18]  Ion Necoara,et al.  Fully Inverse Parametric Linear/Quadratic Programming Problems via Convex Liftings , 2015 .

[19]  D. Dumur,et al.  On the continuity and complexity of control laws based on multiparametric linear programs , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[20]  Mario Sznaier,et al.  Suboptimal control of linear systems with state and control inequality constraints , 1987, 26th IEEE Conference on Decision and Control.

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

[22]  Graham C. Goodwin,et al.  Inverse Minimax Optimality of Model Predictive Control Policies , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[23]  Graham C. Goodwin,et al.  A revisit to inverse optimality of linear systems , 2012, Int. J. Control.

[24]  Sorin Olaru,et al.  Recognition of additively weighted Voronoi diagrams and weighted Delaunay decompositions , 2015, 2015 European Control Conference (ECC).

[25]  K. Ball.  CONVEX BODIES: THE BRUNN–MINKOWSKI THEORY , 1994 .

[26]  Sorin Olaru,et al.  Set-theoretic Fault-tolerant Control in Multisensor Systems: Stoican/Set-theoretic Fault-tolerant Control in Multisensor Systems , 2013 .

[27]  D. Mayne,et al.  Computation of invariant sets for piecewise affine discrete time systems subject to bounded disturbances , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[28]  Morten Hovd,et al.  Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law , 2016, Autom..

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

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

[31]  G. Bitsoris,et al.  Constrained regulation of linear continuous-time dynamical systems , 1989 .

[32]  Mircea Lazar,et al.  The Minkowski-Lyapunov equation for linear dynamics: Theoretical foundations , 2014, Autom..

[33]  P. Olver.  Nonlinear Systems , 2013 .

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

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

[36]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

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

[38]  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).

[39]  M. Krstic,et al.  Inverse optimal design of input-to-state stabilizing nonlinear controllers , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[40]  J. Richalet,et al.  Model predictive heuristic control: Applications to industrial processes , 1978, Autom..

[41]  E. Kerrigan.  Robust Constraint Satisfaction: Invariant Sets and Predictive Control , 2000 .

[42]  Sorin Olaru,et al.  On the complexity of the convex liftings-based solution to inverse parametric convex programming problems , 2015, 2015 European Control Conference (ECC).

[43]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[44]  Morten Hovd,et al.  An inverse optimality argument to improve robustness in constrained control , 2014 .

[45]  E. Gilbert,et al.  Theory and computation of disturbance invariant sets for discrete-time linear systems , 1998 .

[46]  Tor Arne Johansen,et al.  Approximate explicit receding horizon control of constrained nonlinear systems , 2004, Autom..

[47]  Ion Necoara,et al.  On the lifting problems and their connections with piecewise affine control law design , 2014, 2014 European Control Conference (ECC).

[48]  Colin Neil Jones,et al.  On the facet-to-facet property of solutions to convex parametric quadratic programs , 2006, Autom..

[49]  María M. Seron,et al.  A systematic method to obtain ultimate bounds for perturbed systems , 2007, Int. J. Control.

[50]  Sorin Olaru,et al.  Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[51]  Miroslav Krstic,et al.  Inverse optimal stabilization of a rigid spacecraft , 1999, IEEE Trans. Autom. Control..

[52]  Eric Ostertag.  Mono- and Multivariable Control and Estimation: Linear, Quadratic and LMI Methods , 2011 .

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

[54]  Sorin Olaru,et al.  Positive invariant sets for fault tolerant multisensor control schemes , 2008 .

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