Elsevier

Automatica

Volume 48, Issue 10, October 2012, Pages 2620-2626
Automatica

Brief paper
A fast ellipsoidal MPC scheme for discrete-time polytopic linear parameter varying systems

https://doi.org/10.1016/j.automatica.2012.06.065Get rights and content

Abstract

This paper proposes a fast ellipsoidal Model Predictive Control (MPC) strategy to address feedback regulation problems for constrained polytopic Linear Parameter Varying (LPV) systems subject to bounded disturbances. In order to deal with the specific non-convex structure of the state prediction tubes arising in LPV contexts, a new convexification procedure is proposed and, based on the off-line computation of a sequence of inner ellipsoidal approximations of exact one-step controllable sets, a computationally low-demanding MPC algorithm is presented. Comparisons with state-of-the-art MPC control algorithms for LPV systems are reported in a final numerical example where several methods are contrasted in terms of achievable domains of attraction, control performance, numerical burdens and memory requirements.

Introduction

The Linear Parameter Varying (LPV) system paradigm provides an interesting modeling framework which is capable of describing special classes of non-linear and/or time-varying phenomena. Because of its relevance in practical applications, several Model Predictive Control (MPC) schemes have recently been developed for the LPV framework; for a comprehensive literature on the subject see Franzè, Garone, Famularo, and Casavola (2009), Lu and Arkun (2000), Park and Jeong (2004) and Pluymers, Rossiter, Suykens, and De Moor (2005). These strategies are mainly modified versions of existing robust MPC schemes, adapted to the LPV paradigm by exploiting the scheduling parameter availability for prediction and control purposes. As a consequence, they inherit most of the known drawbacks of robust MPC control strategies.

In order to reduce the computational burdens of direct on-line MPC schemes for LPV systems while preserving adequate control performance, the ellipsoidal fast MPC approach introduced in Angeli, Casavola, Franzè, and Mosca (2008) is of interest here and it will be extended to deal with LPV systems. This approach is an adaptation of the general philosophy proposed by Bertsekas and Rhodes in Bertsekas and Rhodes (1971) for systems subject to set-membership disturbances and model uncertainties which inspired several Fast-MPC schemes such as (Besselman et al., 2009, Björnberg and Diehl, 2006, Ding et al., 2007, Wan and Kothare, 2003).

Following this philosophy, the proposed scheme prescribes that a bank of indexed, not necessarily nested, ellipsoids Ei is computed from the outset by exploiting the viability arguments of Kurzhanski and Vályi (1997). Each ellipsoid Ei represents a compact set of states that can be steered in one control move, irrespective of the specific realization of a bounded persistent disturbance and without constraints violation, into another inner set Ei1 and, iteratively in at most i steps, to a suitable robust terminal set which ultimately traps the system trajectories under a suitable terminal control law.

This ellipsoidal family is computed via a dynamic programming approach by accomplishing a sequence of min–max optimization steps. On line, at each time instant, the ellipsoid Ei of smallest index i which contains the actual measured state x(t) is determined and a numerically low-demanding optimization problem is solved in order to compute a control action u(t) able to steer the one-step ahead state prediction vector to the inner set Ei1. Such a condition ensures contraction and viability to the closed-loop trajectories.

A difficulty in dealing with LPV plants when governed by linear scheduled control laws is that the one-step ahead state prediction set is non-convex and depends quadratically on the scheduling parameter. Convex outer approximations of the prediction sets have been proposed in the literature to bypass this numerical obstacle. The most popular approach is to embed the prediction tube into a polytope whose vertices are computed by exploiting a symmetrical property existing between any pairs of elements of the quadratic form linking the scheduling vector to the prediction tube Wang, Tanaka, and Griffin (1996). Here, a refinement of this approach is proposed. It will be shown that this novel convex outer approximation procedure allows the achievement of remarkable larger domains of attraction, at the price of a slightly increased number of LMIs to be checked. Preliminary results on the proposed convexification procedure have been presented in Casavola, Famularo, Franzè, and Garone (2009).

A final numerical example is presented in order to show the properties of the proposed regulation strategy. Its effectiveness is contrasted with state-of-the-art LPV control algorithms in terms of achievable domains of attraction, control performance, numerical burdens and memory requirements.

Section snippets

System description and problem formulation

Consider the discrete-time polytopic LPV system x(t+1)=A(p(t))x(t)+B(p(t))u(t)+Gdd(t) where tZ+{0,1,},x(t)Rn is the state, u(t)Rm the control input and d(t)Rnd an exogenous disturbance. The possibly time-varying vector p(t)Rl is assumed to be measurable at each time instant and belongs to the unit simplex Pl{pRl:i=1lpi=1,pi0}. The system matrices A(p) and B(p) are members of the polytopic matrix family Σ(Pl){(A(p),B(p))=i=1lpi(Ai,Bi),pPl} where the pairs (Ai,Bi) denote the

One-step ahead state prediction set characterization under p-scheduled control laws: an outer approximation approach

Consider the disturbance-free one-step ahead state prediction set X+{x+x+=A(p)x+B(p)u,pPl} for a given state xRn and input vector uRm. The following class of p-scheduled control laws is considered u(t)=j=1lpj(t)uj(t) with uj(t)Rm suitably given control inputs. Under (7), each element x+X+ can be rewritten as x+=i=1j=1lpipjAix+i=1j=1lpipjBiuj,pPl. By standard manipulations, expression (8) becomes x+=[(1lTIn×n)(ppTIn×n)Ã]x+[B̃(ppTIm×m)]ū,pPl where denotes the Kronecker

A fast ellipsoidal MPC scheme

In order to develop a fast MPC scheme for an LPV system along the control paradigm depicted in Problem 1, the main issue is the computation of the indexed one-step robustly controllable regions Ti. In principle, given a robustly controlled-invariant terminal region T, it is possible to compute sets of states which are i-step controllable to T, regardless of disturbance, constraints and scheduling parameter values, via the following recursion: T0TXTi{xX:u(p)U:dD,pPl,(A(p)x+B(p)u(p)+Gdd)

Illustrative example

The aim of this example is to present results on the effectiveness of the proposed MPC strategy and comparisons with other relevant competitors: the on-line polyhedral LPV-BLM strategy of Besselman et al. (2009), the ellipsoidal algorithm of Wan and Kothare (2003) adapted to the proposed LPV framework, the LPV version of the robust RH scheme of Angeli et al. (2008) using the convexification (15), hereafter denoted as LPV-SS, and the Quasi-Min–Max MPC scheme LPV-Quasi of Lu and Arkun (2000): x(t+

Conclusions

A novel fast ellipsoidal MPC strategy for discrete-time polytopic LPV systems subject to bounded disturbances, input and state constraints has been presented. The contribution is twofold: first, the geometrical structure of the scheduling parameter evolution is exploited to achieve a tighter ellipsoidal refinement of the one-step ahead state prediction set. Second, by exploiting ellipsoidal calculus and viability theory, a novel fast ellipsoidal MPC scheme is proposed which exhibits low

Acknowledgments

The authors wish to thank Dr. Thomas Besselmann for providing the Matlab code implementing the MPC scheme described in Besselman et al. (2009).

Alessandro Casavola (Florence, 1958) received the Laurea degree in electrical engineering from the University of Florence, Italy, in 1986 and the Ph.D. degree in systems engineering from the University of Bologna, Italy, in 1990. From 1990 to 1996 he was the Computing Center and Network Administrator at the Department of Mathematics “U. Dini” of the University of Florence, Italy. Since 1998 he has been with the Department of Electronics, Informatics and Systems (DEIS) of the University of

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Alessandro Casavola (Florence, 1958) received the Laurea degree in electrical engineering from the University of Florence, Italy, in 1986 and the Ph.D. degree in systems engineering from the University of Bologna, Italy, in 1990. From 1990 to 1996 he was the Computing Center and Network Administrator at the Department of Mathematics “U. Dini” of the University of Florence, Italy. Since 1998 he has been with the Department of Electronics, Informatics and Systems (DEIS) of the University of Calabria: from 1998 to 2005 as an Associate Professor and since 2005 as a Full Professor. From 2005 to 2011 he was the Chairman of the master degree (laurea specialistica) in Control and Automation Engineering of the University of Calabria. He has participated and led research units in many methodological and applied research projects, funded by national government agencies and private industries. His research activity is testified to by more than 250 scientific publications in prestigious scientific journals and international conferences. Since 2009 he has served as a subject editor for the International Journal of Adaptive Control and Signal Processing. His current research interests include constrained predictive control, control and set-point reconfiguration strategies for fault tolerant systems and supervision of large-scale networked dynamic systems.

Domenico Famularo was born in 1967. He received his Laurea degree in computer engineering from the University of Calabria, Italy, in 1991 and the Ph.D. degree in computational mechanics from the University of Rome, Italy, in 1996. From 1991 to 2000 he was with the DEIS Department of the University of Calabria, Italy as a Research Associate. In 1997 he was a visiting Scholar Researcher at the EECE Department, University of New Mexico, Albuquerque, NM, USA and in 1999 he covered the same position at the EE Systems Department, University of Southern California, Los Angeles, CA. He was a Researcher at the Istituto per il Calcolo e le Reti ad Alte Prestazioni (ICAR)—Consiglio Nazionale delle Ricerche (CNR) and since 2005 he is an Associate Professor, currently with the DEIS Department, University of Calabria. His current research interests include control under constraints, control reconfiguration for fault tolerant systems and networked control systems.

Giuseppe Franzè was born in 1968 in Italy. He received the Ph.D. degree in systems engineering from the University of Calabria, Italy, in 1999. From 1994 to 2002 he was with the DEIS Department of the University of Calabria, Italy, as an Assistant Researcher and since 2002 as an Assistant Professor. His current research interests include constrained predictive control, nonlinear systems, networked control systems, control under constraints and control reconfiguration for fault tolerant systems.

Emanuele Garone (Torino, 1980) obtained his Laurea degree from the University of Calabria, Italy in 2005 and his Ph.D. degree in systems engineering from the University of Calabria, Italy, in 2009. In 2007, he was a finalist in the IEEE CSS CDC Best Student-Paper Award. Since November 2008 to October 2010 he was with the DEIS Department of the University of Calabria, Italy, as an assistant researcher. Since November 2010, he is an assistant professor at the Ecole Polytechnique de Bruxelles, Université Libre de Bruxelles. His current research interests include: model predictive control for Linear Parameter Varying systems, distributed model predictive control, networked control systems, and teams of heterogeneous robots.

The material in this paper was partially presented at the European Control Conference 2009 (ECC’09), August 23–26, 2009, Budapest, Hungary. This paper was recommended for publication in revised form by Associate Editor Fen Wu under the direction of Editor Roberto Tempo.

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