On the optimality of localised distributed controllers

Design of optimal distributed controllers with a priori assigned localisation constraints is a difficult problem. Alternatively, one can ask the following question: given a localised distributed exponentially stabilising controller, is it inversely optimal with respect to some cost functional? We study this problem for linear spatially invariant systems and establish a frequency domain criterion for inverse optimality (in the LQR sense). We utilise this criterion to separate localised controllers that are never optimal from localised controllers that are optimal. For the latter, we provide examples to demonstrate optimality with respect to physically appealing cost functionals. These are characterised by state penalties that are not fully decentralised and they provide insight about spatial extent of the LQR weights that lead to localised controllers.

[1]  Fernando Paganini,et al.  Convex synthesis of localized controllers for spatially invariant systems , 2002, Autom..

[2]  J. K. Hedrick,et al.  Constant Spacing Strategies for Platooning in Automated Highway Systems , 1999 .

[3]  Andrew G. Sparks,et al.  Adaptive output feedback tracking control of spacecraft formation , 2002 .

[4]  Pravin Varaiya,et al.  Smart cars on smart roads: problems of control , 1991, IEEE Trans. Autom. Control..

[5]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[6]  Petros A. Ioannou,et al.  Vehicle following control design for automated highway systems , 1997, 1997 IEEE 47th Vehicular Technology Conference. Technology in Motion.

[7]  Cédric Langbort,et al.  Distributed control design for systems interconnected over an arbitrary graph , 2004, IEEE Transactions on Automatic Control.

[8]  Stephen Barnett.  State-Space and Frequency-Domain Methods in the Control of Distributed Parameter Systems , 1984 .

[9]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[10]  Bassam Bamieh,et al.  Optimal control of arrays of microcantilevers , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[11]  Geir E. Dullerud,et al.  Distributed control design for spatially interconnected systems , 2003, IEEE Trans. Autom. Control..

[12]  Fernando Paganini,et al.  Distributed control of spatially invariant systems , 2002, IEEE Trans. Autom. Control..

[13]  A. Rantzer,et al.  A Separation Principle for Distributed Control , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[14]  Nader Motee,et al.  Optimal Control of Spatially Distributed Systems , 2008, 2007 American Control Conference.

[15]  J.M. Fowler,et al.  A formation flight experiment , 2003, IEEE Control Systems.

[16]  Tryphon T. Georgiou,et al.  Weight Selection in Feedback Design With Degree Constraints , 2008, IEEE Transactions on Automatic Control.

[17]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[18]  D. F. Chichka,et al.  Solar-powered, formation-enhanced aerial vehicle systems for sustained endurance , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[19]  Petros G. Voulgaris,et al.  A convex characterization of distributed control problems in spatially invariant systems with communication constraints , 2005, Syst. Control. Lett..

[20]  Mihailo R. Jovanovic,et al.  On the peaking phenomenon in the control of vehicular platoons , 2008, Syst. Control. Lett..

[21]  Peter Seiler,et al.  Disturbance propagation in vehicle strings , 2004, IEEE Transactions on Automatic Control.

[22]  Geir E. Dullerud,et al.  Distributed control of heterogeneous systems , 2004, IEEE Transactions on Automatic Control.

[23]  Mihailo R. Jovanovic,et al.  On the ill-posedness of certain vehicular platoon control problems , 2005, IEEE Transactions on Automatic Control.

[24]  Petros G. Voulgaris,et al.  Optimal H2 controllers for spatially invariant systems with delayed communication requirements , 2003, Syst. Control. Lett..

[25]  Randal W. Beard,et al.  A coordination architecture for spacecraft formation control , 2001, IEEE Trans. Control. Syst. Technol..

[26]  Andrew G. Sparks,et al.  Spacecraft formation flying: dynamics and control , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[27]  Sanjay Lall,et al.  A Characterization of Convex Problems in Decentralized Control$^ast$ , 2005, IEEE Transactions on Automatic Control.

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

[29]  Bassam Bamieh,et al.  Optimal Control of Arrays of Microcantilevers , 1999 .

[30]  A. Rantzer.  Linear quadratic team theory revisited , 2006, 2006 American Control Conference.

[31]  A. Jadbabaie,et al.  On decentralized optimal control and information structures , 2008, 2008 American Control Conference.