An application of Lie groups in distributed control networks

Abstract Here we introduce a class of linear operators called recursive orthogonal transforms (ROTs) that allow a natural implementation on a distributed control network. We derive conditions under which ROTs can be used to represent SO( n ) for n ⩾4. We propose a paradigm for distributed feedback control based on plant matrix diagonalization. To find an ROT suitable for this task, we derive a gradient flow on the appropriate underlying Lie group. A numerical example is presented.

[1]  Gregory W. Wornell,et al.  A separation theorem for periodic sharing information patterns in decentralized control , 1997 .

[2]  Roger W. Brockett,et al.  Stabilization of motor networks , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[3]  O. Beldiman,et al.  Asymptotic behavior of networked control systems , 1999, Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328).

[4]  R. Brockett,et al.  Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[5]  N. S. Barnett,et al.  Private communication , 1969 .

[6]  Wen-June Wang,et al.  Stabilization and estimation for perturbed discrete time-delay large-scale systems , 1997, IEEE Trans. Autom. Control..

[7]  John B. Moore,et al.  Singular-Value Decomposition via Gradient and Self-Equivalent Flows , 1992 .

[8]  Linda Bushnell,et al.  Error encoding algorithms for networked control systems , 2002, Autom..

[9]  R. Brockett,et al.  Systems with finite communication bandwidth constraints. I. State estimation problems , 1997, IEEE Trans. Autom. Control..

[10]  Tad Hogg,et al.  Market Organizations for Controlling Smart Matter , 1997 .

[11]  R. Brockett.  Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems , 1991 .

[12]  S. Łojasiewicz.  Ensembles semi-analytiques , 1965 .

[13]  Kenneth C. Chou,et al.  Multiscale approach to the control of smart materials , 1995, Smart Structures.

[14]  Peter F. Al-Hokayem.  Stability Analysis of Networked Control Systems , 2003 .

[15]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[16]  Tad Hogg,et al.  Controls for unstable structures , 1997, Smart Structures.

[17]  M. Victor Wickerhauser,et al.  Large-Rank Approximate Principal Component Analysis with Wavelets for Signal Feature Discrimination and the Inversion of Complicated Maps , 1994, J. Chem. Inf. Comput. Sci..

[18]  George Kantor,et al.  Approximate Matrix Diagonalization for Use in Distributed Control Networks , 1999 .

[19]  Kristi A. Morgansen,et al.  Limited communication control , 1999 .

[20]  Tad Hogg,et al.  Learning in Multiagent Control of Smart Matter , 1997 .

[21]  George Kantor,et al.  Efficient Implementation of Controllers for Large Scale Linear Systems via Wavelet Packet Transforms , 1998 .

[22]  Wing Shing Wong,et al.  Systems with finite communication bandwidth constraints. II. Stabilization with limited information feedback , 1999, IEEE Trans. Autom. Control..