Distributed control of linear time-varying systems interconnected over arbitrary graphs.

*(English)*Zbl 1305.93131Summary: We focus on designing distributed controllers for interconnected systems in situations where the controller sensing and actuation topology is inherited from that of the plant. The distributed systems considered are composed of discrete-time linear time-varying subsystems interconnected over arbitrary graph structures. The main contribution of this paper is to provide results on general graph interconnection structures in which the graphs have potentially an infinite number of vertices. This is accomplished by first extending previous machinery developed for systems with spatial dynamics on the lattice inline image. We derive convex analysis and synthesis conditions for design in this setting. These conditions reduce to finite sequences of LMIs in the case of eventually periodic subsystems interconnected over finite graphs. The paper also provides results on distributed systems with communication latency and gives an illustrative example on the distributed control of hovercrafts along eventually periodic trajectories. The methodology developed here provides a unifying viewpoint for our previous and related work on distributed control.

##### MSC:

93C55 | Discrete-time control/observation systems |

94C15 | Applications of graph theory to circuits and networks |

93B36 | \(H^\infty\)-control |

93B50 | Synthesis problems |

##### Keywords:

distributed control; \(H_\infty\) control; time-varying systems; arbitrary graph; eventually periodic systems
PDF
BibTeX
XML
Cite

\textit{M. Farhood} et al., Int. J. Robust Nonlinear Control 25, No. 2, 179--206 (2015; Zbl 1305.93131)

Full Text:
DOI

##### References:

[1] | Raza, Vehicle following control design for automated highway systems, IEEE Control Systems 16 (6) pp 43– (1996) |

[2] | Horowitz, Control design of an automated highway system, Proceedings of the IEEE 88 (7) pp 913– (2000) |

[3] | Wolfe JD Chichka DF Speyer JL Decentralized controllers for unmanned aerial vehicle formation flight Proceedings of the AIAA Guidance, Navigation, and Control Conference 1996 |

[4] | Fowler, A formation flight experiment, IEEE Control Systems 23 (5) pp 35– (2003) |

[5] | Jovanovic, On the ill-posedness of certain vehicular platoon control problems, IEEE Transactions on Automatic Control 50 (9) pp 1307– (2005) · Zbl 1365.93323 |

[6] | Shaw GB The generalized information network analysis methodology for distributed satellite systems PhD Thesis 1998 |

[7] | Ruggiero, Modeling and vibration control of an active membrane mirror, Smart Materials and Structures 18 (9) pp 10pp– (2009) |

[8] | Jovanović, A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure, IEEE Transactions on Automatic Control: Special Issue on Systems Biology 53 pp 75– (2008) · Zbl 1366.92043 |

[9] | Dullerud, Distributed control of heterogeneous systems, IEEE Transactions on Automatic Control 49 (12) pp 2113– (2004) · Zbl 1365.93317 |

[10] | D’Andrea, Distributed control design for spatially interconnected systems, IEEE Transactions on Automatic Control 48 (9) pp 1478– (2003) · Zbl 1364.93206 |

[11] | Recht, Distributed control of systems over discrete groups, IEEE Transactions on Automatic Control 49 (9) pp 1446– (2004) · Zbl 1365.93093 |

[12] | Langbort, Distributed control design for systems interconnected over an arbitrary graph, IEEE Transactions on Automatic Control 49 (9) pp 1502– (2004) · Zbl 1365.93141 |

[13] | Bamieh, Distributed control of spatially invariant systems, IEEE Transactions on Automatic Control 47 (7) pp 1091– (2002) · Zbl 1364.93363 |

[14] | Bamieh, A convex characterization of distributed control problems in spatially invariant systems with communication constraints, Systems & Control Letters 54 (6) pp 575– (2005) · Zbl 1129.93301 |

[15] | Fardad, Frequency analysis and norms of distributed spatially periodic systems, IEEE Transactions Automatic Control 53 (10) pp 2266– (2008) · Zbl 1367.93148 |

[16] | Farhood, LMI tools for eventually periodic systems, Systems and Control Letters 47 (5) pp 417– (2002) · Zbl 1106.93327 |

[17] | Farhood, Duality and eventually periodic systems, International Journal of Robust and Nonlinear Control 15 (13) pp 575– (2005) · Zbl 1100.93019 |

[18] | Ben-Artzi, Inertia theorems for nonstationary discrete systems and dichotomy, Linear Algebra and its Applications 120 pp 95– (1989) · Zbl 0677.93037 |

[19] | Farhood, Model reduction of nonstationary LPV systems, IEEE Transactions on Automatic Control 52 (2) pp 181– (2007) · Zbl 1366.93089 |

[20] | Farhood, Control of nonstationary LPV systems, Automatica 44 (8) pp 2108– (2008) · Zbl 1283.93082 |

[21] | Zhou, Robust and Optimal Control (1996) |

[22] | Packard, The complex structured singular value, Automatica 29 (1) pp 71– (1993) · Zbl 0772.93023 |

[23] | Gahinet, A linear matrix inequality approach to H control, International Journal of Robust and Nonlinear Control 4 (4) pp 421– (1994) · Zbl 0808.93024 |

[24] | Packard, Gain scheduling via linear fractional transformations, Systems and Control Letters 22 (2) pp 79– (1994) · Zbl 0792.93043 |

[25] | Dullerud, A new approach to analysis and synthesis of time-varying systems, IEEE Transactions on Automatic Control 44 (8) pp 1486– (1999) · Zbl 1136.93321 |

[26] | Farhood, LPV control of nonstationary systems: a parameter-dependent Lyapunov approach, IEEE Transactions on Automatic Control 57 (1) pp 209– (2012) · Zbl 1369.93244 |

[27] | D’Andrea R A linear matrix inequality approach to decentralized control of distributed parameter systems Proceedings of the American Control Conference 1998 1350 1354 |

[28] | Lofberg J YALMIP: a toolbox for modeling and optimization in MATLAB Proceedings of the CACSD Conference 2004 http://users.isy.liu.se/johanl/yalmip/ |

[29] | Toh, SDPT3 - a Matlab software package for semidefinite programming, Optimization Methods and Software 11 pp 545– (1999) · Zbl 0997.90060 |

[30] | Mishra A Dullerud GE On an operator-pencil approach to distributed control of heterogeneous systems Proceedings of the IEEE Conference on Decision and Control 2011 7506 7511 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.