Summary
It is known in the literature on Automated Highway Systems that information flow can significantly affect the propagation of errors in spacing in a collection of vehicles. This chapter investigates this issue further for a homogeneous collection of vehicles, where in the motion of each vehicle is modeled as a point mass and is digitally controlled. The structure of the controller employed by the vehicles is as follows: \( U_i (z) = C(z)\sum\nolimits_{j \in S_i } {(X_i - X_j - \tfrac{{L_{ij} z}} {{z - 1}})} \), where U i(z) is the (z- transformation of) control action for the i th vehicle, X i is the position of the i th vehicle, L ij is the desired distance between the i th and the j th vehicles in the collection, C(z) is the discrete transfer function of the controller and S i is the set of vehicles that the i th vehicle can communicate with directly. This chapter further assumes that the information flow is undirected, i.e., i ∈ S j ⇔ j ∈ S i and the information flow graph is connected. We consider information flow in the collection, where each vehicle can communicate with a maximum of q(n) vehicles. We allow q(n) to vary with the size n of the collection. We first show that C(z) cannot have any zeroes at z = 1 to ensure that relative spacing is maintained in response to a reference vehicle making a maneuver where its velocity experiences a steady state offset. We then show that if the control transfer function C(z) has one or more poles located at z = 1, then the motion of the collection of vehicles will become unstable if the size of the collection is sufficiently large. These two results imply that C(1) ≠ 0 and C(1) must be well defined. We further show that if q(n)/n → 0 as n → ∞ then there is a low frequency sinusoidal disturbance of at most unit amplitude acting on each vehicle such that the maximum error in spacing response increase at least as \( \Omega \left( {\sqrt {\tfrac{{n^3 }} {{q^3 (n)}}} } \right) \). A consequence of the results presented in this chapter is that the maximum of the error in spacing and velocity of any vehicle can be made insensitive to the size of the collection only if there is at least one vehicle in the collection that communicates with at least Ω(n) other vehicles in the collection. We also show that there can be at most one vehicle that communicates with Ω(n) vehicles and that any other vehicle in the collection can only communicate with at most p vehicles, where p depends only on the chosen controller and the its sampling time.
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References
D. Swaroop and J. K. Hedrick, String Stability of interconnected systems IEEE Transactions on Automatic Control, vol 41, pp. 349–357, March 1996.
Aniruddha G. Pant, Mesh stability of formations of Unmanned Aerial vehicles, Ph.D Thesis, University of California, Berkeley, 2002.
D. Swaroop, String Stability of interconnected systems: An application to platooning in automated highway systems, Ph.D. Thesis, University of California, Berkeley, 1994.
Singh, Sahjendra N., Rong Zhang, Phil Chandler, and Siva Banda, Decentralized Adaptive Close Formation Control of UAVs, AIAA 2001-0106, 39th AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, Jan. 2001.
Pachter, Meir, John J. DAzzo, and Andrew W. Proud, Tight Formation Flight Control, AIAA Journal of Guidance, Control and Dynamics, Vol. 24, No. 2, MarchApril 2001.
Khatir M., and E.J. Davison, Bounded Stability and Eventual String Stability of a Large Platoon of Vehicles using Non-Identical Controllers, 2004 IEEE Control and Decision Conference, Paradise Island, Dec. 2004, to appear.
Tolga Eren, Brian D. O. Anderson, A. Stephen Morse, Walter Whiteley, and Peter N. Belhumeur. Operations on rigid formations of autonomous agents. Communications in Information and Systems, 2004. to appear.
Peter Joseph Seiler, Coordinated control of Unmanned Aerial Vehicles, Ph.D Thesis, Department of Mechanical Engineering, University of California, Berkeley, 2001.
Swaroop, D., Hedrick, J.K., Chien, C.C. and Ioannou, P.A., “A Comparison of Spacing and Headway Control Laws for Automatically Controlled Vehicles,” Vehicle System Dynamics Journal, Vol. 23, No. 8, pp. 597–625, 1994.
J. A. Fax and R.M. Murray, Information Flow and cooperative control of vehicle formations, Proceedings of the IFAC World Congress, Barcelona, Spain, pp. 2360–2365, July, 2002.
P. J. Seiler, Aniruddha Pant and J. K. Hedrick, Preliminary Investigation of Mesh stability for Linear Systems, IMECE99/DSC-7B-1, 1999.
G. F. Franklin, J. D. Powell and M. L. Workman, Digital Control of Dynamic Systems, Prentice Hall, 1997.
Aniruddha Pant, Pete Seiler, T. John Koo, Karl Hedrick, Mesh Stability of Unmanned Aerial Vehicle Clusters, Proceedings of American Control Conference 2001, pp. 62–68.
Sai Krishna Y, Swaroop Darbha and K.R. Rajagopal Information flow and its relation to the stability of motion of vehicles in a rigid formation, Proceedings of American Control Conference 2005, pp. 1853–1858.
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Yadlapalli, S.K., Darbha, S., Rajagopal, K.R. (2007). Information Flow Requirements for the Stability of Motion of Vehicles in a Rigid Formation. In: Grundel, D., Murphey, R., Pardalos, P., Prokopyev, O. (eds) Cooperative Systems. Lecture Notes in Economics and Mathematical Systems, vol 588. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48271-0_21
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DOI: https://doi.org/10.1007/978-3-540-48271-0_21
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