LARGESCALE SYSTEMS SCALING BASICS WHY ARE THEY DIFFERENT

LARGE-SCALE SYSTEMS SCALING

BASICS

WHY ARE THEY DIFFERENT

NORMS

CHANGE OF THE SYSTEM Controller fixed! Bassam Bamieh Rick Middleton Mihailo Jovanović … Mehran Mesbahi Magnus Egersted Daniel Zelazo … Also: • Controllability • Observability

COMMON PROPERTIES OF LARGE-SCALE SYSTEMS

B. BAMIEH: COHERENCE IN LARGE SCALE NETWORKS Paper: B. Bamieh, M. R. Jovanović, P. Mitra, and S. Patterson, “Coherence in Large-Scale Networks: Dimension-Dependent Limitations of Local Feedback, ” Autom. Control. IEEE Trans. , vol. 57, no. 9, pp. 2235– 2249, 2012. Scaling of effect of white noise when adding agents Effect of dimension of the graph Role of number of integrators Local feedback not sufficient!

ASSUMPTIONS White noise Reason: Fourier transform!

VEHICULAR PLATOON String of vehicles 1…N - circle Regulation error as nearest neighbor interaction Is it enough to look at neighbors? Performace? Limitations? Fundamentally difficult problem! Bad topology = one-dimensional graph 9

MODEL control White-noise disturbance Relative local feedback Absolute global feedback

SIMILAR TO WAVE EQUATION

SIMULATION

DISORDER Intuition: Local feedback can only suppress short-scale disturbances Needed global feedback? In higher dimensions, local feedback is sufficient!

COMMENTS Using only local feedback in we cannot have 1 D, large and yet coherent formations! In higher dimensions the response is bounded! Similar to statistical mechanics Resistive lattice

ROLE OF INTEGRATORS

SPATIAL DIMENSIONS

LQ CONTROLLER Still localized The same scaling problems!

OPEN PROBLEMS What is a dimension of a general graph? Nonlinear feedback? Dynamic feedback? Role of asymmetry? Eigenvalues do not show much!

ASYMMETRIC VEHICULAR PLATOONS EVEN WORSE SCALING

PLATOON 21

MODEL

MOTIVATION - SCALING IN PLATOONS Step responses of platoons with different control Symmetric control Predecessor following Symmetric control Asymmetric control 23

LAPLACIAN OF PLATOONS Path graph – nearest neighbor interaction Only one path among vertices Tridiagonal Laplacian: real, positive eigenvalues Interlacing for principal submatrices Signature similar to totally non-negative matrix All holds for arbitrary weights 24

ASYMMETRY IN CONTROL 25

STEADY-STATE GAIN

HARMONIC INSTABILITY 27

PRODUCT FORMULA 28

EXPONENTIAL SCALING 29

TWO AND MORE INTEGRATORS 30

DESIGN OF STRING-STABLE CONTROLLER 31

KNOWN SUFFICIENT TESTS 32

COMPARISON OF TRANSIENTS WITH ONE INTEGRATOR 33

CONCLUSIONS FOR PLATOONS 34

PAPERS Herman, Martinec, Hurak, Sebek: Nonzero bound on Fiedler eigenvalue causes exponential growth of H-infinity norm of vehicular platoon, IEEE Transactions on Automatic Control Martinec, Herman, Hurak, Sebek: Wave-absorbing vehicular platoon controller, European Journal of Control Martinec, Herman, Sebek: The transfer-function approach to travelling waves in path graphs, submitted, ar. Xiv 1410. 0474 Herman, Martinec, Hurak, Sebek: Scaling of transfer functions in vehicular platoons : the role of asymmetry disputed, submitted, ar. Xiv 1410. 3943 35 http: //aa 4 cc. dce. fel. cvut. cz, ivo. herman@fel. cvut. cz