We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of vehicular platoons, flow control, microelectromechanical systems (MEMS), smart structures, and systems described by partial differential equations with constant coefficients and distributed controls and measurements. For fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator (LQR), 2 and , optimal controllers can be obtained by solving a parameterized family of standard finite-dimensional problems. We show that optimal controllers have an inherent degree of decentralization, and this provides a practical distributed controller architecture. We also prove a general result that applies to partially distributed control and a variety of performance criteria, stating that optimal controllers inherit the spatial invariance structure of the plant. Connections of this work to that on systems over rings, and systems with dynamical symmetries are discussed.

Abstract-We consider distributed consensus and vehicular formation control problems. Specifically we address the question of whether local feedback is sufficient to maintain coherence in large-scale networks subject to stochastic disturbances. We define macroscopic performance measures which are global quantities that capture the notion of coherence; a notion of global order that quantifies how closely the formation resembles a solid object. We consider how these measures scale asymptotically with network size in the topologies of regular lattices in 1, 2 and higher dimensions, with vehicular platoons corresponding to the 1 dimensional case. A common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures, with a dimensions of 3 being necessary to achieve coherence in consensus and vehicular formations under certain conditions. In particular, we show that it is impossible to have large coherent one dimensional vehicular platoons with only local feedback. We analyze these effects in terms of the underlying energetic modes of motion, showing that they take the form of large temporal and spatial scales resulting in an accordion-like motion of formations. A conclusion can be drawn that in low spatial dimensions, local feedback is unable to regulate largescale disturbances, but it can in higher spatial dimensions. This phenomenon is distinct from, and unrelated to string instability issues which are commonly encountered in control problems for automated highways.

We study the linearized Navier-Stokes (LNS) equations in channel flows from an input-output point of view by analysing their spatio-temporal frequency responses. Spatially distributed and temporally varying body force fields are considered as inputs, and components of the resulting velocity fields are considered as outputs into these equations. We show how the roles of Tollmien-Schlichting (TS) waves, oblique waves, and streamwise vortices and streaks in subcritical transition can be explained as input-output resonances of the spatio-temporal frequency responses. On the one hand, we demonstrate the effectiveness of input field components, and on the other, the energy content of velocity perturbation components. We establish that wall-normal and spanwise forces have much stronger influence on the velocity field than streamwise force, and that the impact of these forces is most powerful on the streamwise velocity component. We show this using the relative scaling of the different input-output system components with the Reynolds number. We further demonstrate that for the streamwise constant perturbations, the spanwise force localized near the lower wall has, by far, the strongest effect on the evolution of the velocity field.

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We investigate energy ampli cation in parallel channel ows, where background noise is modeled as stochastic excitation of the linearized Navier-Stokes equations. We show analytically that the energy of three-dimensional streamwise-constant disturbances achieves O(R 3) ampli cation. Our basic technical tools are explicit analytical calculations of the traces of solutions of operator Lyapunov equations, which yield the covariance operators of the forced random velocity elds. The dependence of these quantities on both the Reynolds number and the spanwise wave-number are explicitly computed. We show how the ampli cation mechanism is due to a coupling between normal velocity and vorticity disturbances, which in turn is due to non-zero mean shear and disturbance spanwise variation. This mechanism is viewed as a consequence of the non-normality of the dynamical operator, and not necessarily due to the existence of near resonances or modes with algebraic growth.

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