Bassam Bamieh

Distributed control of spatially invariant systems

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), H/sub 2/ and H/sub /spl infin//, 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.

A general framework for linear periodic systems with applications to H/sup infinity / sampled-data control

The authors present a framework for dealing with continuous-time periodic systems. The main tool is a lifting technique which provides a strong correspondence between continuous-time periodic systems and certain types of discrete-time time-invariant systems with infinite-dimensional input and output spaces. Despite the infinite dimensionality of the input and output spaces, a lifted system has a finite-dimensional state space if the original system does. This fact permits rather constructive methods for analyzing these systems. As a demonstration of the utility of this framework, the authors use it to describe the continuous-time (i.e., intersample) behavior of sampled-data systems, and to obtain a complete solution to the problem of parameterizing all controllers that constrain the L/sup 2/-induced norm of a sampled-data system to within a certain bound. >

A lifting technique for linear periodic systems with applications to sampled-data control

A lifting technique is developed for periodic linear systems and applied to the H∞ and H2 sampled-data control problems.

Coherence in Large-Scale Networks: Dimension-Dependent Limitations of Local Feedback

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-D 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 1-D 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 large-scale 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.

Componentwise energy amplification in channel flows

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. In this paper, we analyse the dynamical properties of the Navier–Stokes (NS) equations with spatially distributed and temporally varying body force fields. These fields are considered as inputs, and different combinations of the resulting velocity fields are considered as outputs. This input–output analysis can in principle be done in any geometry and for the full nonlinear NS equations. In such generality, however, it is difficult to obtain useful results. We therefore concentrate on the geometry of channel flows, and the input–output dynamics of the linearized Navier–Stokes (LNS)

A Convex Characterization of Distributed Control Problems in Spatially Invariant Systems with Communication Constraints

In this paper we consider the problem of distributed controller design in spatially invariant systems for which communication among sites is limited. In particular, the controller is constrained so that information is propagated with a delay that depends on the distance between subsystems—a structure we refer to as “funnel” causality. We show that the problem of optimal design can be cast as a convexproblem provided that the plant has a similar funnel-causality structure, and the propagation speeds in the controller are at least as fast as those in the plant. As an example, we consider the case of the wave dynamics with limited propagation speed control.

On the ill-posedness of certain vehicular platoon control problems

We revisit the vehicular platoon control problems formulated by Levine and Athans and Melzer and Kuo. We show that in each case, these formulations are effectively ill-posed. Specifically, we demonstrate that in the first formulation, the systems stabilizability degrades as the size of the platoon increases, and that the system loses stabilizability in the limit of an infinite number of vehicles. We show that in the LQR formulation of Melzer and Kuo, the performance index is not detectable, leading to nonstabilizing optimal feedbacks. Effectively, these closed-loop systems do not have a uniform bound on the time constants of all vehicles. For the case of infinite platoons, these difficulties are easily exhibited using the theory of spatially invariant systems. We argue that the infinite case is a useful paradigm to understand large platoons. To this end, we illustrate how stabilizability and detectability degrade as functions of a finite platoon size, implying that the infinite case is an idealized limit of the large, but finite case. Finally, we show how to pose H/sub 2/ and H/sub /spl infin// versions of these problems where the detectability and stabilizability issues are easily seen, and suggest a well-posed alternative formulation based on penalizing absolute positions errors in addition to relative ones.

Energy amplification in channel flows with stochastic excitation

We investigate energy amplification in parallel channel flows, 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(R3) amplification. 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 fields. The dependence of these quantities on both the Reynolds number and the spanwise wave number are explicitly computed. We show how the amplification mechanism is due to a coupling between wall-normal velocity and vorticity disturbances, which in turn is due to nonzero 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.

Convergence Rates of Distributed Average Consensus With Stochastic Link Failures

We consider a distributed average consensus algorithm over a network in which communication links fail with independent probability. In such stochastic networks, convergence is defined in terms of the variance of deviation from average. We first show how the problem can be recast as a linear system with multiplicative random inputs which model link failures. We then use our formulation to derive recursion equations for the second order statistics of the deviation from average in networks with and without additive noise. We give expressions for the convergence behavior in the asymptotic limits of small failure probability and large networks. We also present simulation-free methods for computing the second order statistics in each network model and use these methods to study the behavior of various network examples as a function of link failure probability.

A general moment-invariants/attributed-graph method for three-dimensional object recognition from a single image

A consistent development of general moment invariants of affine transformations for two-dimensional image functions is presented. Based on this development, a new general moment-invariants/attributed-graph (MIAG) method is presented for the identification of three-dimensional objects from a single observed image using a model-matching approach. The three-dimensional location and orientation parameters of the object are also obtained as a byproduct of the matching procedure. The scheme presented allows the observed object to be partially Occluded. For identification purposes, a three-dimensional object is represented by an attributed graph describing the geometrical structure and shape of the surface bounding the object. In such a description, two-dimensional general moment invariants of the rigid planar patches (RPP) constituting the object faces are used as attributes or feature vectors which are invariant under three-dimensional motion. With this representation, the identification problem becomes a subgraph isomorphism problem between the observed image and a library model. An algorithm is presented for this matching process, and the results are illustrated by computer simulations.