Distributed control of spatially invariant systems

Abstract

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.

A Decomposition Approach to Distributed Control of Dynamic Deformable Mirrors

“…(2)) is bounded. In order to do so, we will first evaluate the spectrum of the pattern matrix (or better, operator) in the infinite case (N → ) by making use of the theory of spatially invariant systems by Bamieh et al (2002), and then relate this result to finite truncations.…”

Scalable distributed Kalman Filtering for mass-spring systems

“…Kalman Filter design for the entire system reduces to design for a parametrized family of low-order systems, as remarked in [2]. The same goes for controllability/observability analysis, controller design, performance evaluation etc.…”

“…The corresponding nodal dynamics arė decay exponentially with distance at least asymptotically as shown in [2], and should be amenable to spatial truncation.…”

“…A distributed LQG problem on general graphs with localized states and information propagation delays is solved in [6], but the estimator in general has to keep an estimate of the global state in every node. Localized couplings in the plant will often lead to optimal controllers with couplings that decay with distance, see [2], [5]. For an LMI approach to distributed control, see [3] and references therein.…”

“…Thus spatial truncation is a theoretically attractive approximation, see [2]. This paper investigates translation invariant Kalman Filtering and spatial truncation for mass-spring systems with the aim to illustrate actual behavior and to highlight interesting properties.…”