Quantised Control in Distributed Embedded Systems

Problem Presentation Traditional control design is based on ideal assumptions concerning the amount, type andaccuracy of the information flow that can be circulated across the controller. Unfortunately, real implementation plat-forms exhibit non-idealities that may substantially invalidate such assumptions. As a result, the system’s closed-loopperformance can be severely affected and sometimes stability itself is jeopardised. These problems show up withparticular strength when multiple feedback loops share a limited pool of computation and communication resources.In this case the designer is confronted with the challenging task of choosing at the same time the control law and theoptimal allocation policy for the shared resources (control algortihm/system architecture co-design). An intriguinggeneral discussion for this class of problem can be found in [2]. Investigations in this field have been developed eversince in several directions. A first prong of research activities has focused on the problem of resource sharing, i.e.,finding optimal allocation policies for shared computation and communication resources [10, 7, 3]. However, thesepapers do not explicitly cope with quantisation and bit rate constraints that play an important role in complex dis-tributed systems. A remarkable thread of papers has focused on the problem of stabilisation under bit rate constraints[4, 12, 5, 9, 6, 8, 1]. The main concern in these works is to find encoding-decoding schemes that make for an optimaluse of the channel, when the latter is used in a control loop. In this perspective, the authors generally synthesizequantisation schemes instrumental to this goal. Albeit interesting from a theoretical point of view, this approach isto be verified from the standpoint of technological feasibility. In [11] a different view is taken. The authors analysethe attainable control performance when quantisation is a fixed element of the problem. An evident motivation forthis work is the analysis of control systems where actuation and/or sensing are e.g., binary, thresholded or quantisedsensors, actuators or converters.In this work, we make the same assumption as in [11]: control loops are operated by quantised actuators, whichare regarded as given hardware components to build on the top of. For the sake of simplicity, we restrict to the caseof uniform quantisers. Moreover, a limited bandwidth channel is shared between several independent feedback loops.Each loop is used to control a first-order linear and time-invariant plant whose dynamics is described by