A Theory of Synchronization Errors in Interconnected Systems

The most important error-generating events in Network Control Systems are known to be: time-variant communication delays, packet drops, bandwidth limitations, and synchronization errors. Synchronization errors constitute an important and often overlooked network induced non-ideal behavior. Little is known about their effects on system stability and performance robustness. Therefore studying and understanding synchronization errors is the primary objective of this thesis. Most literature either assumes that all subsystems are working synchronously or use a continuous-time model, to circumvent the difficulties of the desynchronized discrete-time approach. Only a few papers proposed a desynchronized discrete time model, using certain idealized assumptions. In this thesis we present two new models that efficiently capture the salient effects of synchronization errors in interconnected discrete-time systems: a state-space based model, and a system description based on infinite dimensional Toeplitz like operators. We first model and perform stability analysis for systems with identical clock frequencies that are operating asynchronously. This class of systems finds applications in high-speed circuitry. The introduced stability analysis utilizes spectral methods, and a proposed fault detection algorithm identifies desynchronized sub-systems. For the case of a two system control loop, it is shown that the nature of the period ratio $frac{T_1}{T_2}$ profoundly affects system behavior. The case of commensurate rates $frac{T_1}{T_2}$ is tackled using both, the state-space approach and the infinite-dimensional operator approach. From the state-space description it follows that the system is equivalent to a periodic, time-variant, discrete-time system. Stability analysis is then performed using the classical spectral techniques. The Toeplitz based approach reveals an underlying periodic matrix block pattern in the infinite-dimensional matrix representation. This periodicity property is utilized to derive equivalent stability conditions. Finally, we investigate the case of irrational ratios $frac{T_1}{T_2}$, using the two above-mentioned approaches. It is shown that the infinite-dimensional matrix representation always admits a sparse factorization. This amazing property will enable the formulation of future stability conditions and computationally efficient algorithms. Based on the state-space model, a sufficient stability condition is derived, that can be made arbitrarily close to being necessary at the expense of increased computations. The case of period ratio uncertainties is also addressed.