Control over networks with fading communication channels - Liang, Xu (2018)
Classical control theory assumes that the communication links connecting plants, sensors and controllers are perfect. However, this is not true in practical applications. The imperfection of communication channels would introduce uncertainties into feedback control systems, which might impact the stability and performance of the corresponding control system. Different issues arise when different communication channels are used in control systems, such as the minimal data rate, tolerable time delay and minimal data rate, tolerable time delay and minimal etc. This thesis focuses on the fading phenomenon in wireless communications and studies how channel fading affects the stability of feedback control systems. In the first part of this thesis, we consider the mean square stabilizability problem of discrete-time linear time-invariant (LTI) systems controlled over fading channels. Firstly, we consider the power constrained fading channel, which suffers from both SNR constraints and the time-varying independent and identically distributed (i.i.d. ) channel fading. We try to characterize channel requirements for the existence of coding and controlling policies that can mean square stabilize the linear system. We show that there is a fundamental limitation on the mean square stabilizability. For scalar systems and two-dimensional systems, necessary and sufficient conditions for the mean square stabilizability are provided. Moreover, time division multiple access (TDMA) and adaptive TDMA communication schemes are designed for high-dimensional systems, which are proved to be optimal under certain situations. Then we proceed to study the mean square stabilizability problem over Gaussian finite-state Markov channels, which suffer from both SNR constraints and the correlated channel fading modeled by a Markov chain. Similarly, the existence of a fundamental limitation for mean square stabilizability is proved. Sufficient stabilization conditions under TDMA communication schemes are derived in terms of the stability of of a Markov jump linear system (MJLS). Besides, for networked control over power constrained Markov lossy channels, one special kind of Gaussian finite-state Markov channels, we present a necessary and sufficient condition for the mean square stabilizability of two-dimensional systems. Moreover, improved sufficient stabilizability conditions are derived based on an adaptive TDMA communication scheme for general high-dimensional systems. In the second part of this thesis, we study the consensusability problem of linear discrete-time multi-agent systems (MASs) over fading networks with both undirected and directed communication topologies. The agents in the MAS communicate with their neighborhoods through fading channels. We aim to characterize requirements on the agent dynamics, channel capacities and the network topology for the existence of a distributed consensus controller. First of all, we study the consensus problem under an undirected graph setting. Sufficient conditions to guarantee mean square consensus are derived with both identical fading networks and non-identical fading networks. The results imply that the consensusability is closely related to the statistics of fading networks, the eigenratio of the graph, and the instability degree of the dynamical system. Then, we consider the mean square consensus problem over fading networks with directed graphs. Sufficient conditions are firstly provided for mean square consensus over identical fading networks. For consensus over non-identical fading networks with directed graphs, compressed in-incidence matrix (CIIM), compressed incidence matrix (CIM) and compressed edge Laplacian (CEL) are proposed to facilitate the modeling and consensus analysis. It is shown that the mean square consensusability is solely determined by the edge state dynamics on a directed spanning tree. As a result, sufficient conditions are provided for mean square consensus over non-identical fading networks with directed graphs in terms of fading parameters, the network topology and the agent dynamics. Moreover, the role of network topology on the mean square consensusability is discussed.