Controllability in Complex Network Systems with Switching Topology

Controllability in complex network systems (CNSs) has attracted much attention due to their extensive applications such as power grids regulation, formation task of unmanned air vehicles and cooperative surveillance. These networks are often dynamically changing due to the complex environment in practice. Hence, the CNSs with switching topology are used to describe such circumstances. Although the existing research on the controllability of CNSs has been paid much attention by many researchers, there are still some further problems worth exploring. For example, the structural controllability of CNSs with periodic switching topology (CNSs-PST), the state controllability involving switching time sequence for CNSs-PST, and the application of controllability in fractional-order switching CNSs under the influence of delay are still not discussed. These problems are necessary and important in CNSs. In this thesis, we will first investigate the relationship between structure controllability and state controllability, and then study the controllability in CNSs with switching topology from the following several topics. The first part of this thesis is aimed to investigate the structural controllability in the CNSs-PST from two aspects. It should be noted that PST has been widely observed in both natural and engineered systems. For instance, in industrial automation or robotic systems, tasks may be periodically allocated, leading to periodic changes in the cooperative topology of automated devices. However, existing studies on CNSs-PST primarily focus on stability issues. To date, how the periodic switching mechanism affects the controllability of CNSs remains unclear, particularly in large-scale CNSs. In light of this, this part first addresses the structual controllability problem in complex network systems with strictly periodic switching topologies (CNSs-SPST), i.e., both the topology structures and edge weights undergo simultaneous periodic switching. We establish the structural controllability conditions based on some existing notions, such as temporal dilation, temporal walk, $n$-walk theory. Then, we further address the structual controllability problem in complex network systems with generally periodic switching topologies (CNSs-GPST), i.e., the topology structures undergo periodic switching and edge weights may undergo non-periodic switching. A condition to judge structural controllability is obtained, which only requires to analyse the topology of the joint graph in a single period for CNSs-GPST. This part is pivotal as it systematically addresses the unexplored relationship between periodic switching mechanisms and structural controllability in CNSs, bridging a critical knowledge gap for both strict and general switching scenarios. Finally, several numerical examples are presented to validate the theoretical results. The second part of this thesis develops new state controllability conditions for CNSs-PST with respect to dwell time. Firstly, a necessary and sufficient condition is proposed to determine how dwell times influence the state controllability by using the algebraic and geometric knowledge of matrix. Next, based on the obtained controllability condition, we further establish several state controllability criteria that will not be affected by dwell time for CNSs-PST under special constraints. This implies that the state controllability of CNSs-PST remains even when the dwell time changes. Compared with existing studies, the results in this section explore the intrinsic connection between dwell time and controllability of switched CNSs. This provides novel insights for stability analysis and controller design of such systems. Moreover, the investigated model serves as a fundamental framework. If practical factors such as time delays and impulsive effects are further incorporated, the proposed results can be extended to broader scenarios. Finally, several numerical examples are presented to confirm the theoretical results. The third part of this thesis is to apply the controllability theory to fractional-order switched multi-agent systems with input delay. Due to the complexity of multi-agent working environment, the dynamic characteristics of some natural or engineering phenomena may not be described by integer order dynamical systems, but can only be explained by the cooperative behavior of intelligent individuals in fractional order dynamics. In light of this, the relevant controllability conditions are established for fractional-order switching multi-agent systems with input delay (FOSMASID) by resorting to the algebraic method. First, we obtain the solution representation of FOSMASID over every subinterval by using the Laplace transform and mathematical induction. Next, by introducing the Gramian matrices over every subinterval, we establish the controllability conditions without requiring all impulse-dependent matrices to be nonsingular. Furthermore, based on the relevant matrix theory, we further establish a controllability condition that is necessary and sufficient by introducing the form of a row of Gramian matrices. In the last, a numerical example with three subsystems is worked out to illustrate the theoretical results.<p></p>