| Structural controllability is critical for researching and controlling large scale networks.In real applications,to design an effective control schemes for a given network,the selection of driver nodes must satisfy network's structural controllability.It will be more flexible and operable for designing control schemes if we can find out multiple selections for driver nodes.In this paper,we investigate the problem how to describe and obtain the solution domain for all selections of driver nodes guaranteeing structural controllability.Different from existing works,we focus on finding all possible selections for driver nodes,rather than only giving one selection.Furthermore,in accord with practical applications,we define the complete selection rule set as the solution domain which is composed of a series of selection rules expressed by intuitive algebraic forms,rather than listing the feasible driver nodes selection schemes one by one.The complete selection rule set explicitly indicates which nodes must be controlled and how many nodes need to be controlled among a node set and thus is particularly helpful for freely selecting driver nodes.Based on two algebraic criteria of structural controllability,we separately develop an input-connectivity algorithm and a relevancy algorithm to deduce selection rules for driver nodes.To reduce the computational complexity,we propose a pretreatment algorithm to efficiently reduce the scale of network's structural matrix,and a rearrangement algorithm to divide the matrix into several independent ones with lower triangular form.A general procedure is proposed to get the complete selection rule set for driver nodes guaranteeing structural controllability.Simulation tests with efficiency analysis of the proposed algorithms are given and the result of applying the proposed procedure to some real networks is also shown,indicating the validity of the procedure.In order to reduce the computational complexity in high-dimensional networks,different methods of partitioning matrix are discussed in this paper,and the procedure for deducing sufficient selection rule set with higher computational efficiency is developed.The resultant selection rule set is not complete,but it provides rules with more succinct expression for driver nodes' selection. |
