Non-negative Edge-consensus Problems For Networked Linear Positive Systems With Input Saturation

Consensus as the most important and fundamental coordinated behavioural regulation of cooperative control has been studied for many years,but most of current works about cooperative control of networked systems are concerned with the nodes of network.However,for some networks,such as traffic networks,social networks and computers-routers networks which focus on material,information or data transmission,the evolutions of edge states are also important.Based on this phenomenon,the research on edge consensus of networked systems has gained an interest of some researchers.Due to some realistic factors,such as edge-states' non-negative constraints,input saturation,edge states' scaling property,the internal information that is difficult or impossible to detect,and large scale networks' global information which is difficult or impossible to obtain,the edge consensus problems need to be further studied.Therefore,this thesis proposes a variety of observer-based edge-consensus protocols to solve edge-consensus problems of networked linear positive system with some realistic factors.The main achievements are given as follows:For the edge-consensus problems of networked continuous-time linear positive systems with unmeasurable edge-states,input saturation and edge-states' non-negative constraints,two kinds of observer-based edge-consensus protocols are proposed,and the corresponding evolution processes are given.Drawing support from line graph theory,positive system theory,ARE-based and DARE-based low-gain feedback technique,and Lyapunov function-based method,the mathematical expressions of feedback gain matrix and observer gain matrix are constructed,and sufficient conditions for guaranteeing non-negative edge-states and bounded inputs are obtained.Moreover,the feedback gain matrix and observer gain matrix which can satisfy all sufficient conditions,are existed and easy to obtain.For the edge-consensus problems of networked discrete-time linear positive systems with edge states' scaling property,unmeasurable edge-states,input saturation and edge-states' non-negative constraints,two kinds of observer-based scaled edge-consensus protocols are designed.Based on line graph theory,the specific evolution processes of two protocols are given.The mathematical expressions of feedback gain matrix and observer gain matrix are obtained by solving MARE and MDARE,respectively.Then by virtue of low-gain feedback technique,positive system theory and Lyapunov stability theory,the scaled consensus conditions which can satisfy the input saturation and edge-states' non-negative constraints,are derived.Moreover,all scaled consensus conditions can be guaranteed when the feedback gain matrix and observer gain matrix are non-positive.For the fully distributed edge-consensus problems of networked continuous-time linear positive systems with edge states' scaling property,unmeasurable edge-states,input saturation and edge-states' non-negative constraints,the main objective is to make edge-consensus protocols,theorems and scaled edge-consensus conditions independent of networks.To be specific,the edge-based adaptive gain functions are introduced,and based on this,two kinds of fully distributed output-feedback adaptive scaled edge-consensus protocols are constructed.Then by solving the basic ARE and DARE which do not contain any information of network,the mathematical expressions of feedback gain matrix and observer gain matrix are constructed.For the different adaptive protocols,the modified Lyapunov functions are designed by virtue of adaptive control theory,respectively.Moreover,based on positive system theory and low-gain feedback technique,the sufficient conditions for guaranteeing non-negative edge-states,bounded inputs and scaled edge-consensus are obtained.In addition,the feedback gain matrix and observer gain matrix which can satisfy all sufficient conditions,are existed and easy to obtain.For the edge-consensus problems of networked continuous-time or discrete-time linear positive systems with input saturation,edge-states' non-negative constraints together with only using the neighbouring edges' actual outputs,the main objective is to design the observer-based edge-consensus protocols which are easier and simpler to implement in practical application.In other words,the main objective is to design the observer-based edge-consensus protocols with only using the neighbouring edges' actual outputs.Drawing support from line graph theory,LMI-based control method,ARE-based and MARE-based low-gain feedback technique,the mathematical expressions of feedback gain matrix and observer gain matrix for networked continuous-time and discrete-time linear positive systems,are derived,respectively.Meanwhile,the consensus conditions for guaranteeing non-negative edge-states and bounded inputs are obtained.However,under the control protocols,the gain matrices which can satisfy all the consensus conditions,may not exist,both the continuous-time and discrete-time networked linear positive systems.Therefore,the main objective of following study is to solve this problem.For the scaled edge-consensus problems of networked continuous-time or discrete-time linear positive systems with input saturation,edge-states' non-negative constraints together with only using the neighbouring edges' actual outputs,the main objective is to overcome the difficulty in finding the feedback gain matrix and observer gain matrix.In other words,the main objective is to design the feedback gain matrix and observer gain matrix which are existed and easy to obtain for networked continuous-time linear positive systems,and for networked discrete-time linear positive systems,the feedback gain matrix and observer gain matrix are also existed and easy to obtain when edge-states' non-negative constraints are not considered,and the feedback gain matrix and observer gain matrix only need to be non-positive when edge-states' non-negative constraints are considered.To be specific,observer gain matrices are obtained by solving DARE and MDARE,rather than the control-based LMI.Moreover,the-nrom of transfer function will be used to analyse the scaled edge-consensus problems.