Consider a multiagent system with the following sensing$/$communication topology. The

graph adjacency matrix can be defined with constant entries aij with aij $=0$ or $1.$ Let

xi be the state of agent i$.$ Initial values x$1(0)=2$; x$2(0)=8$; x$3(0)=6$; x$4(0)=9$ Figure $1$: A sensing$/$communication graph of four agents

part $(1)$ Assume that agents are evolving according to the following continue$-$time

consensus protocol,

$x_i^{}(t)={\displaystyle \underset{j=1}{\overset{4}{}}}{a}_{ij}(x_j(t)-x_i(t)).$

Explain whether consensus can be achieved for agents. If so$,$ what is the consensus

value?

part $(2)$ Assume that agents are evolving according to the following discrete$-$time

consensus protocol,

$x_i(k+1)=x_i(k)+lon{\displaystyle \underset{j=1}{\overset{4}{}}}{a}_{ij}(x_j(k)-x_i(k)),$

where $lon>0$ is the sampling time. Explain whether consensus can be achieved for

agents. If so$,$ what is the consensus value?

part $(3)$ Assume that agents are evolving according to the following discrete$-$time

consensus protocol,

$x_i(k+1)=x_i(k)+\frac{1}{d_i}{\displaystyle \underset{j=1}{\overset{4}{}}}{a}_{ij}(x_j(k)-x_i(k)),$

where $d_i$ is in$-$degree of agent $i.$ Explain whether consensus can be achieved for

agents. If so$,$ what is the consensus value?