I am trying to learn from Stanford’s course EE 263 taught by Stephen Boyd on linear dynamical systems: http://ee263.stanford.edu/archive/. This class is archived, so you don’t have to worry about me asking for homework help on this forum. The class was from 2007.
In homework 6, the first question, problem 9.9 is about a power control algorithm which is modeled as a discrete linear dynamical system. It is an extension of problem 2.1. All of the homework problems can be found here: https://see.stanford.edu/materials/lsoeldsee263/homeworkProblems.pdf. The solutions to the problem I’m referring to can be found here: https://web.stanford.edu/class/archive/ee/ee263/ee263.1082/hw/hw6sol.pdf.
We are asked to explain how the system is both stable, and converges (which are the same thing in this case). I understand that by observing the eigenvalues of the dynamics matrix, $A$, we are able to tell that the system is stable, because the eigenvalues all have absolute value less than 1. But the solution also claims the following:
Also, the SINR at each receiver $i$, given by $S_i$, converges to the same constant value $\alpha \gamma$, which is enough for a successful signal reception. This can be shown by observing that at equilibrium $p_i(t+1) = p_i(t) = \bar{p}_i$, and the power update equation gives $$ \bar{p}_i = \bar{p}_i( \alpha \gamma / S_i(t) ). $$ After cancellation, we obtain the constant value for each SINR, $S_i = \alpha \gamma$.
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I do not understand how to show that the equilibrium is reached at $p_i (t+1)=p_i (t)= \bar p _i$. When I take the dynamics matrix, $A$ to the power $t$, $A^t$, and I take $t \rightarrow \infty$, I do not see $A^t\rightarrow I$ approaching the identity matrix, which is what that expression implies.
Can someone please show me how to show that the equilibrium is reached at $p_i (t+1)=p_i (t)= \bar p _i$? Or, more generally how to find out what a system does at equilibrium would be helpful.