Wireless Ad Hoc and Sensor Networks

52 Wireless Ad Hoc and Sensor Networks2.1.3 Linear SystemsA special and important class of dynamical systems is the discrete-timelinear time invariant (LTI) systemxk ( + 1) = Axk ( ) + Buk ( )yk ( ) = Cxk( )(2.6)with A, B, and C constant matrices of general form (e.g., not restricted toEquation 2.5). An LTI is denoted by (A,B,C). Given an initial state x( 0),the solution to the LTI system can be explicitly written ask−1∑j=0kk−−j 1xk ( ) = Ax( 0) + A Buj ( )(2.7)The next example shows the relevance of these solutions and demonstratesthat the general discrete-time nonlinear systems are even easier tosimulate on a computer than continuous-time systems, as no integrationroutine is needed.Example 2.1.1: Discrete-Time System — Savings Account(Lewis, Jagannathan, and Yesilderek 1999)Discrete-time descriptions can be derived from continuous-time systemsby using Euler’s approximation or system discretization theory. However,many phenomena are naturally modeled, using discrete-time dynamics,including population growth and decline, epidemic spread, economicsystems, and so on. The dynamics of the savings account using compoundinterest are given by the first-order systemxk ( + 1) = ( 1+ ixk ) ( ) + uk ( )where i represents the interest rate over each interval, k is the intervaliteration number, and uk ( ) is the amount of the deposit at the beginningof the kth period. The state xk ( ) represents the account balance at thebeginning of interval k.ANALYSISAccording to Equation 2.7, if equal annual deposits are made ofthe account balance isk−1∑j=0kk−−j 1xk ( ) = ( 1+ i) x( 0) + ( 1+i)duk ( ) = d,