2.4 Time Response of the LTI System4(t-t) .B .u(t).dtX(t) = e4(t-to)X(t.) +Let the initial state X(t.) = O, in other words, the LTI system istaken on the zero-state situation. +d[e-A(t-to) X(t)] = e-4(t-to) Bu(t)dtTaking definite integral on both side, we have" d[e-4(-) X(t)] = e-A(-to) X(t) = [ e-A(r-o) Bu(t)d tToX(t)= [ eA(t-) Bu(t)dtTherefore,Obviously, it is the second term on the right-hand side and it canbe called the forced response or zero-state response of the LTIsystem
2.4 Time Response of the LTI System
2.4 Time Response of the LTI SystemConsider the LTI system with the initial condition X(t.)X(t) = AX(t) + Bu(t)the solution can also be written as the more general cased(t -t). B.u(t) .dtX(t) = Φ(t - t)X(t.)+zero-inputresponsezero-state response4(t-t) .B .u(t).dtX(t) = e4(t-to) X(t.) +
2.4 Time Response of the LTI System
Example 2.6 Determine the solution of the LTI systemdescribedby+[o]01X:X+t≥0X(0) =[x;(0)x2(0)]一u-2where u(t) =l(t) is the unit step functionSolutionFrom Example 2.1 we have obtained that2e'-e-2t2eyeΦ(t) = eAr -e'" + 2e-2tL-2e-+2e-2tTherefore, the solution of the LTI system can also be calculatedX(t) = @(t)X(O) + Φ(t -t)Bu(t)dt