14.1Singularity Function Response:For a system described by the following differential equations:x Ax Bv xxa aa axxb bb bvvsss1211 1221 221211 1221 2212 The following formulas can be used to determine the forced responses to sources consisting of singularity functions:For a ramp source of the form vs gstggt12 for 0t, the solution can be written as:xfmmtnn1212where A Bg1smm12 and A1mmnn1212If the system has a single input, B becomes a vector b and sg becomes a scalar sg14.2For a step input of the form vs 21ggsg for 0t, the forced response is: A Bg x1s fNote this is the same as for DC sources examined earlier.If the system has a single input, B becomes a vector b and sg becomes a scalar sgFor impulse sources of the form vs gstggt ( ) ( )12 the forced solution is zero, since( )t 0 for t > 0. Therefore, the complete solution is simply the natural solution whose coefficients are evaluated based on the following initial conditions:)()( 0xBg0xsIf the system has a single input, B becomes a vector b and sg becomes a scalar sg14.3Example: Derive the formula for each of singularity function responses through proper substitution of a guessed a general response. For the impulse response formula you must integrate both sides from 0- to 0+ after the substitution to get the proper result. Example:Independently find the impulse, step and ramp response for the system output below:sCLCLivivi013115; CLviv 100;show )()4exp()(0tutttvimpulse )()4exp(4)4exp(1161)(0tuttttvStep)()4exp()4exp( 21 21161)(0tutttttvRamp14.4Multiple Source Example:vc1vc212 9 vsis12F13Fvovvvvivccccss12127181614142290 0 vvvivoccss 1 1 0 012 Find the solution if:v u ts9 ( ) and )(5)( tutisShow: )())5351.0exp()1038.0(exp(83.27)(0tutttv Use Matlab to help with computations, it is convenient to set up and perform computation in matrix form. Matlab (short for Matrix Laboratory) is designed to docomputations this way.14.5% Define differential equation matricesa = [-(7/18), (1/6); (1/4), -(1/4)];b = [2, (2/9); 0, 0];% Natural Solution - find roots of Char. Eqn.% det|pI-a| = 0 = 72p^2 + 46p + 4rts = roots([72 46 4])rts = -0.5351 -0.1038% Roots are real and distinct so natural soln. is of% the form vc1(t) = C1*exp(rts(1)*t)+C2*exp(rts(2)*t)% vc2(t) = D1*exp(rts(1)*t)+D2*exp(rts(2)*t)% Forced soln for vsfc1 = -inv(a)*b*[5; 9]fc1 = 54.0000 54.0000% Initial conditions for x dotx0p = [0; 0] % System at restx0p = 0 0x0dotp = a*x0p+b*[5; 9]x0dotp =14.6 12 0% Solution has form: vc1(t) = [fc1(1) + C1*exp(rts(1)*t)+C2*exp(rts()*t)]% vc2(t) = [fc1(2) + D1*exp(rts(1)*t)+D2*exp(rts(2)*t)]% Form of derivative equation: vc1dot(t) = [rts(1)*C1*exp(rts(1)*t)+rts(2)*C2*exp(rts(2)*t)]% vc2dot(t) = [rts(1)*D1*exp(rts(1)*t)+rts(2)*D2*exp(rts(2)*t)]% Solve Resulting system of equation to find C1 and C2s1 = [1 1; rts(1), rts(2)];r1 = [x0p(1)-fc1(1); x0dotp(1)];cc = inv(s1)*r1cc = -14.8254 -39.1746% Solve Resulting system of equation to find D1 and D2s2 = [1 1; rts(1), rts(2)];r2 = [x0p(2)-fc1(2); x0dotp(2)];dd = inv(s2)*r2dd = 13.0021
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