If f(t)=est then u(t)= H(s)est

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Presentation transcript:

If f(t)=est then u(t)= H(s)est Transfer Function: f(t): Input, u(t): Output Example 7.1 (continued) If f(t)=est then u(t)= H(s)est Transfer Function: s3H(s)est+4s2H(s)est+14sH(s)est+20H(s)est=3est+sest a=[1,4,14,20];roots(a) (s3+4s2+14s+20)H(s)=3+s Eigenvalues: -1±3i, -2 Exponential/Harmonic Input: s=-0.2+2.7i; hs=(s+3)/(s^3+4*s^2+14*s+20); abs(hs), angle(hs) RESONANCE

H(s) Input-Output relationship in s domain: x(t) y(t) Y(s)=X(s) H(s) Impulse Response: y(t)=h(t) Impulse function Δ(s)=1 Example 7.1 (Continued): p1=[1,3]; p2=[1,4,14,20]; [r,p,k]=residue(p1,p2)

z=-0.05+0.1833i; 2*abs(z), phase(z) Impulse response of the system ξ=0.3162 (s=-1±3i), for the system Δt=0.099, t∞=6.283 clc;clear; t=0:0.099:6.283; yt=0.3801*exp(-t).*cos(3*t-1.837)+0.1*exp(-2*t) plot(t,yt) Steady-state response is: For stable systems, response returns to zero or reaches a finite value determined by the input amplitude. STABILITY

p1=[1,3]; p2=[1,4,14,20,0]; [r,p,k]=residue(p1,p2) Step Input Response: Example 7.1 (Continued): u(t): Step function 1 p1=[1,3]; p2=[1,4,14,20,0]; [r,p,k]=residue(p1,p2)

clc;clear; t=0:0.099:6.283; yt=0.1202*exp(-t).*cos(3*t+2.5536)-0.05*exp(-2*t)+0.15; plot(t,yt)

Final value theorem: Example 7.1 (continued) Step input response