Eigenvalues: p1=-4.526, p2,3=-0.4993±2.7883i, p4=-0.4753 2.1 Step Input Response: Roots([2.1 12.6 31.74 88.2 36]) Eigenvalues: p1=-4.526, p2,3=-0.4993±2.7883i, p4=-0.4753 Steady-state response Laplace transform of unit step input Final value theorem: Stability: If the step response reaches to a constant value, the system is said to be stable. Steady State Error: ess=1-css ess=0 [r,p,k]=residue(nh,[dh,0]) r = -0.1402 -0.3976 + 0.1912i -0.3976 - 0.1912i -0.0646 1.0000 z=r(3);a2=2*abs(z),fi2=angle(z)

Step Response of a Second Order System: ωn : Undamped natural frequency ξ: Damping Ratio clc,clear wn=1;ksi=0.2; tp=2*pi/wn;dt=tp/20;ts=tp/ksi; t=0:dt:ts;w=wn*sqrt(1-ksi^2); a=wn/w;sigma=ksi*wn;fi=-acos(ksi)-pi/2; c=a*exp(-sigma*t).*cos(w*t-fi)+1; plot(t,c)

tr : Rise Time td : Delay Time ts : Settling time (%5) cmax : Peak value, tmax: Peak time, cmax-css:Maksimum overshoot css : Steady State Response, 1-css : Steady State Error Re - σ 

Design Criteria of Control Systems: Stability Steady State Error ess=1-css → 0 Sensitivity to a disturbance [css]d → 0 Overshoot , typical value % 5, damping ratio ξ=0.7 Settling time : tss (Depends on the application) Transfer Function of a PID Controller Sensitivity to disturbance At the first stage apply P control: observe the stability ess, [css]d If necessary apply PI control, it eliminates/reduces the steady state error If necessary apply PD kontrol, it reduces the overshoot Apply PID control, useful for all criteria