2.1 Step Input Response: Eigenvalues: p 1 =-4.526, p 2,3 = ±2.7883i, p 4 = Final value theorem: Stability: If the step response reaches to a constant value, the system is said to be stable. Steady State Error: e ss =1-c ss e ss =0 [r,p,k]=residue(nh,[dh,0]) r = i i z=r(3);a2=2*abs(z),fi2=angle(z) Roots([ ]) Laplace transform of unit step input Steady-state response

Step Response of a Second Order System: ω n : 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)

c max : Peak value, t max : Peak time, c max -c ss :Maksimum overshoot c ss : Steady State Response, 1-c ss : Steady State Error t s : Settling time (%5) t r : Rise Time t d : Delay Time Re - σ 

Design Criteria of Control Systems: Stability Steady State Error e ss =1-c ss → 0 Sensitivity to a disturbance [c ss ] d → 0 Overshoot, typical value % 5, damping ratio ξ=0.7 Settling time : t ss (Depends on the application) At the first stage apply P control: observe the stability e ss, [c ss ] 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 Transfer Function of a PID Controller Sensitivity to disturbance