Dynamic Response Unit step signal: Step response: y(s)=H(s)/s, y(t)=L-1{H(s)/s} Unit impulse signal: δ(t)1 Impulse response: h(t)= L-1 {H(s)} In Matlab: use “step”, “impulse”, “lsim”, etc

Time domain response specifications Defined based on unit step response Defined for closed-loop system Steady-state value yss Steady-state error ess Settling time ts = time when y(t) last enters a tolerance band

By final value theorem In MATLAB: num = [ .. .. .. .. ] b0 = num(length(num)), or num(end) a0 = den(length(den)), or den(end) yss=b0/a0

If numerical values of y(t) available, abs(y – yss) < tol means inside band abs(y – yss) ≥ tol not inside e.g. t_out = t(abs(y – yss) ≥ tol) contains all those time points when y is not inside the band. Therefore, the last value in t_out will be the settling time. ts=t_out(end)

Peak time tp = time when y(t) reaches its maximum value. Peak value ymax = y(tp) Hence: ymax = max(y); tp = t(y = ymax); Overshoot: OS = ymax - yss Percentage overshoot:

Delay time td = the time when y(t) first reaches 50% of yss If ymax is reached as t→∞, there is no peak time and there is no overshoot. Delay time td = the time when y(t) first reaches 50% of yss Not frequently used Some people use a percentage different from 50% t50=t(y<=0.5*yss); td=t50(end)

Rise time tr = the time it takes for y(t) to go from 0. 1yss to 0 Rise time tr = the time it takes for y(t) to go from 0.1yss to 0.9yss for the first time. Rise time captures how fast a system responds to changes in a reference input td, tp has similar effect

If t50 = t(y >= 0.5·yss), this contains all time points when y(t) is ≥ 50% of yss so the first such point is td. td=t50(1); Similarly, t10 = t(y >= 0.1*yss) & t90 = t(y >= 0.9*yss) can be used to find tr. tr=t90(1)-t10(1)

90%yss tr≈0.45 10%yss ts ts tp≈0.9sec td≈0.35

±5% ts=0.45 yss=1 ess=0 O.S.=0 Mp=0 tp=∞ td≈0.2 tr≈0.35

tp=0.35 O.S.=0.4 Mp=40% yss=1 es=0 ts≈0.92 td≈0.2 tr≈0.1

Steady-state tracking & sys. types Unity feedback control: plant + e r(s) y(s) C(s) G(s) - + e r(s) Go.l.(s) y(s) -

r(t) t

r(t) t

r(t)=R·1(t) r(s)=R/s r(t)=R·t·1(t) r(s)=R/s2 r(t)=R·1/2·t2 r(s)=R/s3 type 0 (N=0 a0≠0) Kp=b0/a0 ess=R/(1+Kp) Kv=0 ess=∞ Ka=0 type 1 (N=1 a0=0 a1≠0 b0≠0 ) Kp= ∞ ess=0 Kv=b0/a1 ess=R/Kv type 2, N=2 a0=a1=0 a2≠0,b0≠0 Kv= ∞ Kp=b0/a2 ess=R/Ka type≥3, N ≥ 3 a0=a1=a2=0 b0≠0 Ka= ∞ ref. input sys. type

Example of tank + C H -

r(s) + Kps+KI s + - r(s) e ωn2 s(s+2ξ ωn) 1 Ts+1 e.g.

example e(s) y(s) r(s) G(s)