Dynamic Response

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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. Dynamic Response. Defined based on unit step response Defined for closed-loop system - PowerPoint PPT Presentation

Transcript of Dynamic Response

  • Dynamic ResponseUnit step signal:

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

  • Defined based on unit step responseDefined for closed-loop system

    Steady-state value yss

    Steady-state error ess

    Settling time ts= time when y(t) last enters a tolerance bandTime domain response specifications

  • By final value theoremIn 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) < tolmeans inside bandabs(y yss) tolnot 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 - yssPercentage overshoot:

  • 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 reaches50% of yss

    Not frequently used

    Some people use a percentage different from 50%


  • 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 >= 0.5yss),this contains all time points wheny(t) is 50% of yssso the first such point is td.


    Similarly,t10 = t(y >= 0.1*yss)&t90 = t(y >= 0.9*yss)can be used to find tr.tr=t90(1)-t10(1)

  • tp0.9sec10%yss90%ysstr0.45td0.35tsts

  • tr0.355% ts=0.45yss=1ess=0O.S.=0Mp=0tp=td0.2

  • tr0.1td0.2ts0.92tp=0.35O.S.=0.4Mp=40%yss=1es=0

  • Steady-state tracking & sys. typesUnity feedback control:

    G(s) C(s)+-r(s)ey(s)plant Go.l.(s)+-r(s)ey(s)

  • r(t)t

  • r(t)t0

  • sys.typeref.input

  • Example of tank H+- C

  • +r(s)

  • example G(s)r(s)e(s)y(s)