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Dynamic Response

This article explains the concepts of dynamic and steady-state responses in control systems. Dynamic Response refers to the system's behavior immediately after an input is applied, while Steady State Response characterizes the long-term behavior as time approaches infinity (t → ∞). We delve into important metrics such as settling time, peak time, overshoot, and steady-state error. MATLAB techniques for evaluating these responses are discussed, alongside critical specifications derived from the Final Value Theorem. Gain insight into how these responses impact control system design and analysis.

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Dynamic Response

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  1. Dynamic Response • Steady State Response: the part of resp. when t→∞ • Transient response: the part of resp right after the input is being applied. • Both are part of the total resp. total resp = z.i. resp + z.s.resp. z.i. resp = “Output due to i.c. when input ≡ 0” z.s. resp = “Output due to input excitation when all i.c. are set=0 at t=0”

  2. us(t) 1 0 Typical test signal • Unit step signal: • Unit impulse:δ(t) δ(t) t

  3. Unit ramp: • Unit acc. signal: r(t) t a(t) 0.5 0 1 t

  4. Exponential signal: • sinusoidal: 1 0 t

  5. 1 s 1 s y(s)=H(s) u(s)= H(s) • Unit step response: In Matlab: step • Unit impulse resp: Matlab: impulse y(s)=H(s) u(s)=1 H(s)

  6. 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

  7. 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 Time domain response specifications

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

  9. 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)

  10. 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:

  11. 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(end)

  12. 90%yss 10%yss tr≈0.45 ts ts tp≈0.9sec td≈0.35

  13. ±5% ts=0.45 yss=1 ess=0 O.S.=0 Mp=0 tp=∞ td≈0.2 tr≈0.35

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

  15. 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) -

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