Second Order Systems

Transfer Function

H (s) = \frac{ω_{n}^{2}}{s^{2} + 2 ζ ω_{n} s + ω_{n}^{2}}, ω_{n} > 0

Poles

s = - ζ ω_{n} \pm ω_{n} \sqrt{ζ^{2} - 1}

$\zeta$ damping ratio $\omega_n$ is the undamped natural frequency.

Underdamped System: $0\ge\zeta<1$

Critically Damped System: $\zeta=1$

Overdamped System: $\zeta\gt1$

$0\ge\zeta\ge1$

Define poles at,

s = - σ \pm j ω_{d}

such that,

σ = ζ ω_{n}, ω_{d} = ω_{n} \sqrt{1 - ζ^{2}}

$\omega_d$ is the damped natural frequency.

The transfer function becomes:

H (s) = \frac{ω_{n}^{2}}{(s + σ + j ω_{d}) (s + σ - j ω_{d})} = \frac{ω_{n}^{2}}{(s + σ)^{2} + ω_{d}}

Step Response

y (t) = 1 - e^{- σ t} (c o s (ω_{d} t) + \frac{σ}{ω_{d}} s i n (ω_{d} t))

$\zeta\gt1$

Poles are at,

- ζ ω_{n} \pm ω_{n} \sqrt{ζ^{2} - 1}

The transfer function becomes:

H (s) = \frac{ω_{n}^{2}}{(s + ζ ω_{n} + ω_{n} \sqrt{ζ^{2} - 1}) (s + ζ ω_{n} - ω_{n} \sqrt{ζ^{2} - 1})}

Step Response

y (t) = 1 - k_{1} e^{(- ζ ω - ω_{n} \sqrt{ζ^{2} - 1}) t} - k_{2} e^{(- ζ ω + ω_{n} \sqrt{ζ^{2} - 1}) t}

$k_1$ $k_2$ .

Performance Measures

$t_r$

Rise time is the time required to go from 10% to 90% of the final value.

t_{r} \approx \frac{2.16 ζ + 0.6}{ω_{n}}

$0.3 \gt \zeta \gt 0.8$ .

Cruder approximation:

t_{r} \approx \frac{1.8}{ω_{n}}

$t_p$

Peak time is the time at which the maximum value is reached.

t_{p} = \frac{π}{ω_{d}} = \frac{π}{ω_{n} \sqrt{1 - ζ^{2}}}

$M_p$

Peak value is the maximum value the output reaches.

M_{p} = 1 + e^{- σ \frac{π}{ω_{d}}} = 1 + e^{- \frac{π ζ}{\sqrt{1 - ζ^{2}}}}

$\%OS$

Overshoot percentage is the percentage by which the response overshoots from the steady state value in the first peak.

% O S = e^{- \frac{π ζ}{\sqrt{1 - ζ^{2}}}} \times 100

$t_s$

Settling time is the time required to get within 2% of the final value and stay there.

t_{s} \approx \frac{4}{σ} = \frac{4}{ζ ω_{n}}

Second Order Systems

Transfer Function

Poles

Underdamped & Critically Damped System, 0≥ζ≥10\ge\zeta\ge1

Step Response

Overdamped System, ζ>1\zeta\gt1

Step Response

Performance Measures

Rise Time, trt_r

Peak Time, tpt_p

Peak Value, MpM_p

Overshoot Percentage, %OS\%OS

Settling Time, tst_s

$0\ge\zeta\ge1$

$\zeta\gt1$

$t_r$

$t_p$

$M_p$

$\%OS$

$t_s$