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Time-Domain Analysis

First-Order Systems

Transfer function:

G(s) = K / (tau*s + 1)

Where K is the DC gain and tau is the time constant.

Step response:

y(t) = K * (1 - e^(-t/tau))

Key characteristics:

  • At t = tau: output reaches 63.2% of final value
  • At t = 3*tau: 95% of final value
  • At t = 5*tau: 99.3% (considered settled)
  • Rise time (10%-90%): t_r = 2.2 * tau
  • No overshoot, no oscillation

Impulse response:

y(t) = (K/tau) * e^(-t/tau)

Second-Order Systems

Standard form:

G(s) = omega_n^2 / (s^2 + 2*zeta*omega_n*s + omega_n^2)

Parameters:

  • omega_n: natural frequency (rad/s) -- frequency of oscillation if undamped
  • zeta: damping ratio -- determines oscillatory character

Poles:

s = -zeta*omega_n +/- omega_n*sqrt(zeta^2 - 1)

Response Classification by Damping Ratio

| zeta | Behavior | Poles | |------|----------|-------| | 0 | Undamped | Purely imaginary: +/- jomega_n | | 0 < zeta < 1 | Underdamped | Complex conjugate: -sigma +/- jomega_d | | 1 | Critically damped | Repeated real: -omega_n | | > 1 | Overdamped | Two distinct negative real |

For the underdamped case (most common in practice):

sigma = zeta * omega_n          (exponential decay rate)
omega_d = omega_n*sqrt(1-zeta^2)  (damped frequency)

Underdamped Step Response

y(t) = 1 - (e^(-sigma*t) / sqrt(1-zeta^2)) * sin(omega_d*t + phi)

where phi = arccos(zeta)

Transient Response Specifications

Rise time (0% to 100%, first crossing):

t_r = (pi - phi) / omega_d

Approximate: t_r ≈ 1.8 / omega_n for 0.3 < zeta < 0.8.

Peak time (time to first peak):

t_p = pi / omega_d

Percent overshoot:

%OS = 100 * exp(-zeta*pi / sqrt(1 - zeta^2))

Inversely: zeta = -ln(%OS/100) / sqrt(pi^2 + ln^2(%OS/100))

| zeta | %OS | |------|-----| | 0.1 | 73% | | 0.3 | 37% | | 0.5 | 16% | | 0.7 | 4.6% | | 1.0 | 0% |

Settling time (within 2% of final value):

t_s = 4 / (zeta * omega_n) = 4 / sigma

For 5% criterion: t_s = 3 / sigma.

Design Implications

Settling time depends only on the real part of the poles (sigma). Overshoot depends only on zeta. These define regions in the s-plane:

  • Settling time spec -> vertical line at Re(s) = -sigma_min
  • Overshoot spec -> lines from origin at angle beta = arccos(zeta)
  • Desired poles lie in the intersection of these regions

Effect of Additional Poles and Zeros

Extra pole: a third pole at s = -p makes the system slower. If p > 5*sigma, its effect is negligible (dominant pole approximation).

Extra zero: a zero at s = -z increases overshoot and speeds up the response. If z > 10*sigma, its effect is negligible.

Non-minimum phase zero (right-half plane, s = +z): causes initial undershoot -- the response initially goes in the wrong direction before recovering. This fundamentally limits achievable bandwidth.

Steady-State Error Analysis

For unity feedback with open-loop G(s) = K * prod(s+z_i) / (s^N * prod(s+p_j)):

System type = N (number of integrators).

Step input R(s) = 1/s:

e_ss = 1 / (1 + lim(s->0) G(s))
     = 1 / (1 + Kp)

Zero for type >= 1.

Ramp input R(s) = 1/s^2:

e_ss = 1 / lim(s->0) s*G(s) = 1/Kv

Zero for type >= 2.

Parabolic input R(s) = 1/s^3:

e_ss = 1 / lim(s->0) s^2*G(s) = 1/Ka

Zero for type >= 3.

Steady-State Error with Disturbance

For a disturbance D(s) at the plant input:

e_ss_disturbance = -lim(s->0) s * G_plant(s) / (1 + G_c(s)*G_plant(s)) * D(s)

High controller gain at low frequencies reduces disturbance effects.

Sensitivity Analysis

Sensitivity function:

S_T^G = (dT/T) / (dG/G) = (G/T) * (dT/dG)

For closed-loop system T = G/(1+GH):

S_T^G = 1 / (1 + G*H)

This is the sensitivity function S(s). The complementary sensitivity is:

T(s) = G*H / (1 + G*H)

Fundamental constraint: S(s) + T(s) = 1 for all s.

Implications:

  • Good tracking (T near 1) means poor disturbance at sensor (S near 0) and vice versa
  • Cannot have both S and T small simultaneously at any frequency
  • Bode sensitivity integral (waterbed effect): for stable open-loop systems with relative degree >= 2:
integral_0^inf ln|S(j*omega)| d_omega = 0

Reducing sensitivity in one frequency range necessarily increases it elsewhere.

Higher-Order System Approximation

Dominant pole approximation: if a system has poles much closer to the imaginary axis than others, the transient response is dominated by those poles. Approximate by a lower-order system using only dominant poles.

Conditions for validity:

  1. Non-dominant poles are at least 5x further from the imaginary axis
  2. Zeros near non-dominant poles provide approximate cancellation
  3. DC gain is preserved in the approximation

Pole-zero cancellation: if a pole and zero are close together, their effects approximately cancel. However, internal stability must still be verified -- cancelled unstable poles cause unbounded internal states.

Test Signals

| Signal | Time domain | Laplace | Purpose | |--------|------------|---------|---------| | Impulse | delta(t) | 1 | Reveals natural modes | | Step | u(t) | 1/s | Steady-state error, transient specs | | Ramp | tu(t) | 1/s^2 | Tracking accuracy | | Parabola | t^2/2u(t) | 1/s^3 | Acceleration tracking | | Sinusoid | sin(omega*t) | omega/(s^2+omega^2) | Frequency response |