G(s) = 0.1s + 1s (s+1) (0.02s+1)²
Magnitude in dB and phase are both additive. Sketch each standard element, then add the straight lines.
Rules used: pole at ωc → slope −20 dB/dec above ωc, phase 0° → −90° between ωc/10 and 10ωc. Zero: the opposite sign. Integrator: −20 dB/dec everywhere, 0 dB at ω = 1, phase −90° everywhere.
Send a sinusoid into an asymptotically stable LTI system. Once the transient has died out, the output is a sinusoid at the same frequency, with amplitude |G(iω)| and phase shift arg G(iω).
G(s) = 1s² + 0.6s + 1 u(t) = sin(ωt)
The Bode plot is a catalogue of such experiments, one per frequency. The steady-state formula needs asymptotic stability; for an unstable system the Bode plot is only a plot of the complex function G(iω).
A = 001−2 B = 01 C = 11 D = 0
G(s) = C(sI−A)−1B = s + 1s (s + 2)
Poles at 0 and −2 (eigenvalues of A), zero at −1. The input u(t) = 5e−t has U(s) = 5/(s+1): its pole sits exactly on the zero.
A zero at s = z means the input ezt can be completely blocked: with the right initial state the states move, but the output stays zero.
Internal stability: eigenvalues of A. BIBO stability: poles of G(s). They agree unless an unstable pole is cancelled by a zero, which hides it from the output.
A = −1021 B = 0−1 C = −13 → G(s) = −3s + 1(s+1)(s−1)
A = −1501 B = 11 C = 10 → G(s) = s − 1(s+1)(s−1) = 1s + 1
Shown: impulse response, i.e. u = δ(t), which is the same as u = 0 with x(0) = B.
closed loop: K GP(s)1 + K GP(s) poles: 1 + K GP(s) = 0