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Second-order step response
Second-order step response
G=ωn²/(s²+2ζωn s+ωn²) · 欠/临界/过阻尼 · %OS超调
G(s) = ωn² / (s² + 2ζωn·s + ωn²) · %OS = 100·e^(−ζπ/√(1−ζ²))
Transfer-function poles in the s-plane
Transfer-function poles in the s-plane
s平面极点→响应 · 实轴指数/复数振荡/RHP不稳定
s = σ ± jω · impulse response character h(t) = e^(σt)·cos(ωt)
Settling time & performance metrics
Settling time & performance metrics
tr · tp=π/(ωn√(1−ζ²)) · ts≈4/(ζωn) · Mp · ±2%带
tr · tp = π/(ωn√(1−ζ²)) · ts ≈ 4/(ζωn) (2%) · Mp = e^(−ζπ/√(1−ζ²))
Routh–Hurwitz stability criterion
Routh–Hurwitz stability criterion
劳斯阵 · 首列同号则稳定 · 变号数=右半平面根
Build the Routh array from the characteristic polynomial — first-column sign changes count the unstable roots
Gain margin & phase margin
Gain margin & phase margin
−180°处GM · 0dB处PM · 开环频响 · 稳定裕度
Open-loop Bode plot — how much extra gain and phase lag the loop tolerates before it goes unstable
Closed-loop disturbance rejection
Closed-loop disturbance rejection
闭环抑制扰动 · ess∝1/(1+L) · 环路增益越大误差越小
Feedback attenuates an output disturbance — the residual error shrinks as 1/(1+L)
Lead / lag compensator
Lead / lag compensator
超前z<p加相位 · 滞后z>p加低频增益 · 极零配置
C(s) = K·(s+z)/(s+p) · lead z<p adds phase · lag z>p adds low-frequency gain
Pole placement by state feedback
Pole placement by state feedback
状态反馈u=−Kx · A−BK极点到期望 · 需可控
ẋ = Ax + Bu, u = −Kx ⇒ closed loop A−BK · double integrator plant
Steady-state error & system type
Steady-state error & system type
系统型别0/1/2 · Kp/Kv/Ka · 阶跃/斜坡/抛物ess
unity feedback · Kp = lim G, Kv = lim sG, Ka = lim s²G (s→0) · ess = 1/(1+Kp) | 1/Kv | 1/Ka
State-space model
State-space model
ẋ=Ax+Bu,y=Cx+Du · A/B/C/D · 状态轨迹
ẋ = A x + B u · y = C x + D u · forward-Euler simulation
Controllability & observability
Controllability & observability
可控阵[B AB…]满秩 · 可观阵[C;CA;…]满秩 · 秩判定
Cₒ = [B AB] · Oᵦ = [C ; CA] · full rank n ⇔ controllable / observable
Block-diagram reduction
Block-diagram reduction
串联G1G2 · 并联G1±G2 · 反馈G/(1+GH) · 化为单一传函
series: G₁·G₂ · parallel: G₁±G₂ · feedback: G/(1+GH)