control-systems

name	control-systems
description	Control systems — Laplace transforms, transfer functions, block diagrams, PID tuning, Bode plots, Nyquist criterion, stability margins, state-space, root locus.
metadata	{"priority":8,"promptSignals":{"phrases":["control system","PID","transfer function","Bode plot","Nyquist","stability margin","state space","root locus","closed loop","gain margin"],"minScore":3}}

Control Systems — Complete Skill

Laplace Transform

Common Pairs

f(t)	F(s)
δ(t) unit impulse	1
u(t) unit step	1/s
t (ramp)	1/s²
e^(-at)	1/(s+a)
sin(ωt)	ω/(s²+ω²)
cos(ωt)	s/(s²+ω²)
t·e^(-at)	1/(s+a)²
e^(-at)sin(ωt)	ω/((s+a)²+ω²)

Useful Theorems

Initial value: lim_{t→0} f(t) = lim_{s→∞} sF(s) Final value: lim_{t→∞} f(t) = lim_{s→0} sF(s) [only if stable] Differentiation: L{df/dt} = sF(s) - f(0⁻) Integration: L{∫f dt} = F(s)/s

Transfer Function G(s)

G(s) = Y(s)/U(s) [output/input, zero initial conditions] From ODE: m·ÿ + c·ẏ + k·y = F(t) → G(s) = 1/(ms²+cs+k)

Standard 2nd order form: G(s) = ω_n² / (s² + 2ζω_n s + ω_n²) ω_n = √(k/m), ζ = c/(2√(km)) Natural freq, damping from physical system

Poles: s = -ζω_n ± jω_n√(1-ζ²) (complex for ζ<1) Zeros: values of s where G(s)=0

Block Diagram Algebra

Series: G_total = G₁ × G₂ Parallel: G_total = G₁ + G₂ Feedback loop: G_cl = G(s) / (1 + G(s)H(s)) where G(s) = forward path, H(s) = feedback path

Closed-loop standard form (unity feedback): T(s) = G(s)/(1+G(s)) = C(s)/R(s)

Error: E(s) = R(s)/(1+G(s)) Steady-state error: e_ss = lim_{s→0} s × E(s)/s = lim_{s→0} 1/(1+G(s)) [step input]

System Type and Steady-State Error

Type number = number of pure integrators (1/s poles) in open-loop G(s)H(s)

Input	Type 0	Type 1	Type 2
Step	1/(1+Kp)	0	0
Ramp	∞	1/Kv	0
Parabola	∞	∞	1/Ka

Kp = lim_{s→0} G(s) (position constant) Kv = lim_{s→0} s·G(s) (velocity constant) Ka = lim_{s→0} s²·G(s) (acceleration constant)

PID Controller

G_c(s) = K_p + K_i/s + K_d·s = K_p(1 + 1/(T_i·s) + T_d·s)

K_p = proportional gain (faster response, reduces error but can cause overshoot) K_i = integral gain (eliminates steady-state error, reduces stability margin) K_d = derivative gain (predictive, reduces overshoot, amplifies noise) T_i = K_p/K_i (integral time constant), T_d = K_d/K_p (derivative time)

Ideal PID: G_c(s) = K_p(1 + 1/T_i s + T_d s) Filtered derivative: G_c(s) = K_p(1 + 1/T_i s + T_d s/(T_d s/N + 1)) (N=5-20, avoids noise amp)

Ziegler-Nichols Tuning (Ultimate Gain Method)

1. Set K_i = K_d = 0; increase K_p until sustained oscillation → K_u, T_u
2. PID: K_p = 0.6K_u, T_i = T_u/2, T_d = T_u/8
3. PI: K_p = 0.45K_u, T_i = T_u/1.2 Gives ~25% overshoot — often too aggressive, use as starting point

Cohen-Coon Tuning (Process Reaction Curve)

Fit step response to FOPTD model: K, τ (time constant), θ (dead time) PID: K_p = (τ/(K·θ))(1.33 + θ/4τ), T_i = θ(32 + 6θ/τ)/(13 + 8θ/τ), T_d = 4θ/(11 + 2θ/τ)

Bode Plot Analysis

G(jω) = |G(jω)| ∠G(jω): magnitude and phase vs. frequency (log scale)

Gain [dB] = 20 log₁₀|G(jω)| Phase [deg] = ∠G(jω)

Building blocks: Gain K: constant dB, 0° phase Integrator 1/s: -20 dB/decade, -90° phase First-order 1/(τs+1): -20 dB/decade above ω=1/τ, -90° phase (total) Second-order 1/(s²/ω_n² + 2ζs/ω_n + 1): -40 dB/decade above ω_n, -180° phase total Zero (τs+1): +20 dB/decade above ω=1/τ, +90° phase (total)

Asymptotic approximation: add contributions of each element

Stability (Bode Stability Criterion)

For open-loop G(jω)H(jω):

Gain Margin (GM): GM = 1/|GH(jω_pc)| where ω_pc = phase crossover frequency (∠GH = -180°) GM [dB] = -20 log₁₀|GH(jω_pc)| Target: GM > 6 dB (robust: GM > 10 dB) GM < 0: unstable

Phase Margin (PM): PM = 180° + ∠GH(jω_gc) where ω_gc = gain crossover frequency (|GH| = 1) Target: PM > 45° (robust: PM > 60°) PM < 0: unstable

Bandwidth ω_BW: frequency where |T(jω)| = -3 dB (closed-loop bandwidth) Speed of response: t_rise ≈ 1.8/ω_BW, t_settling ≈ 4/(ζω_n)

Nyquist Criterion

Plot G(jω)H(jω) in complex plane as ω: 0→∞ Count encirclements N of (-1+j0) point Z = N + P (Z = closed-loop RHP poles, P = open-loop RHP poles) For stability: Z = 0 → N = -P (N clockwise encirclements = P)

For minimum-phase system (P=0): N = 0 (no encirclement of -1) for stability

Root Locus

Plot closed-loop poles s as gain K varies 0→∞ Open-loop: G(s)H(s) = K × N(s)/D(s) Closed-loop poles: 1 + G(s)H(s) = 0 → D(s) + K·N(s) = 0

Rules:

• Start (K=0): at open-loop poles
• End (K=∞): at open-loop zeros (or ∞ along asymptotes)
• Asymptote angles: (2k+1)×180°/(n-m) for k=0,1,...
• Asymptote centroid: σ_a = (Σpoles - Σzeros)/(n-m)
• Breakaway: d/ds[K(s)]=0

Design: add poles/zeros (lag/lead compensators) to shape root locus

State-Space Representation

ẋ = Ax + Bu y = Cx + Du

A = system matrix [n×n], B = input matrix [n×m] C = output matrix [p×n], D = feedthrough matrix [p×m]

Transfer function: G(s) = C(sI-A)⁻¹B + D

Stability: eigenvalues of A (= system poles) must have Re(λ_i) < 0

Controllability: C_matrix = [B, AB, A²B,..., A^(n-1)B]; rank = n → fully controllable Observability: O_matrix = [C; CA; CA²;...; CA^(n-1)]; rank = n → fully observable

LQR (optimal control): minimize J = ∫(xᵀQx + uᵀRu)dt Solve: Riccati equation AᵀP + PA - PBR⁻¹BᵀP + Q = 0 → K = R⁻¹BᵀP Control law: u = -Kx

Common Control Architectures

Cascade control: inner (fast) + outer (slow) loops; inner loop acts like ideal actuator to outer Feedforward: add known disturbance directly to control signal (before feedback) Smith Predictor: for dead-time systems; predicts output ahead by θ (dead time) IMC (Internal Model Control): robust, handles model uncertainty

Discrete-Time Control (Digital)

Z-transform: z = e^(sT_s), T_s = sampling period Bilinear (Tustin): s = 2(z-1)/(T_s(z+1)) → maps s-domain design to z-domain Sample rate: f_s ≥ 10-20× bandwidth (f_BW) for good performance Nyquist: f_s > 2× f_BW (minimum)

Output

Provide: transfer function G(s), closed-loop poles, GM [dB], PM [°], bandwidth ω_BW [rad/s], PID gains (K_p, T_i, T_d), step response: overshoot [%], rise time [s], settling time [s], steady-state error.