servo-control

name	servo-control
description	Servo control system design — PID tuning (Ziegler-Nichols, IMC, pole placement), cascade control (inner velocity/outer position loop), feedforward, velocity and current loops, servo motor selection (torque-speed curve, back-EMF, BEMF constant), encoder feedback (resolution, velocity estimation), bandwidth, settling time, position error, friction compensation, cogging, and notch filters for resonance.
metadata	{"priority":7,"promptSignals":{"phrases":["servo control","servo motor","servo loop","position control","velocity loop","servo tuning"],"minScore":3}}

Servo Control System Design — Complete Skill

Servo System Architecture

Cascade Control Structure

Three-loop cascade (typical machine tool / robotics):

1. Current loop (torque loop): innermost; bandwidth 1–10 kHz; controls motor current/torque
2. Velocity loop: middle; bandwidth 100–2,000 Hz; controls angular velocity
3. Position loop: outermost; bandwidth 10–200 Hz; controls shaft position

Loop bandwidth hierarchy: ω_current >> ω_velocity >> ω_position (factor of 5–10× between loops) Position loop cannot exceed ~1/3 velocity loop bandwidth (stability constraint) Example: current = 5,000 Hz; velocity = 1,000 Hz; position = 200 Hz

Inner loop saturation: Current loop limits peak torque = I_peak × K_T (motor torque constant) Velocity loop limits peak acceleration = T_peak / J_load Position loop tracks within limits set by inner loops

Motor Model and Transfer Functions

DC/BLDC Motor Electrical Model

Armature circuit: V_a = L_a × (dI_a/dt) + R_a × I_a + K_e × ω [V; L_a = inductance; R_a = resistance; K_e = back-EMF constant]

Torque equation: T_motor = K_T × I_a [K_T = torque constant ≈ K_e for SI units (V·s/rad)]

Mechanical equation: J_total × dω/dt = T_motor - T_friction - T_load [J_total = motor + load inertia referred to motor shaft]

Transfer function (electrical time constant τ_e = L_a/R_a << mechanical; neglect L_a): ω(s)/V_a(s) = K_T / (J_total × R_a) × 1 / (s + (K_T × K_e + B×R_a)/(J_total × R_a)) ≈ K_m / (τ_m × s + 1) [motor gain K_m; mechanical time constant τ_m = J_total × R_a/(K_T × K_e)]

Numerical example: K_T = 0.5 N·m/A; K_e = 0.5 V·s/rad; R_a = 2 Ω; J_total = 0.001 kg·m²; B = 0.001 N·m·s/rad τ_m = 0.001 × 2 / (0.5 × 0.5) = 0.008 s; K_m = 0.5/(0.5×0.5) = 2 (rad/s)/V

Current Loop Design

PI Controller in Current Loop

Plant (electrical): G_I(s) = 1/(L_a × s + R_a) [current to voltage] PI controller: C_I(s) = K_pI × (1 + 1/(τ_I × s)) = K_pI × (τ_I × s + 1)/(τ_I × s)

Zero-pole cancellation (common tuning): Set τ_I = L_a/R_a = τ_e [cancels motor electrical pole] Then closed-loop: T_I(s) = ω_cI / (s + ω_cI) [ω_cI = K_pI/(L_a); first-order response] Choose ω_cI = 2π × f_c_current (desired current bandwidth)

Practical values: f_c_current = 1,000–5,000 Hz (switching frequency limits practical bandwidth) K_pI = ω_cI × L_a; τ_I = L_a/R_a

Velocity Loop Design

PI Controller (Velocity)

Plant (after current loop): G_vel(s) ≈ K_T / (J_total × s) [integrator; torque → speed] PI controller: C_vel(s) = K_pV × (1 + 1/(τ_V × s))

Open-loop transfer function: L_vel(s) = K_pV × K_T / J_total × (τ_V × s + 1)/(τ_V × s²)

Loop gain crossover: set ω_c_vel where |L_vel(jω)| = 1 For stability, set τ_V = 4/ω_c_vel (gives phase margin ≈ 65°) K_pV = ω_c_vel × J_total / K_T

Velocity observer (instead of discrete differentiation): Luenberger: [ω̂_dot; d̂_dot] = A×[ω̂;d̂] + B×I_a + L×(ω_meas - ω̂) Observer bandwidth 5–10× velocity loop bandwidth

Encoder-based velocity estimation: Δω = Δθ / Δt [from encoder count change; noisy at low speed] Better: M/T method — count pulses over fixed time; measure time over fixed pulses Resolution: 1 count/T_sample; minimum detectable velocity = (1 count)/(encoder resolution × T_sample)

Position Loop Design

P or PD Controller

Plant (after velocity loop): G_pos(s) ≈ 1/s [integrator; speed → position] P controller: C_pos(s) = K_pp

Open-loop: L_pos(s) = K_pp / s Phase margin = 90° for P only — always stable but steady-state position error with velocity demand Crossover: ω_c_pos = K_pp (rad/s) → position bandwidth f_BW = K_pp/(2π)

PD with velocity feedforward: C_pos(s) = K_pp + K_dp × s [improves phase margin; reduces position lag] Alternatively: feedforward V_ff = ω_ref (reference velocity) fed directly to velocity loop Position error at constant velocity: e_ss = v / K_pp [m or rad; reduce by increasing K_pp]

Tracking error: For constant acceleration a: e_acc = a / (K_pp × K_pV × K_T/J_total) [m/rad]

Servo Motor Selection

Torque-Speed Curve

DC servo motor regions:

• Continuous torque: T_cont at rated speed ω_rated; defines thermal limit (RMS torque)
• Peak torque: T_peak = 2–4 × T_cont (short duration only; current-limited)
• Speed-torque knee: T × ω = const (field weakening for PM BLDC above base speed)

Back-EMF limit: V_bus = K_e × ω_max + I × R_a → ω_max = (V_bus - I_rated × R_a) / K_e Verify supply voltage adequate for maximum speed at rated torque

Torque requirements: T_accel = J_total × α_max + T_friction + T_gravity [N·m; α = angular acceleration] J_total = J_motor + J_load/n² + J_coupling [n = gear ratio; reflected load inertia]

Inertia matching: Optimal: J_load/n² = J_motor (equal inertia matching) Ratio: J_load/(J_motor × n²) = 1–5 (acceptable); >10 may limit achievable bandwidth

RMS torque (thermal sizing): T_rms = √(Σ T_i² × t_i / t_cycle) ≤ T_continuous [duty cycle calculation]

Motor Sizing Example

Axis spec: m_load = 50 kg, v_max = 1 m/s, a_max = 10 m/s², rack-pinion (r = 0.05 m), n_gearbox = 1 J_load = m × r² = 50 × 0.05² = 0.125 kg·m² T_accel = J_load × α_max = 0.125 × (10/0.05) = 25 N·m [α = a/r = 10/0.05 = 200 rad/s²] ω_max = v_max/r = 1/0.05 = 20 rad/s = 191 RPM Select motor: T_peak ≥ 25 N·m; ω_rated ≥ 20 rad/s; J_motor ≈ J_load/5 = 0.025 kg·m²

Advanced Compensation

Feedforward Control

Velocity feedforward: u_ff_vel = K_ff_v × ω_ref [feedforward gain ≈ 1.0 for model-based; reduces position lag at speed]

Acceleration feedforward: u_ff_acc = J_total/K_T × α_ref [torque feedforward; significantly improves tracking] Reduces position error during acceleration from e = a/K_vel² to near zero

Friction Compensation

Coulomb friction: τ_friction = τ_c × sign(ω) [step change at velocity reversal; causes limit cycles at zero speed] Compensation: add τ_ff_friction = τ_c × sign(ω_ref) to torque command

Stiction (stick-slip): Static friction > kinetic; causes "stick" at velocity reversal Dither injection: small oscillating signal at high frequency to break stiction Integral windup at zero: clamp integrator during hold; reset on motion command

Coulomb + Viscous + Stribeck model: τ_f(ω) = [τ_c + (τ_s - τ_c)×exp(-(ω/ω_s)²)] × sign(ω) + B×ω [τ_s = stiction; ω_s = Stribeck velocity; B = viscous damping]

Notch Filter for Mechanical Resonance

Problem: load compliance (flexible coupling, ball screw resonance) creates resonance at ω_n_res Without filtering: position loop bandwidth limited to < ω_n_res / 5 → slow response

Notch filter: N(s) = (s² + 2ζ_z × ω_n × s + ω_n²) / (s² + 2ζ_p × ω_n × s + ω_n²) At ω_n: gain = ζ_z/ζ_p; set ζ_z << ζ_p → deep notch at resonance Phase margin must be re-evaluated after notch; check Bode plot

Cogging torque (BLDC motors): Periodic torque ripple at N_slots × ω; compensate with feedforward table indexed by θ Amplitude: 1–15% of rated torque; causes position ripple at high resolution

Settling Time and Position Error

First-order position closed-loop: τ_settle = 1/ω_c_pos; settle to 2% in 4τ; 0.1% in 7τ For ω_c_pos = 100 rad/s (16 Hz): τ_settle = 10 ms; 2% settle = 40 ms

Steady-state position error (type 1 system, P controller): Step position: e_ss = 0 (perfect tracking) Constant velocity: e_ss = v_cmd / K_pp [position lag proportional to speed] With feedforward: e_ss → 0 (if gains perfectly matched)

Standards and References

Standard	Scope
NEMA MG1	Motors and generators specifications
IEC 61800-3	Adjustable speed electrical power drive systems
ISO 230-4	Machine tool accuracy — circular tests
ANSI/ASA S2.19	Mechanical vibration — measurement of resonance
Ogata "Modern Control Engineering"	Control textbook reference
Ellis "Control System Design Guide"	Practical servo tuning reference

Output

Provide: mechanical system (load inertia J_load [kg·m²]; friction τ_c [N·m]; maximum speed ω_max [rpm]; maximum acceleration α [rad/s²]; gear ratio n), motor selection (T_peak [N·m]; T_cont [N·m]; ω_rated [rpm]; J_motor [kg·m²]; K_T [N·m/A]; K_e [V·s/rad]; R_a [Ω]; L_a [mH]; inertia ratio J_load/n²/J_motor), torque sizing (T_accel [N·m]; T_friction; T_gravity; T_peak margin; T_rms vs. T_cont), drive selection (bus voltage V_bus [V]; peak current I_peak [A]; bus voltage check vs. K_e × ω_max), loop gains (current: K_pI; τ_I; f_BW_I [Hz] | velocity: K_pV; τ_V; f_BW_V [Hz] | position: K_pp; f_BW_pos [Hz]), settling time (2% settle time [ms]; 0.1% settle [ms]), position error at max speed (e_ss [mm or deg]; feedforward gain recommended), compensation (friction τ_c [N·m]; notch filter if resonance < 5× position BW; frequency ω_n_res [Hz]), and applicable standard (NEMA MG1; IEC 61800; ISO 230-4).