DC Motor PID Control

This example demonstrates how to build a transfer function-based DC motor speed closed-loop control system, employing feedforward compensation and anti-windup PID control strategies to achieve high-precision, fast-response motor speed regulation.

The system uses a dual-channel drive architecture to eliminate steady-state error while ensuring dynamic performance.

Theoretical Basis

DC Motor Transfer Function Modeling

A DC motor is composed of an armature circuit coupled with a mechanical rotating subsystem.

Armature Circuit Voltage Equation:

V_a(t) = R_a i_a(t) + L_a \frac{di_a(t)}{dt} + K_e \Omega(t)

Mechanical Torque Balance Equation:

K_t i_a(t) = J \frac{d\Omega(t)}{dt} + B \Omega(t)

Where $V_a$ is the drive voltage, $\Omega$ is the rotor angular velocity, $K_t$ is the torque constant, $K_e$ is the back-EMF constant, is the moment of inertia, and is the friction coefficient.

Transfer Function Derivation: Eliminating the current variable yields the standard second-order form:

\begin{aligned} G(s) &= \frac{\Omega(s)}{V_a(s)} \\ &= \frac{K_t}{J L_a s^2 + (J R_a + B L_a)s + (R_a B + K_t K_e)} \\ &= \frac{K_{motor}}{a_2 s^2 + a_1 s + a_0} \end{aligned}

Feedforward + PID Dual-Channel Control

The system adopts a parallel drive structure, balancing response speed and disturbance rejection.

Feedforward Channel:

V_{ff}(s) = \omega_{ref}(s) \cdot K_{ff}, \quad K_{ff} = \frac{1}{K_{rpm}}

Directly generates the baseline drive voltage based on the reference speed, achieving fast response without overshoot.

PID Channel:

V_{pid}(s) = \left(K_p + \frac{K_i}{s} + K_d \frac{N s}{s + N}\right) \cdot e(s)

Compensates for model errors and external disturbances through Proportional-Integral-Derivative regulation.

Total Control Output:

V_{control}(s) = V_{ff}(s) + V_{pid}(s)

Anti-Windup Mechanism

When the control voltage exceeds the saturation limit, the anti-windup loop is activated to suppress integrator saturation:

\dot{x}_i = K_i e - K_w \cdot \text{sat}(V_{calc} - V_{sat})

Block Function	Parameter	Value	Description
Target Speed	`omega_ref`	`2000`	Target Speed (rpm)
	`t_step`	`1.0`	Step Time (s)
Motor TF	`Kt`	`0.3`	Torque Constant (N·m/A)
	`Ke`	`0.05`	Back-EMF Constant (V/(rad/s))
	`Ra`	`1.2`	Armature Resistance (Ω)
	`La`	`0.008`

Note: K_rpm_per_volt = Kt / (Ra*B + Kt*Ke) * 60/(2*pi) is the DC gain of the motor (rpm/V), automatically calculated from raw parameters.

Performance Metrics

Steady-State Performance

Steady-State Error: ≤ 1% × Target Speed
- Target: 2000, Steady-State: 2004
Voltage Utilization: ≤ 70% (Reserved dynamic margin)
- Limit: 24V, Max Control Voltage: ~13V

Dynamic Performance

Overshoot: ≤ 5%
Settling Time: ≤ 1.5s (Within ±2% error band)
Anti-Windup Recovery: ≤ 0.3s

Results Analysis

Typical Response Process:

0-0.2s: Feedforward channel dominates, approaching target speed rapidly.
0.2-1.0s: PID channel fine-tunes, integrator eliminates residual error.
>1.5s: Enters steady state, PID output approaches zero, feedforward maintains speed.

Observed Effects:

Speed tracks the setpoint smoothly with static error < 10 rpm.
Voltage output is smooth with no significant saturation.
Anti-windup mechanism intervenes quickly during load steps.

Applications

Engineering Prototyping: Development of 24V DC motor speed control systems.
Algorithm Research: PID parameter auto-tuning, intelligent anti-windup strategies.

Configuration Code

# 1 Motor Physical Parameters (SI Units)
# Modify these parameters to adapt to different motors
Ra = 1.2;           # Armature Resistance (Ω) - Affects steady-state voltage demand
La = 0.008;         # Armature Inductance (H) - Affects electrical response speed
Kt = 0.3;           # Torque Constant (N·m/A) - Affects torque output capability
Ke = 0.05;          # Back-EMF Constant (V/(rad/s)) - Key: Smaller Ke means higher max speed
J = 0.002;          # Moment of Inertia (kg·m²) - Affects mechanical response speed
B = 0.001;          # Friction Coefficient (N·m/(rad/s)) - Affects low-speed performance
 
# 2 Transfer Function Coefficient Calculation
# Automatically calculate numerator and denominator based on physical parameters
K_motor = Kt;                           # TF Numerator = Torque Constant
a2 = La * J;                            # 2nd-order term: Electro-mechanical coupling
a1 = Ra * J

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