Robust Performance Assessment of a Simulink-Tuned PID Controller for a DC Motor: A Case Study in Noise Rejection

Technical Summary: Robust Performance Assessment of a Simulink-Tuned PID Controller for a DC Motor

Problem Statement
Direct Current (DC) motors are ubiquitous in industrial and consumer applications due to their linear torque characteristics and ease of control. However, achieving precise speed or position regulation in practical environments requires robust feedback strategies that can compensate for load disturbances, parameter variations, and, critically, measurement noise. While the Proportional-Integral-Derivative (PID) controller remains the dominant industrial control algorithm, its derivative term acts as a high-pass filter, inherently amplifying high-frequency sensor noise. This amplification can lead to excessive actuator wear, mechanical vibrations, and increased power consumption. A significant gap exists in many educational and introductory contexts where PID performance is evaluated under idealized, noise-free conditions, failing to account for the stochastic perturbations inherent in physical hardware implementations.

Methodology
This study employs a simulation-based approach within the MATLAB/Simulink environment to evaluate a classically tuned PID controller applied to a separately excited DC motor. The methodology is structured into three primary phases:

1. Plant Modeling: The DC motor is modeled as a linear time-invariant (LTI) system using coupled electromechanical differential equations. The open-loop transfer function relates angular velocity output to armature voltage input, utilizing specific physical parameters (e.g., $R=2.0\,\Omega$, $L=0.5\,H$, $J=0.02\,kg\cdot m^2$).
2. Controller Synthesis: A classical PID controller with a first-order derivative filter was implemented to mitigate noise amplification. The controller parameters ($K_p$, $K_i$, $K_d$, and filter coefficient $N$) were derived using the automated PID Tuner tool in Simulink. The tuning objective balanced rapid reference tracking with minimal overshoot, targeting a phase margin of approximately 60 degrees.
3. Robustness Assessment: To simulate realistic operating conditions, a stochastic disturbance source was introduced into the feedback path. This was achieved using a Band-Limited White Noise block configured with a noise power of $1\times 10^{-7}$ and a sample time of 0.001 seconds. The simulation was executed in two distinct phases: a nominal test (noise-free) to establish baseline metrics, and a robustness test (noise-injected) to evaluate disturbance rejection capabilities.

Key Contributions
The paper provides a comprehensive, quantitative comparison of a PID-controlled DC motor system under ideal versus noisy conditions. Its primary contributions include:

• Precise Parameter Documentation: The study explicitly reports the final controller gains ($K_p = 8.3463$, $K_i = 36.9358$, $K_d = 0.39908$, $N = 731.767$) obtained via automated tuning.
• Dual-Phase Validation: It establishes a rigorous benchmark by quantifying transient response metrics (rise time, settling time, overshoot) in a noise-free environment and subsequently verifying system integrity when subjected to additive measurement noise.
• Noise Rejection Analysis: The work analyzes the specific impact of Band-Limited White Noise on the control signal and tracking error, demonstrating the efficacy of the derivative filter in bounding high-frequency fluctuations without compromising stability.

Results
The simulation results yield the following quantitative and qualitative findings:

• Nominal Performance (Noise-Free): The tuned controller achieved a rise time of 0.243 seconds, a settling time (2% criterion) of 0.817 seconds, and an overshoot of 7.04%. The system exhibited zero steady-state error and robust stability margins, including a phase margin of 69 degrees at a crossover frequency of 6.4 rad/s.
• Robustness Performance (With Noise): Upon activation of the noise source, the system maintained unconditional closed-loop stability. While the output signal exhibited a bounded, high-frequency ripple consistent with the noise power spectral density, the macroscopic transient envelope remained virtually identical to the nominal case.
• Steady-State Behavior: The integral action effectively eliminated steady-state bias, resulting in a mean tracking error of zero despite the presence of noise. However, the controller output voltage displayed persistent high-frequency fluctuations, illustrating the trade-off between transient damping and noise amplification.
• Stability: Crucially, the noise injection did not induce divergence, limit cycling, or instability. The system preserved its stability margins, confirming that the tuned parameters are resilient to the specific level of stochastic perturbation modeled.

Significance and Claims
The authors conclude that the derived PID tuning parameters, in conjunction with the implemented derivative filter, offer a robust and practically viable control solution for DC motor applications. The study claims that the controller successfully preserves transient performance and stability even in the presence of realistic sensor noise artifacts. By validating the design against stochastic disturbances, the paper asserts that the Simulink-based tuning methodology yields a controller that is not merely an academic exercise but a design capable of withstanding real-world signal imperfections. The findings suggest that with appropriate filtering and tuning, PID controllers can effectively balance the competing demands of rapid reference tracking and noise rejection in industrial electromechanical systems.