⚡ electrical engineering

Cascade PID Control of an Inverted Pendulum on a Cart System: Simulation and Experimental Analysis

This study evaluates the performance of a cascade PID control strategy for an inverted pendulum on a cart through both Simscape-based simulation and Arduino-driven experimentation, revealing that while the approach achieves stabilization, it faces significant limitations in disturbance rejection and tracking compared to an LQR-based alternative due to real-world factors like sensor noise and friction.

Original authors: Khalid Mehrab, Md Zamiul Alam, Shadman Tahmid Haque

Published 2026-05-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Khalid Mehrab, Md Zamiul Alam, Shadman Tahmid Haque

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to balance a broomstick on the palm of your hand while walking down a hallway. That is essentially what this paper is about, but instead of a broom and a hand, the researchers built a robot cart with a stick on top and tried to teach it to balance itself while moving.

Here is a breakdown of their work using simple analogies:

The Big Challenge: The Wobbly Stick

The system they built is called an "inverted pendulum." Think of it like a tightrope walker who is upside down. If they stand still, they fall over immediately. They need constant, tiny adjustments to stay upright.

This is tricky because the stick (the pendulum) and the cart are "glued" together in their behavior. If you push the cart forward to catch the falling stick, the stick swings backward. If you stop the cart too hard, the stick might tip over. It's a delicate dance where one move affects the other.

The Solution: The "Manager and the Worker" (Cascade PID)

The researchers used a control strategy called Cascade PID. To understand this, imagine a construction site with two people:

The Worker (Inner Loop): This person's only job is to keep the stick from falling. They are fast, reactive, and panic a little if the stick tilts even a tiny bit. They don't care where the cart is; they just care that the stick stays vertical.
The Manager (Outer Loop): This person tells the Worker where to walk. If the cart needs to move 10 centimeters to the right, the Manager says, "Okay, move there." But the Manager trusts the Worker to handle the balancing act while moving.

The "Cascade" part means the Manager talks to the Worker, and the Worker talks to the motor. This two-step process helps keep the system stable.

The Two Tests: The Video Game vs. The Real World

The team tested this idea in two ways:

1. The Simulation (The Perfect Video Game)
First, they built a digital version of the cart in a computer program (Simulink). In this "video game" world, there is no dust, no friction, and no shaky hands.

The Result: The digital cart balanced perfectly. It moved to the target spot, and the stick stayed upright. It was smooth and predictable, like a character in a well-coded video game.

2. The Experiment (The Real-Life Mess)
Next, they built a physical robot using an Arduino (a small computer), a motor, and sensors.

The Reality Check: Real life is messy. The wheels had friction, the sensors were a bit noisy (like static on a radio), and the motor didn't respond exactly like the math predicted.
The Result: The robot did balance, which was a success! However, it behaved differently than the computer model. The "Manager" and "Worker" had to be tuned differently. For example, in the computer, they used negative numbers to tell the cart to move backward, but the physical robot's software didn't understand negative numbers, so they had to find a workaround.

The Problems They Found

Even though the robot worked, the researchers hit some walls:

The "Too Light" Stick: In their first setup, the stick was very light (like a feather). When they gave it a little push (a disturbance), the robot couldn't recover. It was like trying to balance a feather on your finger while someone blows on it; it's too sensitive.
The Short Hallway: When the robot tried to balance a heavier stick or recover from a push, it needed to move the cart back and forth a lot. The track they built was too short. The cart would hit the end of the rail before it could finish balancing.
The "Overshoot": Sometimes, when the cart tried to move to a new spot, it would swing past the target and have to come back, like a car braking too late at a stop sign.

The Better Solution: The "Smart Manager" (LQR)

To fix the "overshoot" and the difficulty in recovering from pushes, they tried swapping out the "Worker" (the angle controller) for a more advanced algorithm called LQR.

The Analogy: If the PID controller is a worker following strict rules, the LQR is a worker who can "think ahead." It calculates the most efficient way to balance and move simultaneously.
The Result: This new setup reduced the wild swinging (overshoot) and helped the system handle disturbances better, though it still required a long track to work perfectly.

The Takeaway

The paper concludes that while the simple "Manager and Worker" (Cascade PID) approach works well for keeping the stick up, it has limits.

It struggles with heavy disturbances unless the stick is heavy enough.
It needs a long track to do its job.
Real-world robots are messier than computer simulations, so you have to tune them differently.
Using smarter math (like LQR) for the balancing part can make the whole system smoother and more efficient.

In short: They proved you can balance a stick on a moving cart using a two-step control system, but real life requires more space, heavier sticks, and sometimes smarter algorithms than a simple computer simulation suggests.

Technical Summary: Cascade PID Control of an Inverted Pendulum on a Cart System

Problem Statement
The inverted pendulum on a cart represents a classic benchmark in control theory, characterized as a highly nonlinear, multivariable, and underactuated Single-Input Multi-Output (SIMO) system. The primary challenge lies in stabilizing the inherently unstable upper equilibrium (balancing the pendulum upright) while simultaneously regulating the cart's horizontal position using a single control force. While advanced control strategies such as Linear Quadratic Regulators (LQR), fuzzy logic, and neural networks exist, the Proportional-Integral-Derivative (PID) controller remains the industry standard due to its simplicity and robustness. However, applying standard PID to this SIMO system is non-trivial; researchers often employ cascade structures to decouple the fast dynamics of the pendulum angle from the slower dynamics of the cart position. A significant gap exists in understanding the practical discrepancies between idealized simulations and real-world experimental implementations, particularly regarding sensor noise, unmodeled friction, and the limitations of cascade PID in disturbance rejection and large position commands.

Methodology
The study employs a dual-approach methodology involving both simulation and physical experimentation to evaluate a cascade PID control architecture.

Modeling and Simulation: A nonlinear model of the system was developed using Simscape Multibody within MATLAB/Simulink. This approach utilized physical parameters (cart mass: 0.5672 kg, pendulum mass: 0.0374 kg, length: 0.38 m) without requiring analytical linearization. The control structure featured an inner loop for pendulum angle stabilization and an outer loop for cart position tracking.
Experimental Setup: A physical prototype was constructed comprising a DC motor-driven cart, a rotary encoder for angle measurement, an ultrasonic sensor for position tracking, and an Arduino MEGA for real-time control. The system utilized an H-bridge (L298N) for motor actuation.
Control Implementation: The cascade PID controller was tuned in Simulink using the PID Tuner and manual adjustments to achieve specific performance metrics (e.g., <10% cart overshoot, <15% angular deviation). In the experimental phase, the controller was implemented using the Arduino IDE's built-in PID library. The system was tested under reference inputs (e.g., 10 cm cart movement) and subjected to external disturbances.
Comparative Analysis: To assess the limitations of the PID approach, a comparative simulation was conducted where the inner angle loop was replaced with an LQR-based controller while maintaining the outer PID position loop.

Key Results

Simulation vs. Experiment: Simulation results demonstrated effective stabilization with a pendulum overshoot of 10.44° and a cart settling time of approximately 7.3 seconds. Experimental results confirmed successful real-time stabilization but revealed significant deviations from simulation. Notably, the experimental gains differed substantially from simulated values (e.g., Angle Controller $K_p$ was 30 in experiment vs. 204.26 in simulation).
Sources of Discrepancy: The authors attribute the differences between simulation and experiment to four factors: the absence of an explicit filter coefficient in the Arduino PID library, the difference between force-based actuation in simulation and PWM-based actuation in hardware, sensor noise and vibrations, and unmodeled frictional effects in the physical rail and timing belt.
Disturbance Rejection Limitations: The cascade PID controller struggled to reject disturbances with the initial system parameters (pendulum mass 0.0374 kg). Successful disturbance rejection required increasing the pendulum mass to 0.6 kg and length to 0.5 m. Even with these modifications, the system exhibited large transient cart displacements, necessitating a track length of approximately 160 cm to prevent the cart from falling off the rail.
LQR Comparison: Replacing the inner PID loop with an LQR controller significantly improved performance. This hybrid approach reduced the maximum position overshoot to approximately 125 cm and shortened the settling time to roughly 20 seconds, while also reducing force demands and initial angular overshoot.

Significance and Claims
The paper provides practical insights into the applicability and limitations of cascade PID control for inverted pendulum systems. It highlights that while cascade PID is effective for stabilization under idealized conditions, its performance is heavily constrained by physical realities such as track length, sensor noise, and unmodeled dynamics. The study modestly claims that while the control structure is applicable, the inner loop places high demands on the system, often resulting in large outer-loop overshoots that may be physically unfeasible in constrained environments. The comparative analysis suggests that while PID is a standard solution, integrating optimal control methods like LQR for the inner loop can offer superior disturbance rejection and reduced overshoot. The authors conclude that future work could explore controlling the cart's acceleration and the pendulum's angular rate separately using nested PID loops to further enhance performance.