⚡ electrical engineering

Prescribed Performance Sway Suppression Control for Overhead Cranes under Friction and Actuator Saturation

This paper proposes a unified collocated partial feedback linearization control framework for overhead cranes that integrates online friction estimation, anti-saturation recovery, and prescribed performance functions to achieve rapid sway suppression and asymptotic stability under actuator limits and friction, outperforming conventional baselines in simulations.

Original authors: Trong Nghia Phan, Huu Cuong Nguyen, Quang Hieu Ngo

Published 2026-07-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Trong Nghia Phan, Huu Cuong Nguyen, Quang Hieu Ngo

Original paper licensed under CC BY 4.0 (https://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 a giant construction crane moving a heavy load of bricks from one spot to another. The goal is simple: get the cart (the trolley) to the new spot quickly and stop it exactly there. But there's a catch: the load is hanging on a rope, and if you move the cart too fast or stop too abruptly, the bricks will swing wildly like a pendulum. If they swing too much, they could hit something, or the whole system could become unstable.

This paper presents a new "brain" (a control system) for these cranes that solves three specific headaches that usually make this job difficult: friction, motor limits, and uncontrolled swinging.

Here is how the authors explain their solution using simple concepts and analogies:

1. The Three Big Problems

The authors identify three things that usually ruin a smooth crane ride:

The Sticky Floor (Friction): The cart moves on rails. Sometimes the wheels stick, and sometimes they slide. This "stick-slip" behavior makes the cart jitter and prevents it from stopping exactly where you want it to. It's like trying to slide a heavy box across a floor that is sometimes icy and sometimes covered in sandpaper.
The Strained Muscle (Actuator Saturation): The motor driving the cart has a maximum strength. If the computer asks for more power than the motor can give, the motor hits a "wall" (saturation). It's like a runner sprinting at 100% effort; they can't go any faster, no matter how hard they try. If the system doesn't handle this carefully, the cart might overshoot or get stuck.
The Wild Swing (Sway): Even if you stop the cart perfectly, the hanging load might still be swinging. The goal is to stop the swing before the cart even stops moving.

2. The Solution: A "Smart Coach" with Three Tricks

The authors built a control system that acts like a highly skilled coach who knows exactly how to handle the cart and the swinging load. They combined three techniques into one unified system:

Trick A: The "Memory" Coach (Friction Compensation)

The system doesn't just guess how sticky the floor is; it learns it in real-time.

How it works: As the cart moves, the system uses a mathematical "memory" (called Recursive Least Squares) to constantly update its understanding of the friction. It's like a driver who feels the road getting slippery and immediately adjusts their braking pressure without thinking about it.
The Result: The cart stops exactly where it should, without that annoying "jitter" caused by sticky wheels.

Trick B: The "Safety Valve" (Anti-Saturation)

When the motor hits its maximum power limit, the system doesn't panic.

How it works: Imagine a car with a speed governor. If you hit the limit, a normal system might keep pressing the gas pedal, causing the engine to strain. This new system has a "safety valve" that instantly adjusts the command. It says, "Okay, we are at max power; let's back off slightly so we don't get stuck in that limit."
The Result: The motor recovers quickly from its limits, preventing the cart from overshooting the target.

Trick C: The "Invisible Cage" (Prescribed Performance)

This is the most unique part. The system doesn't just try to stop the swing eventually; it promises the swing will never get too big, even for a split second.

How it works: Imagine an invisible, shrinking cage around the swinging load. At the start, the cage is wide to allow for movement. As time goes on, the cage gets smaller and smaller, forcing the load to settle down. The system is mathematically guaranteed to keep the load inside this shrinking cage at all times.
The Result: The load never swings dangerously high, even during the fastest parts of the move.

3. How They Tested It

The authors didn't just build this on paper; they simulated it using a standard, realistic crane model (the "Inteco 3D crane"). They compared their new "Smart Coach" against three other common methods:

The Basic Linear Controller (LQR): Good, but gets confused when the motor hits its limit.
The Standard Non-Linear Controller (PFL): Good at handling the physics, but doesn't have the "invisible cage" to limit the swing.
The Simple PD Controller: A basic, old-school method that doesn't account for friction or motor limits.

4. The Results: Who Won?

The "Smart Coach" (their new PFL+PPF system) won every category:

Smoother Stops: It stopped the cart with almost no overshoot.
Least Swinging: The load swung the least (only about 7.4 degrees).
- The basic linear controller let it swing 11.7 degrees.
- The simple PD controller let it swing a wild 19.2 degrees and kept oscillating (shaking) for a long time because it couldn't handle the friction.
Fastest Calm Down: It stopped the swinging energy in just 4 seconds, faster than any other method.
Robustness: Even when they changed the weight of the load, the length of the rope, or blew a "wind" on the load, the system kept the swing inside its safe "cage" without needing to be re-tuned.

Summary

In short, this paper describes a new way to control cranes that treats the cart and the swinging load as a team. By teaching the computer to learn the friction, respect the motor's limits, and force the swing into a shrinking safety zone, they created a system that is faster, safer, and more precise than previous methods. It ensures that heavy loads move quickly but never swing dangerously.

Technical Summary: Prescribed Performance Sway Suppression Control for Overhead Cranes under Friction and Actuator Saturation

Problem Statement
Overhead cranes are critical underactuated systems where a single control input (trolley force) must simultaneously achieve precise trolley positioning and payload sway suppression. Practical deployment is hindered by three concurrent challenges that often degrade performance or induce instability when addressed in isolation:

Trolley–Rail Friction: Causes steady-state errors, stick-slip oscillations, and limit cycles.
Actuator Saturation: Motor input limits can lead to integrator windup and degraded transient responses.
Transient Sway Constraints: The absence of explicit bounds on the sway angle during motion can allow unsafe swing amplitudes before convergence.

Existing literature typically addresses these issues separately (e.g., linear methods lacking friction compensation, or nonlinear methods ignoring saturation and transient constraints). Furthermore, while Partial Feedback Linearization (PFL) is a standard approach for underactuated systems, integrating friction estimation, anti-saturation mechanisms, and prescribed performance constraints within a unified collocated PFL framework has not been previously achieved.

Methodology
The authors propose a unified control framework based on Collocated Partial Feedback Linearization (PFL) that integrates three key components:

System Modeling and Friction Estimation:
- The crane is modeled as a 2D underactuated system with a regularized Coulomb–viscous friction model to eliminate discontinuities at zero velocity.
- Recursive Least Squares (RLS) is employed for online estimation of the motor gain and friction coefficients ( $b_\pm, c_\pm$ ). The system dynamics are rearranged into a state-affine form to facilitate this regression.
Control Architecture:
- Collocated PFL: The directly actuated trolley channel is exactly linearized. The control law solves for the input force to track a virtual acceleration input $v$ , while the payload sway (non-collocated channel) is shaped indirectly through the resulting internal dynamics.
- Anti-Saturation Modification: A feedback loop is introduced where the saturation error ( $e_s = u - u^*_c$ ) is fed back into the control command with a gain $\gamma$ . This ensures rapid recovery from actuator limits without altering the structural stability of the internal dynamics.
- Prescribed Performance Control (PPF): To enforce explicit transient bounds on the sway angle $\theta(t)$ , a Prescribed Performance Function $\rho(t)$ is defined. A two-step backstepping procedure is used to inject the PPF constraint into the relative-degree-two sway dynamics. This involves generating a virtual velocity reference from the normalized PPF error and designing the acceleration channel to track it, ensuring the sway angle remains strictly within $-\rho(t) < \theta(t) < \rho(t)$ .
Stability Analysis:
- A two-stage Lyapunov analysis is conducted.
- Stage 1 (Transient): Input-to-State Stability (ISS) arguments establish the uniform boundedness of tracking errors during the parameter identification (RLS) and saturation transients.
- Stage 2 (Asymptotic): A Lyapunov–LaSalle argument proves asymptotic convergence of the trolley position and sway angle to zero, while guaranteeing the PPF constraint is never violated. The analysis confirms that the cross-coupling terms in the backstepping design cancel analytically, removing the need for restrictive gain conditions on the damping term.

Key Contributions
The paper identifies three principal contributions:

Unified Constrained Nonlinear Framework: The first integration of online RLS friction estimation, anti-saturation modification, and prescribed-performance sway regulation within a single collocated PFL architecture.
Formal Transient-Performance Bound: The explicit enforcement of a time-varying safety envelope for the payload swing angle throughout the entire motion, ensuring the angle never exceeds a predefined bound.
Rigorous Two-Stage Stability Analysis: A formal proof establishing boundedness during the identification phase and asymptotic convergence with constraint satisfaction thereafter, specifically addressing the coupled effects of friction, saturation, and PPF.

Simulation Results
Simulations were conducted using physical parameters from a benchmark Inteco 3D crane ( $M=1.67$ kg, $m=0.73$ kg, $l=1.0$ m). The proposed PFL+PPF controller was compared against:

PFL (Nonlinear): PFL without PPF.
LQR+sat: A saturated Linear Quadratic Regulator with hard clipping but no recovery mechanism.
PD Energy: A PD-based controller with no friction compensation or saturation handling.

Key Performance Metrics:

Peak Sway Angle: PFL+PPF achieved the lowest peak sway at 7.4°, compared to 8.9° (PFL), 11.7° (LQR+sat), and 19.2° (PD Energy).
Energy Dissipation: The proposed controller dissipated sway energy fastest, reaching negligible levels within 4 seconds, outperforming the others (4.5s for PFL/LQR, and non-convergent for PD).
Constraint Satisfaction: Only the PFL+PPF controller strictly maintained the sway angle within the prescribed time-varying bound $\pm\rho(t)$ at all times.
Friction Compensation: The PD baseline exhibited a persistent limit-cycle oscillation due to uncompensated friction, whereas all friction-compensated controllers converged asymptotically.
Robustness: The PFL+PPF controller maintained performance and respected the PPF bound under:
- ±30% payload mass variation.
- ±20% rope length variation.
- A 0.3 N wind impulse disturbance.
- No controller retuning was required for these variations.

Significance and Claims
The authors claim that this work fills a critical gap by integrating friction compensation, saturation recovery, and transient sway constraints into a single control law. Unlike previous approaches that treat these nonlinearities separately, the proposed framework demonstrates that these components can be unified within a collocated PFL structure.

The paper modestly notes that the current study is limited to a 2D planar model with single-axis friction. The authors state that future work will extend this framework to 3D crane configurations with double-pendulum dynamics and time-varying rope lengths, and will pursue hardware-in-the-loop validation on the Inteco 3D platform. The significance of the current work lies in providing a rigorous, practically verifiable foundation for handling these complex, coupled nonlinearities before such extensions.