⚡ electrical engineering

Robust Quadcopter Motion Control Using Output Feedback

This paper proposes a robust output feedback control strategy for quadcopters that utilizes a geometric approach to transform the system model and an extended observer method to synthesize the control law.

Original authors: Stanislav Kim, Anton Pyrkin, Oleg Borisov

Published 2026-05-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Stanislav Kim, Anton Pyrkin, Oleg Borisov

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 a quadcopter (a drone with four spinning propellers) as a very sensitive, high-speed dancer. The goal of this paper is to teach this dancer how to move to a specific spot and hold that pose perfectly, even if the music is loud, the floor is slippery, or the dancer doesn't know exactly how heavy their own costume is.

Here is the breakdown of the paper's solution, explained through simple analogies:

The Problem: Dancing in the Dark

The authors start by saying that controlling a drone is incredibly hard. The drone's movement is described by complex math that changes constantly.

The Challenge: Usually, to control a dancer perfectly, you need to know exactly where they are, how fast they are moving, and how fast they are spinning. This is like having a camera watching every single joint of the dancer.
The Limitation: In the real world, we often can't measure everything. We might know where the drone is (its position), but we can't easily measure its speed or how fast it's tilting. It's like trying to guide a dancer in the dark; you can see where they are, but you can't feel their momentum.
Previous Attempts: Past methods tried to simplify the math by pretending the drone moves in a straight line (linearization), but this often fails when the drone actually tries to do complex turns. Other methods worked well but required those "perfect cameras" (full state measurement) that we don't always have.

The Solution: The "Super-Observer" and the "Double-Buffer"

The authors propose a new way to control the drone using only the information we can see (the output), without needing to measure the hidden speeds. They do this in three main steps:

1. Re-organizing the Dance (The Normal Form)

First, the authors take the messy, complicated math describing the drone and rearrange it into a "normal form."

The Analogy: Imagine the drone's movement is a tangled ball of yarn. The authors untangle it and lay it out in a straight, orderly line. They also add a "double-buffer" system. In electronics, a buffer smooths out signals. Here, they mathematically "double integrate" the control input.
What this means: Instead of telling the drone "push harder now," the system treats the command as a smooth, gradual instruction that builds up over time. This makes the math much more stable and predictable, turning a chaotic system into a neat, linear one.

2. The "Ghost" Dancer (The Extended Observer)

Since we can't measure the speed and rotation directly, the authors create a "Ghost Dancer" inside the computer.

The Analogy: Imagine you are trying to guide a friend in a foggy room. You can see them, but you can't see how fast they are running. So, you create a mental image (a "Ghost") of your friend. You watch where your real friend is, and you constantly compare it to your Ghost.
How it works: The computer runs a special algorithm called an Extended Observer. This algorithm guesses what the hidden speeds and rotations should be based on the position it can see. It uses a "high-gain" setting, which is like having a very sharp, attentive eye. If the real drone drifts even a tiny bit from where the Ghost expects it to be, the Ghost immediately updates its guess.
The Result: The controller doesn't need to measure the speed directly; it just asks the Ghost, "What do you think the speed is?" and uses that answer to make decisions.

3. The Safety Net (Robust Control)

Finally, they design the actual control law (the instructions sent to the motors).

The Analogy: Even with a good Ghost, the real world is messy. There might be wind gusts or the drone might be slightly heavier than expected. The authors add a "saturation" function.
What this means: Think of this as a safety valve. If the Ghost gets confused or the math suggests a crazy, impossible move, the controller gently caps the instruction so the drone doesn't spin out of control. This ensures that even if the drone is imperfect or the environment is rough, it will still eventually settle down and stop moving (asymptotic stability).

The Bottom Line

The paper claims to have solved the problem of controlling a drone when you can't measure all its moving parts.

They simplified the math to make the drone's behavior predictable.
They built a "Ghost" in the computer that guesses the missing information (speed and rotation) based on what is visible.
They created a control system that uses these guesses to steer the drone safely to its target, proving mathematically that the drone will stay stable even with errors and uncertainties.

In short, they figured out how to drive a car perfectly using only the view through the windshield, without needing to look at the speedometer or the tachometer, by using a smart computer to guess the speed for you.

Technical Summary: Robust Quadcopter Motion Control Using Output Feedback

Problem Statement
The paper addresses the challenge of controlling quadcopter motion in the presence of parametric, structural, functional, and signal uncertainties. While previous research has utilized model linearization or nonlinear control strategies, existing solutions often suffer from limitations such as degraded transient performance due to model inaccuracies, the exclusion of yaw angle dynamics, or cumbersome parameter tuning. Crucially, prior works that provided complete dynamic models for positioning tasks (specifically reference [3]) relied on full state measurement, requiring knowledge of output derivatives that are often unavailable or noisy in practical applications. The core problem tackled here is the synthesis of a robust control law for quadcopter motion where the derivatives of the output variables are not measurable, utilizing only output feedback.

Methodology
The authors propose a control synthesis approach based on a geometric transformation and an extended observer method. The methodology proceeds through the following steps:

Model Transformation: The multidimensional nonlinear quadcopter model, which includes position, velocity, and attitude dynamics with viscous friction and gravitational effects, is transformed into a normal form. This is achieved by defining deviations from desired positions and orientations and applying a specific change of variables.
Input Augmentation: To handle the time-varying gain coefficients inherent in the transformed model, the control input is subjected to double integration. This process renders the gain coefficient stationary, allowing the system to be represented in a standard normal form with a state vector $\zeta \in \mathbb{R}^{16}$ and a control input vector $U$ .
State Feedback Synthesis: A state feedback control law is first derived for the transformed normal form. This law utilizes feedback linearization, canceling nonlinearities and placing the closed-loop poles via gain selection ( $\gamma_1, \gamma_2, \gamma_3, \gamma_4$ ) based on standard characteristic polynomials (e.g., Butterworth or Newton).
Output Feedback Synthesis: Recognizing that the full state vector and specific nonlinear functions are not available for measurement, the authors replace the state feedback with an output feedback law. This involves:
- Extended Observer: An observer is designed to estimate the unmeasurable states and the unknown nonlinearities. The observer utilizes a high-gain approach ( $\kappa$ ) and a cascade structure to estimate the state vector $\hat{\zeta}$ and a lumped disturbance term $\sigma$ .
- Control Law: The final control input is generated using the estimated states and the observer's disturbance estimate, passed through a smooth saturation function to ensure boundedness.

Key Contributions

Extension of Previous Work: The paper extends the complete dynamic positioning results of reference [3] to the case where output derivatives are unmeasurable, removing the dependency on full state measurement.
Geometric Transformation: The successful transformation of the complex, time-varying quadcopter model into a stationary normal form via double integration of the control input.
Robust Output Feedback Law: The synthesis of a control law that guarantees semi-global asymptotic stability without requiring knowledge of the full state vector or the exact inverse of the input gain matrix and nonlinear functions.

Results and Claims
The paper claims that the proposed control law guarantees semi-global asymptotic stability of the closed-loop system. The theoretical results indicate that the equilibrium point of the quadcopter motion (deviations from the desired position and orientation) is asymptotically stable.

Significance
The significance of this work lies in its practical applicability to quadcopter control where sensor limitations prevent full state observation. By relying on an extended observer rather than full state feedback, the proposed method overcomes a major implementation barrier found in previous nonlinear control strategies. The approach simplifies the control tuning procedure compared to earlier methods while maintaining robustness against model uncertainties and external disturbances, all without requiring the measurement of velocity or angular rate derivatives. The work is supported by the Ministry of Science and Higher Education of the Russian Federation.