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Integro-differential equations in angular stabilization of drone motion by distributed feedback control

This paper proposes a universal approach to stabilize drone motion using distributed feedback control with unbounded memory by reducing integro-differential equations to systems of ordinary differential equations, demonstrating that complex exponential kernels can achieve new exponential stability results.

Original authors: Alexander Domoshnitsky, Oleg Kupervasser, Anatoly Polonsky

Published 2026-07-22

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Alexander Domoshnitsky, Oleg Kupervasser, Anatoly Polonsky

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 broom on the palm of your hand. If you only look at where the broom is right now, you might react too late, or overcorrect, and the broom will fall. But what if you could remember every wobble the broom made over the last hour? That "memory" of past movements could help you predict exactly how to move your hand to keep it upright. This is the heart of a field called control theory, which is basically the science of making machines behave the way we want them to. In the real world, machines like drones don't just react to the present; they are influenced by their past, by wind gusts that happened seconds ago, and by the "lag" in their sensors.

Usually, when engineers try to fix a wobbly drone, they use simple rules: "If you tilt left, push right." But this paper explores a much smarter, albeit more complex, idea: using a "distributed feedback" system. Think of this as giving the drone a brain that doesn't just look at a single snapshot of its position, but instead looks at a long, continuous movie of its entire flight history. The challenge is that mathematically, keeping track of an infinite movie is incredibly hard to calculate, especially if the "memory" of the drone is supposed to be unbounded (meaning it remembers everything, forever). The authors of this paper wanted to see if they could use this super-long memory to stabilize a drone's flight in a way that simple, short-term rules couldn't.

The paper, titled "Integro-differential equations in angular stabilization of drone's motion by distributed feedback control," proposes a clever mathematical trick to solve this problem. The authors, Alexander Domoshnitsky, Oleg Kupervasser, and Anatoly Polonsky, tackle the difficult math behind "integro-differential equations." In plain English, these are equations that mix how fast something is changing now with a sum of everything that happened before. The authors show that instead of getting stuck trying to solve these messy equations directly, you can transform them into a different, larger system of standard equations (ordinary differential equations) that are much easier to handle.

Here is the magic part: they prove that even if the drone's memory is theoretically infinite, you don't need to store every single past position in a giant hard drive. Instead, their method creates a "shadow system" of variables that implicitly "absorbs" all that history. It's like having a magic sponge that soaks up the entire history of the flight without needing to list every drop of water. By doing this, they can use standard, well-known tools to check if the drone will stay stable.

The researchers tested this idea on a specific problem: keeping a drone steady in the vertical plane (preventing it from pitching up or down uncontrollably). They found that for certain types of unstable drones—specifically those where the natural physics make them wobble or dive—using a control system based on a single "exponential memory" (a simple type of memory that fades over time) isn't enough to make them stable. However, by combining two different types of memory kernels (think of this as mixing two different flavors of "remembering" the past), they discovered they could stabilize even the most difficult cases.

In their simulations, they didn't just guess; they ran the numbers. For example, they modeled a drone flying at a speed where the Mach number is 1.6 (faster than sound) at a height of 12 kilometers. In this scenario, the drone's natural physics were unstable (with a specific parameter $\gamma_2$ calculated as -30.192). By applying their new control method with specific parameters (like a memory decay rate $\alpha_2$ of 6 and specific control gains $\beta_1$ and $\beta_2$ ), they showed through computer simulation that the drone's wobbles would die down exponentially. The results, visualized in their graphs, show the drone's angle and speed settling into a smooth, stable path rather than spiraling out of control.

The paper also explicitly rules out some simpler approaches. They show that if you try to stabilize a drone using only a single memory term on the drone's speed (rather than its position), it simply won't work for certain unstable configurations; the math proves it's impossible to make it stable that way. They also note that while their method works beautifully for "linear approximations" (simplified models of the drone's behavior), the real world is messy. However, their core finding is that by expanding the "memory" of the control system and using their new mathematical reduction technique, you can achieve a level of stability that was previously thought difficult or impossible with standard methods.

Ultimately, this work suggests that giving a drone a "long memory" and knowing how to process that memory efficiently is a powerful way to keep it flying straight. It turns a complex, history-heavy math problem into a manageable one, opening the door for more robust and reliable autonomous flight systems that can handle the unpredictable nature of the sky.

Technical Summary: Integro-differential equations in angular stabilization of drone's motion by distributed feedback control

Problem Statement
The paper addresses the challenge of angular stabilization for unmanned aerial vehicles (UAVs), specifically focusing on the motion of a drone in a vertical plane. While traditional flight control often relies on discrete state measurements or standard proportional-integral-derivative (PID) logic, this work investigates the use of distributed feedback control in the form of integral operators with potentially unbounded memory.

The authors note that while integral control is intuitively appealing for noise reduction (averaging measurements) and utilizing historical state data for better control, the resulting mathematical models are integro-differential equations. These equations present significant theoretical and numerical difficulties:

Theoretical Complexity: Standard stability analysis methods often struggle with the integral terms, particularly when memory is unbounded.
Numerical Instability: Integral terms accumulate calculation errors over time, making long-term numerical solutions unreliable.
Gap in Literature: There is a scarcity of research applying integral delay feedback control specifically to drone angular stabilization, particularly using kernels that allow for the utilization of long observation times.

Methodology
The core methodological contribution of the paper is a technique to reduce the analysis of integro-differential equations to the analysis of systems of ordinary differential equations (ODEs).

Model Construction: The authors derive a mathematical model for drone longitudinal motion in a vertical plane. This involves linearizing the equations of motion around a steady state to obtain a system of four equations describing perturbations in velocity, angle of attack ( $\alpha$ ), pitch angle ( $\vartheta$ ), and height. Under specific simplifying assumptions (negligible height dependence, slow velocity changes), this reduces to a second-order differential equation for the angle of attack and pitch dynamics.
Reduction to ODEs: The authors propose a universal approach (Theorem 3.1) based on the expansion of the integral kernel $k(t,s)$ $k (t, s)$ into a series of orthogonal basis functions. By representing the kernel as a sum of exponential terms (or finite linear combinations thereof), the integral operator is replaced by auxiliary state variables governed by first-order ODEs.
- If the kernel is an infinite series, the resulting system is infinite-dimensional.
- If the kernel is a finite linear combination of exponentials (e.g., $q=1$ or $q=2$ ), the system reduces to a finite-dimensional ODE system.
Stability Analysis: Once reduced to an ODE system, the authors apply classical Hurwitz criteria to determine the necessary and sufficient conditions for exponential stability. This allows for the identification of "stability zones" in the parameter space of the control gains ( $\beta$ ) and decay rates ( $\alpha$ ).

Key Contributions and Results

Reduction of Complexity: The paper demonstrates that integro-differential equations with unbounded memory can be transformed into equivalent systems of ODEs. This avoids the accumulation of numerical errors associated with direct integration and allows the use of standard ODE stability tools.
Stabilization of Unstable Systems: The authors analyze three specific cases where the underlying drone dynamics are unstable (i.e., the characteristic equation of the uncontrolled system has roots with non-negative real parts):
1. Case I ( $\gamma_1 > 0, \gamma_2 < 0$ ): Stabilization is achievable using a single integral control term acting on the state $x(s)$ .
2. Case II ( $\gamma_1 < 0, \gamma_2 > 0$ ): Stabilization is impossible with a single integral term acting on $x(s)$ but is achievable if the control acts on the derivative $x'(s)$ or if the control structure is modified.
3. Case III ( $\gamma_1 < 0, \gamma_2 < 0$ ): This represents the most unstable scenario. The authors show that a single integral term is insufficient. However, by using a linear combination of two exponential kernels (a more complex control structure), exponential stability can be achieved.
Parameter Zones: The paper derives explicit inequalities (based on Hurwitz criteria) that define the regions of control parameters ( $\alpha, \beta$ ) required to ensure stability for each case.
Application to Drone Flight: The theoretical results are applied to the specific drone model derived in Section 2. The authors calculate specific control parameters for a drone flying at Mach 1.6 and an altitude of 12 km.
- They demonstrate that for a scenario where the uncontrolled system is unstable ( $\gamma_2 < 0$ ), a control law involving a combination of an integral of the state and a weighted integral of the state with exponential decay can stabilize the system.
- Numerical Verification: Using MATLAB's dde23 solver, the authors simulate the system with the derived parameters. The results (Figure 5) confirm that the system exhibits exponential stability, with state variables converging to zero.

Significance and Claims
The paper claims to offer a "universal approach" to studying the stability of integro-differential equations, specifically in the context of feedback control with unbounded memory.

Novelty in Control: The authors emphasize that this is likely the first time in mathematical literature that feedback control in the form of integral terms with exponential kernels has been explicitly used and analyzed for drone angular stabilization.
Handling Unbounded Memory: Unlike previous approaches that required small memory lengths to be tractable, this method treats unbounded memory not as an obstacle but as a feature that aids stability analysis by allowing the "absorption" of all previous states into the auxiliary ODE system.
Practical Utility: The work bridges the gap between complex theoretical integro-differential models and practical engineering applications. By reducing the problem to finite-dimensional ODEs (using simple exponential kernels), the authors provide a computationally feasible method for designing stabilizing controllers for UAVs that utilize distributed feedback.
Enhanced Capabilities: The paper suggests that while simple kernels (single exponential) work for some cases, more complex kernels (linear combinations of exponentials) significantly enhance stabilization capabilities, allowing for the control of systems that would otherwise be impossible to stabilize with simpler integral controls.

The authors conclude that their approach provides new, "unexpectable" results regarding the exponential stability of these equations and offers a concrete framework for stabilizing drone flight in the vertical plane using distributed feedback.

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