Securing Control Systems: Encrypting Nonlinear Dynamics

Author: Denis Avetisyan

Researchers have developed a method to implement encrypted control for complex nonlinear systems by approximating dynamics with autoregressive models.

This work presents an ARX-based reformulation of nonlinear observers that enables continuous encrypted control through finite-impulse-response approximation and homomorphic encryption.

While secure computation offers pathways to protect control systems, limitations in performing recursive operations on encrypted data hinder the implementation of complex nonlinear controllers. This paper, ‘ARX-Implementation of encrypted nonlinear dynamic controllers using observer form’, addresses this challenge by presenting a method to reformulate dynamic control laws into an autoregressive with exogenous inputs (ARX) model. This approach enables continuous encrypted control by approximating recursive computations with finite operations on past inputs and outputs, achieved through an observer-based representation akin to finite-impulse-response approximation. By decoupling control from unbounded recursion, can this method unlock truly scalable and secure cyber-physical systems?

The Illusion of Control: Why Traditional Estimation Fails

Accurate state estimation forms the bedrock of effective control and predictive modeling across a vast range of applications, from aerospace navigation and robotic manipulation to economic forecasting and biological systems analysis. However, when dealing with nonlinear systems – those where the relationship between input and output isn’t a simple straight line – traditional estimation techniques like the Kalman filter often falter. These methods rely on linear approximations, which can introduce significant errors as the system deviates from those linear boundaries, leading to divergence and instability. The complexity arises because nonlinearities introduce interactions between state variables that linear methods cannot adequately capture, necessitating computationally intensive alternatives or innovative approaches to maintain accuracy and robustness in estimation processes. This challenge drives ongoing research into methods capable of effectively handling the inherent intricacies of nonlinear dynamics.

Nonlinear systems, ubiquitous in real-world phenomena from fluid dynamics to biological networks, often defy prediction using techniques designed for simpler, linear models. This limitation arises because nonlinearities introduce behaviors – such as chaos, limit cycles, and multiple equilibria – that linear approximations fundamentally fail to capture. Consequently, employing linear estimation methods on these systems introduces significant inaccuracies; even small errors in initial conditions or measurements can be rapidly amplified, leading to divergence from the true system state and potential instability in control applications. The inability to accurately represent the system’s dynamics compromises the reliability of predictions and hinders effective control strategies, necessitating the development of specialized estimation techniques tailored to the complexities of nonlinear dynamics.

Stability as Emergence: Designing for Convergence

Reliable state estimation necessitates that the observer – the algorithm used to approximate the system’s internal state – converges to the actual, true state of the system. This convergence is not simply a desired characteristic, but a fundamental requirement for accurate and dependable performance. If the observer does not converge, the estimated state will diverge from the true state, rendering the estimation process useless for control, prediction, or monitoring purposes. Stability, in the context of observer design, refers to this guaranteed convergence; a stable observer will consistently approach and maintain an accurate estimate of the system state, even in the presence of disturbances or modeling errors. Without stability, the estimated state can oscillate indefinitely or drift without bound, compromising the overall system’s functionality and potentially leading to instability.

Lyapunov analysis formally verifies observer stability by demonstrating that a chosen Lyapunov function decreases over time, ensuring convergence to the true system state. This involves defining a scalar function $V(e)$ , where $e$ represents the estimation error. Stability is then assessed by examining the time derivative $\dot{V}(e)$ . Class-K functions, $\alpha(e)$ , provide lower bounds- $\alpha_1(e) \le V(e) \le \alpha_2(e)$ -establishing bounded estimation error for bounded disturbances. Class-KL functions, $\beta(e,t)$ , extend this by defining bounds on the estimation error as a function of both error magnitude and time, allowing for transient behavior analysis and demonstrating asymptotic stability – that is, $e(t) \rightarrow 0$ as $t \rightarrow \in fty$ .

Observer form selection is critical in state estimation as it dictates how measured output data is integrated into the state reconstruction process. Typically expressed within a state-space framework, the observer equation $\dot{\hat{x}} = Ax + Bu + L(y - C\hat{x})$ defines the time derivative of the estimated state $\hat{x}$ based on the system dynamics (A, B), the measurement (y), and the observer gain matrix (L). The choice of this form, alongside the gain L, directly impacts convergence speed and stability. Different observer forms – such as the Luenberger observer, Kalman filter, or extended Kalman filter – offer varying computational complexities and performance characteristics based on system and measurement noise properties. Proper form selection, therefore, is predicated on a detailed understanding of the system dynamics, measurement model, and desired estimation accuracy.

ARX: A Minimal Representation of Dynamic Control

The Auto-Regressive with eXogenous inputs (ARX) model offers a streamlined approach to dynamic controller representation by approximating system behavior using a finite number of past control inputs and outputs. Traditional dynamic controllers often rely on recursive calculations to determine present control actions based on historical states; however, the ARX model bypasses this need through a direct mapping of past inputs and outputs to the current control signal. This is achieved by formulating the controller as a linear regression, effectively replacing iterative calculations with a finite series of multiplications and additions. Consequently, the ARX model reduces computational complexity and offers a more efficient alternative for real-time control applications where minimizing latency is critical.

Finite-Impulse-Response (FIR) approximation techniques are integral to the practical implementation of the ARX model by converting the inherent infinite-impulse-response characteristics into a computationally manageable form. This is achieved by truncating the recursive calculations and representing the dynamic controller as a weighted sum of past inputs and outputs, effectively limiting the number of operations required. The resulting discrete-time approximation reduces computational complexity from potentially infinite calculations per time step to a finite number directly proportional to the filter order. This simplification is critical for real-time control applications where processing speed and resource constraints are paramount, allowing for timely execution of control algorithms even on embedded systems with limited processing power.

Our implementation of the Auto-Regressive with eXogenous inputs (ARX) model achieves a reformulation of nonlinear dynamic controllers, facilitating encrypted control operations without compromising computational efficiency. This is accomplished by representing the control logic as a finite number of multiplications, thereby eliminating the need for state recursion inherent in traditional implementations. This approach allows for the secure execution of control algorithms, as all intermediate states are avoided, and sensitive data remains confined within encrypted multiplicative operations. The resulting system significantly reduces computational complexity, enabling real-time control applications while maintaining a high level of security.

Securing the System: Control Through Obfuscation

The Autoregressive with eXogenous input (ARX) model offers a unique architecture that facilitates ‘Encrypted Control’ – a method of maintaining system security by processing information directly in its encrypted state. Traditionally, control systems require decryption for computation, creating a vulnerability; however, the ARX structure, when coupled with techniques like homomorphic encryption, allows for operations on encrypted signals without prior decryption. This approach safeguards sensitive data throughout the entire control loop, mitigating risks associated with potential breaches during computation or transmission. By performing control calculations – such as state estimation and feedback adjustments – directly on encrypted data, the system effectively conceals its operational parameters and internal states, bolstering resilience against adversarial attacks and unauthorized access.

Homomorphic encryption represents a transformative approach to data security, enabling computations to be performed directly on encrypted data – essentially, processing information without ever revealing its content. This remarkable capability bypasses the traditional need for decryption before processing and re-encryption afterward, thereby eliminating a critical vulnerability point. The system leverages mathematical algorithms that maintain data confidentiality even during complex operations, such as those involved in control systems. Consequently, sensitive data remains protected throughout the entire computational pipeline, ensuring privacy and integrity – a significant advantage in applications where data security is paramount, and allowing for secure control schemes without compromising confidential information.

Simulation results demonstrate a compelling characteristic of the ARX model: as the model’s order increases, the impact of both external perturbations and errors introduced during controller reformulation diminishes to a negligible level. This robustness stems from the higher-order model’s enhanced ability to represent and counteract disturbances, effectively filtering out inaccuracies that might otherwise compromise system performance. The findings suggest that carefully selecting an appropriate ARX order allows for secure control schemes, even in the presence of noise or imperfect information, without sacrificing accuracy or stability – a critical feature for reliable operation in complex and potentially adversarial environments.

Validation and the Path Forward: Emergent Behavior in Complex Systems

Practical implementation of the proposed Adaptive Recursive Least Squares (ARX) controller was successfully demonstrated utilizing a Single-Link Flexible Joint Robot as a physical plant. This validation moved the controller beyond simulation, proving its ability to manage a real-world robotic system with inherent flexibility and dynamic complexities. The robot’s movements, guided by the ARX controller, exhibited stable and accurate tracking performance, confirming the controller’s robustness and applicability to robotic applications demanding precise control despite model uncertainties. This achievement establishes a crucial stepping stone toward deploying the ARX controller in more complex robotic systems and dynamic environments, offering a viable solution for advanced motion control.

Simulations reveal a compelling relationship between the performance of the Adaptive Recursive Least Squares (ARX) controller and its order, denoted as $N$ . As $N$ increases, the resulting plant trajectories exhibit a marked convergence towards those achieved by the ideal, nominal closed-loop system. This indicates that higher-order ARX models possess a greater capacity to accurately represent the plant’s dynamic behavior and effectively compensate for disturbances. The observed trend suggests that, while computational complexity rises with increasing $N$ , a corresponding improvement in trajectory tracking accuracy is consistently achieved, highlighting a crucial trade-off for control system designers seeking optimal performance. This ability to closely mimic the nominal system’s response demonstrates the ARX controller’s potential for robust and precise control of complex robotic systems.

Ongoing research endeavors are directed toward refining the robustness of this system against practical limitations. Specifically, investigations are underway to lessen the detrimental effects of quantization errors – the loss of precision when converting continuous data into discrete digital representations – which can introduce inaccuracies into the control process. Simultaneously, exploration of more sophisticated encryption schemes is being pursued to further enhance data security and protect sensitive information during transmission and processing. These advancements aim to bridge the gap between theoretical performance and real-world implementation, paving the way for secure and reliable robotic control systems in various applications.

The pursuit of secure control systems, as detailed in this work, reveals a fascinating interplay between complexity and simplification. This research elegantly demonstrates how intricate nonlinear dynamics can be approximated through ARX models, facilitating encrypted computation. This mirrors a broader principle: order doesn’t need architects; it emerges from local rules. As Epicurus observed, “It is not the desire for pleasure and avoidance of pain that makes people happy, but the absence of disturbance.” Similarly, this method doesn’t impose control, but rather allows it to emerge through secure, localized operations on past data, sidestepping the vulnerabilities inherent in centralized, recursive calculations. The system is a living organism where every local connection matters, and this approach prioritizes the health of those connections by minimizing reliance on potentially compromised global states.

Beyond the Horizon

The pursuit of encrypted control, as demonstrated by this work, reveals a fundamental tension. Attempts to impose direct, hierarchical control on complex dynamical systems are inherently brittle. The reformulation of controllers into autoregressive structures, amenable to finite-impulse approximation and homomorphic encryption, is less about achieving control in the traditional sense, and more about influencing system behavior through localized operations. This shift is not merely a technical maneuver; it reflects a deeper understanding that order doesn’t need architects. The resilience of any such system will not stem from centralized command, but from the robustness of these local rules.

Remaining challenges are predictable. The fidelity of the ARX approximation introduces unavoidable error, and the computational cost of homomorphic encryption remains substantial. However, the more interesting limitations lie elsewhere. This approach excels at maintaining stability under observation, but actively shaping system behavior-guiding it towards novel states-will require navigating the trade-off between encryption overhead and expressive control. Future work should explore how to leverage the inherent noise within encrypted computations to introduce controlled stochasticity, effectively turning the limitations of the system into a feature.

Ultimately, the goal is not to build perfectly secure control systems, but to create systems capable of adapting and persisting in the face of uncertainty. The true measure of success will not be the absence of interference, but the system’s ability to absorb it and continue functioning. It’s a subtle distinction, but one that reflects a growing appreciation for the emergent properties of complex systems-properties that cannot be engineered, only encouraged.

Original article: https://arxiv.org/pdf/2512.21244.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/