DynaMark: A Reinforcement Learning Framework for Dynamic Watermarking in Industrial Machine Tool Controllers

TL;DR

DynaMark is a reinforcement learning framework that adaptively applies dynamic watermarking to industrial machine controllers, effectively detecting tampering with minimal energy use and maintaining control performance in real-time.

Contribution

It introduces a novel RL-based approach that models dynamic watermarking as an MDP, enabling online adaptation without system knowledge, improving detection and efficiency.

Findings

Reduces watermark energy by 70% compared to baselines

Maintains detection delay within one sampling interval

Validates effectiveness on digital twin and physical testbed

Abstract

Industry 4.0's highly networked Machine Tool Controllers (MTCs) are prime targets for replay attacks that use outdated sensor data to manipulate actuators. Dynamic watermarking can reveal such tampering, but current schemes assume linear-Gaussian dynamics and use constant watermark statistics, making them vulnerable to the time-varying, partly proprietary behavior of MTCs. We close this gap with DynaMark, a reinforcement learning framework that models dynamic watermarking as a Markov decision process (MDP). It learns an adaptive policy online that dynamically adapts the covariance of a zero-mean Gaussian watermark using available measurements and detector feedback, without needing system knowledge. DynaMark maximizes a unique reward function balancing control performance, energy consumption, and detection confidence dynamically. We develop a Bayesian belief updating mechanism for real-time detection confidence in linear systems. This approach, independent of specific system assumptions, underpins the MDP for systems with linear dynamics. On a Siemens Sinumerik 828D controller digital twin, DynaMark achieves a reduction in watermark energy by 70% while preserving the nominal trajectory, compared to constant variance baselines. It also maintains an average detection delay equivalent to one sampling interval. A physical stepper-motor testbed validates these findings, rapidly triggering alarms with less control performance decline and exceeding existing benchmarks.

Tables

Table 1. TABLE I: ARX(1,1) parameters identified from watermark-free continuous data.

Motor commands	$A$	$B\left(\mathrm{mm}/\mathrm{mV}\right)$	$Q\left({\mathrm{mm}}^2\right)$	$\overline{𝐲}\left[\mathrm{mm}\right]$
M1) 4000∘, 200 RPM	$1$	$0.0075$	$5.57\times {10}^{-6}$	46.94
M2) 8000∘, 300 RPM	$1$	$0.0108$	$9.81\times {10}^{-6}$	91.38
M3) 4000∘, 300 RPM	$1$	$0.0107$	$9.38\times {10}^{-6}$	46.92
M4) 0∘, 200 RPM	$1$	$0.0076$	$5.57\times {10}^{-6}$	2.48

Table 2. TABLE II: Optimal watermarking variance from stepper-motor LTI approximation via problem in  [ 11 ] under varied loss budgets.

$\delta$	$100\%J_0$	$50\%J_0$	$20\%J_0$	$10\%J_0$	$5\%J_0$
$U_{\star }\left(m{V}^2\right)$	$0.8655$	$0.4327$	$0.1731$	$0.0865$	$0.0432$

Table 3. TABLE III: Parameters setup for the numerical study.

Description	Value
$A$	$1.0$
$B$	$0.010$
$Q$	$1.3741\times {10}^{-13}$
Initial state	${\mu }_0=0,\Sigma =0$
Controller	Proportional, $K_p=1.0$, $\overline{𝐲}=0.012$
Replay-attack prior	$\sigma \sim \mathrm{Ber}\left(0.05\right)$,  $\tau \mid \sigma =1\sim \mathrm{Geom}\left(1/T\right)$
Attacker lag	$\Delta t=0$ (full-history replay)
$w_{\beta }$	$50$ samples
$T$	$1000$

Table 4. TABLE IV: Online configuration of parameters.

Batch Size (points)	Pause Duration (ms)	Buffer Flush
30	10	Yes
100	10	No
100	20	Yes
100	50	Yes
500	20	Yes
100	10	Yes

Table 5. TABLE V: DDPG implementation details for the numerical study and the physical stepper-motor testbed.

Hyperparameter / setting	Numerical study (Section IV-A)	Physical testbed (Section IV-B)
Actor / critic architecture	3 fully-connected layers ($32$ neurons each)	3 fully-connected layers ($64$ neurons each)
Activation & normalization	Leaky-ReLU + layer norm	Leaky-ReLU + layer norm
Number of critics	2 (double-Q to curb bias)	2 (double-Q to curb bias)
Optimizer & learning rate	RMSprop, $1\times {10}^{-3}$	RMSprop, $1\times {10}^{-3}$
Gradient clipping	${\Vert g\Vert }_2\le 1$	${\Vert g\Vert }_2\le 1$
Target-network update	$\tau =5\times {10}^{-3}$ (soft)	$\tau =5\times {10}^{-3}$ (soft)
Replay buffer size	$1\times {10}^6$ transitions	$1\times {10}^6$ transitions
Mini-batch size	$128$	$128$
Exploration noise	OU($\mu =0$, ${\sigma }_0=0.99$, $\theta =0.15$)	OU($\mu =0$, ${\sigma }_0=0.99$, $\theta =0.15$)
Noise decay	$\sigma \leftarrow 0.995\sigma$	$\sigma \leftarrow 0.995\sigma$
Training episodes	$200$	$30$
Steps / episode	${10}^3$ env. steps	$4.1\times {10}^4$ fast-time steps
Reward weights	${\omega }_1={\omega }_2=0.35,{\omega }_3=0.30$	${\omega }_1={\omega }_2=0.35,{\omega }_3=0.30$
Type-I error level	$\alpha =0.10$	$\alpha =0.10$

Equations

$${\mathbf{y}}_{t+1}=A{\mathbf{y}}_t+B{\mathbf{u}}_t+{\mathbf{w}}_t,$$

$${\hat{\mathbf{y}}}_t=A{\mathbf{y}}_{t-1}+B{\mathbf{u}}_{t-1}.$$

$${\mathbf{r}}_t={\mathbf{y}}_t-{\hat{\mathbf{y}}}_t.$$

$${\mathbf{u}}_t^{\prime }={\mathbf{u}}_t+{\varphi }_t.$$

$${\theta }_t$$

$${\psi }_t$$

$${\mathcal{Z}}_t=\sum _{k=0}^{t-1}{A}_{k}B{U}_{t-k-1}{B}^{\top }{A}_k^{\top },$$

$${\mu }_t$$

$${\mathcal{W}}_t$$

$${\hat{\mathbf{y}}}_t=A{\mathbf{y}}_{t-1}+B{\mathbf{u}}_{t-1}+B{\varphi }_{t-1}.$$

$${\mathcal{W}}_t={\mathcal{Y}}_t+Q.$$

$$g_t={\mathbf{r}}_t^{\top }{Q}^{-1}{\mathbf{r}}_t.$$

$${\mathbf{u}}_t^A=-{\mathbf{u}}_t+{\varphi }_t.$$

$${\mathbf{u}}_t^A=-{\mathbf{u}}_t-{\varphi }_t.$$

$${\mathbf{y}}_t^A={\mathbf{y}}_{t-\Delta t},\tau \le t\le \tau +T_r,\Delta t=\tau -t_1.$$

$$d_t=\frac{d_{t-1}p(I_t|\sigma =1)}{d_{t-1}p(I_t|\sigma =1)+(1-d_{t-1})p(I_t|\sigma =0)}.$$

$${\mathbf{u}}_t^{\prime }$$

$${\mathbf{y}}_{t+1}$$

$${\hat{\mathbf{y}}}_{t+1}$$

$${\mathbf{r}}_{t+1}$$

$$g_{t+1}$$

$$d_{t+1}\text{ according to Eq. eqrefeq:dt}.$$

$${\beta }_t=\sum _{k=1}^t{\mathcal{F}}_{t|\tau =k}(\tilde{g})p(\tau =k|\sigma =1)+\mathcal{H}(\tilde{g})p(\tau >t|\sigma =1),$$

$$r_t(s,a)=-{\omega }_1\Vert {\varphi }_t{\Vert }_1-{\omega }_2\Vert {\mathbf{y}}_{t+1}^{wom}-{\mathbf{y}}_{t+1}{\Vert }_2+{\omega }_3|0.5-d_{t+1}|.$$

$$⟨{\mathbf{\tau }}_t,{\mathbf{y}}_t,{\mathbf{u}}_t,{\mathbf{\varphi }}_t,\mathrm{attac}{\mathrm{k}}_{\mathrm{flag}}⟩,$$

$${\hat{\mathbf{y}}}_t=A{\mathbf{y}}_{t-1}+B{\mathbf{u}}_{t-1}$$

$$\mathbb{E}({\hat{\mathbf{y}}}_t)$$

$$={\mu }_t+{\theta }_t-B{\varphi }_{t-1}$$

$${\hat{\mathbf{y}}}_t^A=A{\mathbf{y}}_{t-1}^A-B{\mathbf{u}}_{t-1}+B{\varphi }_{t-1}$$

$${\mathbf{r}}_t^A=$$

$$=$$

$$-A{\mathbf{y}}_{t-1}^A+B{\mathbf{u}}_{t-1}-B{\varphi }_{t-1}$$

$$=$$

$${\hat{\mathbf{y}}}_t^A=A{\mathbf{y}}_{t-1}^A+B{\mathbf{u}}_{t-1}$$

$${\mathbf{r}}_t^A=$$

$$=$$

Full text

I Introduction

The digital transformation, real-time analytics, and Artificial Intelligence (AI) are advancing manufacturing toward interconnected Industry 4.0 ecosystems, but cybersecurity falls short, exposing legacy plant floor assets to sophisticated threats [1, 2, 3]. Notable incidents like the 2014 German steel-mill breach [4], WannaCry shutdowns at auto plants [5, 6], and the 2019 LockerGoga attack on Norsk Hydro [6] highlight the vulnerability of Machine Tool Controllers (MTCs) in managing Computer Numerical Control (CNC) machinery and other equipment on the plant floor. Compounding this vulnerability, MTCs have proprietary, closed architectures, limiting insight into their mechanisms and restricting efforts to understand and mitigate their security risks [7]. Among cyberattacks, replay attacks are especially dangerous as they need no model knowledge; attackers can just record and replay legitimate measurement streams, bypassing intrusion detection and risking part quality and catastrophic damage [8, 9]. A typical method to detect replay attacks is using authentication signals like watermarking, which are unknown to attackers [8].

Physical watermarking verifies system integrity and authenticity, similar to how traditional watermarks prevent piracy and confirm ownership. Watermarking embeds unique authentication signals into the system to distinguish legitimate from replayed data [10]. However, the effectiveness of this method depends heavily on the careful design and implementation of these signals. High detection accuracy may degrade control performance, as overly sensitive detection mechanisms can disrupt normal controller operation [9, 11]. Static or poorly tuned watermarks hinder performance or fail to address evolving attacks [9, 11, 12]. This tradeoff motivates an adaptive watermarking paradigm, an approach that offers greater flexibility but increases complexity, posing additional challenges for proprietary systems [13]. A promising approach uses adaptive watermarking with system data to detect replay attacks, balancing detection accuracy and system responsiveness.

I-A Related Works

I-A1 MTC Cybersecurity

The transition of manufacturing towards IoT-enabled, data-driven Industry 4.0 workflows has expanded the cyberattack surface of MTCs [1, 6, 3, 5]. MTC vulnerabilities stem from outdated systems that lack regular security updates, low operator awareness, and dense network connections, making them prone to cyberattacks [3, 6]. Manufacturing facilities must integrate IoT-specific security measures, including multi-layered authentication, tamper-resistant encryption, and real-time surveillance, into their operational technology infrastructure [3]. AI and Machine Learning (ML) algorithms are capable of analyzing controller data to detect nuanced anomalies, concurrently adjusting defense policies. Furthermore, contemporary cryptographic techniques ensure the security of data flows over their entire life-cycle [2]. Viewing cybersecurity as a core design element, rather than just an operational expense, is crucial for the resilience of smart manufacturing and MTC operations [3, 6].

I-A2 Attacks on MTCs

Cyberattacks on MTCs have the potential to extend their impact beyond mere data breaches, leading to substantial physical damage. Among the possible attacks on MTC, deception attacks are dangerous because they exploit the trust between the cyber and physical components [8, 13]. In a deception attack, an adversary changes system data to cause harmful actions by the system or its users. Three major types of deception attacks on MTCs [13] that are commonly discussed in the literature are as follows:

• •

Flip attacks: Flip attacks jeopardize the integrity and reliability of data and control signals in industrial control systems and MTCs. These attacks flip the sign of data streams or control signals. In MTCs, the actuator’s control action sign is reversed, causing significant errors and instability due to accumulating opposing signals over time [6].

• •

Injection attacks: An injection attack compromises data integrity in integrated systems by injecting deceptive data into data streams. This false data can mislead the control mechanism, causing harmful actions. It exploits the trust between sensors, controllers, and actuators, creating discrepancies between the perceived and actual states [1, 6, 14].

• •

Replay Attacks: In a Replay Attack, adversaries capture and retransmits valid signals to trick the system into wrong actions without needing system knowledge. Attackers record real data during normal operation and replay it later, causing the system to act on outdated or incorrect information, with unintended effects. These attacks are intuitive but hard to detect with conventional mechanisms focused on integrity and authenticity [8]. MTCs are highly vulnerable to replay attacks due to their minimal security protocols, closed architectures, and interaction with physical processes.

I-A3 Replay Attack Detection

A common method for replay attack detection uses authentication signals or watermarking, presumed to be unknown to adversaries [8, 15]. This method enables prompt detection by disrupting the watermark during an attack. Physical watermarking is classified into input-added, which alters input signals, and output-added, which affects output signals [16]. This ensures detection of any interference, regardless of alterations to inputs or outputs. Watermarking-based cyberattack detection was proposed by Mo et al. [17], introducing Dynamic Watermarking (DWM) to detect cyberattacks in controlled systems. By embedding authentication signals into control inputs, they improved detection rates while analyzing trade-offs with control performance. Subsequent research has split into two complementary streams.

1. (i)

Operator–specified, constant-covariance DWM frameworks show that simply superimposing random Gaussian signals can expose tampering even in noisy or partially observed plants: single-input and multi-input-multi-output extensions with robustness to non-Gaussian disturbances [18, 19, 20], linear time-invariant (LTI) to linear time-variant (LTV) generalizations [21, 13], lightweight key-based recursion for unsecured channels [22], and multi-layer industrial deployments that cut false alarms and localize faults [23]. While effective, these schemes assume LTI–Gaussian models and keep the operator picked watermark intensity constant, creating a static detectability–performance trade-off. 2. (ii)

Systematic watermarking strategies to balance between detection accuracy and control performance address the challenges associated with the first direction. These control-theoretic optimization approaches choose a covariance offline for LTI systems by employing Linear-Quadratic-Gaussian (LQG) cost or maximizing the Kullback–Leibler (KL) divergence. This results in constant yet provably optimal signals regarding cost or detection delay [24, 25, 9, 26]. Additionally, with online system identification, these methods lead to convergent rank-one updates when the initial LTI parameters are unknown [11]. These designs rely on an LTI-Gaussian model and fix the watermark power at a single offline value, unable to adjust when plant dynamics change. This limits their effectiveness against replay attacks in time-varying or slightly nonlinear MTCs, highlighting the need for an adaptive, online approach.

I-B Research Gaps & Contributions

Despite cybersecurity advancements, detecting replay attacks in MTCs remains challenging. The review of existing literature identifies three unresolved shortcomings that are particularly pronounced in proprietary, closed-architecture MTCs:

1. G1:

LTI-Gaussian dependence. Most watermarking frameworks assume stationary LTI dynamics and i.i.d. Gaussian noise. Minor time variations, dynamic changes, and unmodeled nonlinearities common in modern plant floors and MTCs undermine detection and performance assurances. 2. G2:

Static watermark statistics. Offline-optimized covariances from the system identification phase cannot adapt to dynamic changes, leading to a fragile trade-off that either hinders control performance or misses intrusion detection. 3. G3:

Limited expressiveness. Constant watermark signals exhibit limited detection capabilities due to their stationary nature. They are unable to leverage measurement data or focus their impact on frequency bands prone to vulnerabilities.

We address the research gaps G1–G3 by presenting DynaMark, a reinforcement learning (RL) framework that learns and adapts DWM signals online without requiring prior knowledge of the plant model. DynaMark utilizes readily available measurements (such as position) and iteratively refines a compact neural policy. This policy is designed to adjust both the covariance and the spectral characteristics of the introduced watermark, with the objectives of maximizing replay-attack detectability and adhering to control-quality constraints. The key contributions of this paper are as follows.

1. (i)

We formulate the dynamic watermarking as a Markov Decision Process (MDP) and designed an RL-based DMW algorithm that generates adaptive watermarking signals for replay attack detection on MTCs. This allows the watermark to adapt to the detector’s confidence about the system state and adjust watermarking intensity dynamically to achieve a balance between control performance and detection power. 2. (ii)

The unique design of the reward function considers control performance, energy consumption, and detection power simultaneously. This achieves a flexible design that accommodates different needs for these three aspects in the operations. 3. (iii)

We derive a Bayesian belief update mechanism, which is used to characterize the detection confidence online and taken as an input by the RL, under the linear system dynamics assumption. Although the method does not rely on any assumptions about system dynamics, this lays the theoretical foundation for the MDP when the system can be represented or approximated by a linear dynamics. 4. (iv)

The effectiveness of the proposed framework is demonstrated on a digital twin (DT) and a real-world physical stepper-motor testbed, where DynaMark achieves faster detection and lower control performance degradation than the benchmark watermarking schemes.

The remainder of this paper is organized as follows: Section II elucidates the modeling assumptions pertaining to MTC system dynamics, the watermarking scheme, attack models on MTCs, and the associated theoretical results that are formulated to characterize the detection power. Section III details the DynaMark methodology and the RL-based algorithm. Section IV presents experimental findings obtained from the DT and the physical testbed, alongside comparisons with constant-covariance benchmarks. Finally, the article concludes with a discussion in Section V.

II Problem Setup

In this section, we introduce the system dynamics of the MTCs, the watermarking scheme, and three possible attack models on MTCs. Fig. 1 presents the flow chart that depicts the dynamic monitoring and control process of machine tools. We provide a detailed explanation of each block in the flowchart.

II-A System Model

A sensor network is monitoring the system shown in Fig. 1. We consider the dynamics of the machine tools as a stochastic linear dynamic system of order $n$ described by the equation

[TABLE]

where $\mathbf{y}_{t}\in\mathbb{R}^{n}$ is the collection of all sensor measurements at time $t$, $\mathbf{u}_{t}\in\mathbb{R}^{c}$ is the vector of the control action generated by the control logic based on the observed $\mathbf{y}_{t}$, and $\mathbf{w}_{t}\in\mathbb{R}^{n}$ is a zero mean independently and identically distributed ($i.i.d$) Gaussian noise, i.e., $\mathbf{w}_{t}\sim\mathcal{N}(0,Q)$ and $Q$ is a positive semidefinite (PSD) covariance matrix [18, 19]. $A\in\mathbb{R}^{n\times n}$ is the state (or system) matrix, and $B\in\mathbb{R}^{n\times c}$ is the input matrix. Further, we assume that the initial condition $\mathbf{y}_{0}\sim\mathcal{N}(\mu_{0},\Sigma)$ independent of the process noise, and $\Sigma$ is a PSD covariance matrix. This representation is a specific instance of the well-known LTI state-space model as the sensors are honest [18, 19]. We model machine tools’ dynamics through measurements because their physical nature limits direct state observation and parameter estimation $A$, $B$, $Q$.

A feedback control system applies correction to the controlled variable. Control logic is a function $f:\mathbb{R}^{n}\to\mathbb{R}^{c}$ defined by parameters $\eta$ as $\mathbf{u}_{t}=f(\mathbf{y}_{t};\eta)$. In addition, we consider the estimator of the state $\mathbf{y}_{t}$ to be

[TABLE]

The residual of the dynamic system in (1) is given by

[TABLE]

The detector, as shown in Fig. 1, calculates the residuals and triggers an alarm based on predefined confidence levels and statistical metrics. These metrics are further discussed in detail in Section II-B. The detector determines if control logic should continue generating signals, indicating normal operation, or stop, suggesting an attack.

Remark 1**.**

In practice, the parameters governing system dynamics ($A$, $B$, and $Q$) can be estimated from the measured signals $\{\mathbf{y}_{t}\}$ and control inputs $\{\mathbf{u}_{t}\}$ recorded during MTCs’ normal operation, using classical system identification methods. Additionally, for inherently nonlinear dynamic systems, piecewise linear modeling and estimation can capture the nonlinearities, where parameters are no longer time-invariant. This is demonstrated by our case study in Section IV-B.

II-B Watermarking Scheme

The main idea of a physical watermark is to inject a random noise, $\phi_{t}$, which is called the watermark signal, into the system (1). This signal is used to excite the system and check whether the system responds to the watermark signal in accordance with the dynamic model of the system. In MTCs, we consider injecting watermarks into the control actions, i.e.,

[TABLE]

In practice, watermarks are randomly drawn from a predetermined distribution at each time step $t$, which is typically assumed to be Gaussian. This distribution is carefully defined by practitioners to maintain controller performance and avoid disrupting operations while enhancing detection capabilities. However, in dynamic watermarking, the distribution or its parameters are adjusted based on the system state and detection performance. In this study, we assume the watermark signals $\{\phi_{t}\}$ are independent zero-mean Gaussian random variables with a covariance $U_{t}$, which changes dynamically.

For ease of derivation and clarity in notation, we define $A_{k}=A^{k}$. Under normal conditions, $\mathbf{y}_{t}=\theta_{t}+\psi_{t},$ where

[TABLE]

Notice that $\theta_{t}\sim\mathcal{N}(0,\mathcal{Z}_{t})$, where

[TABLE]

and, since control actions are determined, $\psi_{t}\sim\mathcal{N}(\mu_{t},\mathcal{W}_{t})$ where

[TABLE]

Then, it holds that $\mathbf{y}_{t}\sim\mathcal{N}(\mu_{t},\mathcal{Z}_{t}+\mathcal{W}_{t})$. Similarly, for the state estimator, $\widehat{\mathbf{y}}_{t}$ is given by

[TABLE]

Given the distribution of $\mathbf{y}_{t}$, $\mathbb{E}(\widehat{\mathbf{y}}_{t})=A\mu_{t-1}+B\mathbf{u}_{t-1}$. Substituting Eq. (8), it follows that $\mathbb{E}(\widehat{\mathbf{y}}_{t})=\mu_{t}$, and $\mathbf{Var}(\widehat{\mathbf{y}}_{t})=A\mathcal{W}_{t-1}A^{\top}+\mathcal{Z}_{t}$. Defining $\mathcal{Y}_{t}=A\mathcal{W}_{t-1}A^{\top}$, $\widehat{\mathbf{y}}_{t}\sim\mathcal{N}(\mu_{t},\mathcal{Y}_{t}+\mathcal{Z}_{t})$ holds. Furthermore, it can be inferred by (9),

[TABLE]

Considering the residuals in (3) under watermarking and the distributions of $\mathbf{y}_{t}$ and $\widehat{\mathbf{y}}_{t}$, it holds that $\mathbb{E}(\mathbf{r}_{t})=0$, and $\mathbf{Cov}(\mathbf{y}_{t},\widehat{\mathbf{y}}_{t})=\mathcal{Y}_{t}+\mathcal{Z}_{t}$. Then, by Eq. (11), $\mathbf{Var}(\mathbf{r}_{t})=Q$ that implies, for the system defined in (1), the residuals follow $\mathbf{r}_{t}\overset{i.i.d}{\sim}\mathcal{N}(0,Q)$. Therefore, the probability of obtaining the measurement $\mathbf{y}_{t}$ is computed as $f_{\mathbf{y}_{t}}(y)=(2\pi)^{-n/2}\det(Q)^{-1/2}\exp\left(-1/2g_{t}\right),$ where

[TABLE]

It is easy to show that $g_{t}\sim\chi^{2}_{n}$ with $n=|\mathbf{y}_{t}|$ degrees of freedom. When this probability is low, it means the system is likely to be subject to a certain anomaly or attack. Therefore, the test for detecting an attack involves checking $g_{t}\lessgtr\tilde{g}$ where $\tilde{g}$ is an appropriate threshold. If $g_{t}$ exceeds the threshold, the $\chi^{2}$ detector will trigger an alarm. Given that watermark signals $\{\phi_{t}\}$ are known to the operator, and assuming the detector does not use them for state estimation, the following results show the distribution of test statistic $g_{t}$.

Lemma 1**.**

Let the watermark signals be known. If $\widehat{\mathbf{y}}_{t}=A\mathbf{y}_{t-1}+B\mathbf{u}_{t-1}$, then $g_{t}\sim\chi^{2}(\lambda)$ with noncentrality parameter $\lambda=\phi_{t}^{\top}B^{\top}B\phi_{t}$ and $n$ degrees of freedom. ∎

The proof of Lemma 1 is deferred to the Appendix -A.

II-C Residuals Analysis under Attack

This section examines how an attacker might disrupt the system described in Section II-A. We theoretically assess the feasibility of attacks on the controlled system. Additionally, we examine how watermarking can improve detection under specific conditions.

II-C1 Residuals Distribution under Flip Attacks

A flip attack flips the sign of the control actions. Under this attack, it holds $\mathbf{u}_{t}^{A}=-\mathbf{u}_{t},$ that feeds into the system in Eq. (1) and the estimator in Eq. (2). Two flip attack scenarios can occur under the watermarking scheme: (i) The flip attack happens before injecting the watermarking signal, i.e.,

[TABLE]

(ii) The flip attack happens after injecting the watermarking signal, i.e.,

[TABLE]

During a flip attack starting at $\tau$, for $t\geq\tau$, the residuals in both cases are given as $\mathbf{r}_{t}^{A}=\mathbf{y}_{t}^{A}-\widehat{\mathbf{y}}_{t}^{A}=\mathbf{w}_{t-1}$, where $\mathbf{y}_{t}^{A}=A\mathbf{y}_{t-1}^{A}+B\mathbf{u}_{t-1}^{A}+\mathbf{w}_{t-1}$ and $\widehat{\mathbf{y}}_{t}^{A}=A\mathbf{y}_{t-1}^{A}+B\mathbf{u}_{t-1}^{A}$. Residuals during an attack match the normal condition distribution, showing watermark signals do not affect flip attack detection. The following theorems define conditions the state estimator must meet for successful detection with watermarks.

Theorem 1**.**

Let the watermark signals be known. Assume the system with $\chi^{2}$ detector is subjected to a flip attack following Eq. (14). If under attack $\widehat{\mathbf{y}}_{t}^{A}=\widehat{\mathbf{y}}_{t-1}^{A}-B\mathbf{u}_{t-1}+B\phi_{t-1}$, then, $\mathbf{r}_{t}^{A}\sim\mathcal{N}(-2B\phi_{t-1},Q)$ and $g_{t}^{A}\sim\chi^{2}(\lambda)$ with a noncentrality parameter $\lambda=4\phi_{t-1}^{\top}B^{\top}B\phi_{t-1}$ and $n$ degrees of freedom. ∎

Theorem 2**.**

Given $\{\phi_{t}\}$, and the system with a noncentral $\chi^{2}$ detector defined by Eq. (12) is subject to a flip attack following Eq. (13) or (14). If under attack $\widehat{\mathbf{y}}_{t}^{A}=A\widehat{\mathbf{y}}_{t-1}^{A}+B\mathbf{u}_{t-1}$, then $\mathbf{r}_{t}^{A}\sim\mathcal{N}\left(-2B\mathbf{u}_{t-1}\pm B\phi_{t-1},Q\right)$ and $g_{t}^{A}\sim\chi^{2}(\lambda)$, with a noncentrality parameter of $\lambda=(-2B\mathbf{u}_{t-1}\pm B\phi_{t-1})^{\top}(-2B\mathbf{u}_{t-1}\pm B\phi_{t-1})$ and $n$ degrees of freedom. A positive value pertains to Eq. (13) whereas a negative value corresponds to Eq. 14. ∎

The proof of Theorem 1 and Theorem 2 is deferred to Appendices -B and -C, respectively. Theorems 1 and 2 show that, with certain detector assumptions, watermarking signals improve the detection of flip attacks on MTCs.

II-C2 Residuals Distribution under Injection Attacks

Injection attacks occur on the sensor measurements, i.e., $\mathbf{y}_{t}^{A}=\mathbf{y}_{t}+\mathbf{a}_{t},$ where $\mathbf{a}_{t}\in\mathbb{R}^{n}$ is the data injected by the attacker. Then, $\mathbf{y}_{t}^{A}$ feeds into the controller and the estimator in Eq. (2) that affects the system in Eq. (1). During the injection attack, $\mathbf{u}_{t}^{A}=f(\mathbf{y}_{t}+\mathbf{a}_{t};\eta)$, $\widehat{\mathbf{y}}_{t+1}^{A}=A\mathbf{y}_{t}^{A}+B\mathbf{u}_{t}^{A}$, and $\mathbf{y}_{t+1}^{A}=A\mathbf{y}_{t}^{A}+B\mathbf{u}_{t}^{A}+\mathbf{a}_{t+1}+\mathbf{w}_{t}$. Residuals during an injection attack are $\mathbf{r}_{t+1}^{A}=\mathbf{r}_{t+1}+\mathbf{a}_{t+1}$. Therefore, the residuals under attack differ from normal conditions, enabling the $\chi^{2}$ detector to detect injection attacks. The residuals under attack remain unchanged even if $\phi_{t}$ is added to the control signal. Therefore, the $\chi^{2}$ detector can still detect an injection attack without watermarking signals.

II-C3 Residuals Distribution under Replay Attacks

Replay attacks use captured legitimate measurement signals to trick the detector into seeing normal operations. The attacker model assumes these capabilities: (i) access to real-time sensor readings $\{\mathbf{y}_{t}\}_{t=t_{1}}^{t=t_{1}+T_{r}}$, and (ii) the ability to modify control actions $\mathbf{u}_{t}$ at any time. The replay attacker (i) records a sequence of sensor measurements $\{\mathbf{y}_{t}\}_{t=t_{1}}^{t=t_{1}+T_{r}}$, with $T_{r}$ ensuring extended replay capability, and (ii) alters control actions, $\mathbf{u}_{t}$, to a desired input from $\tau$ to $\tau+T_{r}$, where $\tau$ is the attack onset. During the replay attack,

[TABLE]

We denote the residuals during the replay attack by $\mathbf{r}_{t|\tau}$. The following lemma describes the distribution of the residuals under a replay attack.

Lemma 2**.**

With the watermark defined in Eq. (4), it holds that $\mathbf{r}_{t|\tau}\sim\mathcal{N}(\mathbf{m}_{t|\tau},\mathcal{S}_{t|\tau})$. At the onset of the attack ($t=\tau$), $\mathbf{m}_{\tau|\tau}=\mathbf{\mu}_{t_{1}}-\mathbf{\mu}_{\tau}$ and $\mathcal{S}_{\tau|\tau}=\mathcal{W}_{t_{1}}+\mathcal{Z}_{t_{1}}+\mathcal{Y}_{\tau}+\mathcal{Z}_{\tau}$. For $t>\tau$, $\mathbf{m}_{t|\tau}=0$ and $\mathcal{S}_{t|\tau}=Q+B(U_{t-\Delta t-1}+U_{t-1})B^{\top}$. ∎

The proof is deferred to the Appendix -D. Lemma 2 derives the distribution of test statistics under replay attacks, denoted as $g_{t|\tau}$.

Theorem 3**.**

With watermarking defined in Eq. (4), under a replay attack, the test statistics in Eq. (12) follow a generalized $\chi^{2}$ distribution, $g_{t|\tau}\sim\tilde{\chi}(\mathbf{\omega}_{t|\tau},\mathbf{\kappa},\mathbf{\lambda}_{t|\tau},s,m)$, where $s=0$, $m=0$, $\mathbf{\omega}_{t|\tau}=(\Lambda_{t|\tau}(1),\dots,\Lambda_{t|\tau}(n))^{\top}$, $\mathbf{\kappa}=\vec{\boldsymbol{1}}$, and $\mathbf{\lambda}_{t|\tau}=(\mathbf{b}_{t|\tau}(1)^{2},\dots,\mathbf{b}_{t|\tau}(n)^{2})^{\top}$. Additionally, $\mathcal{S}_{t|\tau}^{1/2}Q^{-1}\mathcal{S}_{t|\tau}^{1/2}=P_{t|\tau}^{\top}\Lambda_{t|\tau}P_{t|\tau}$ and $\mathbf{b}_{t|\tau}=P_{t|\tau}\mathcal{S}_{t|\tau}^{-1/2}\mathbf{m}_{t|\tau}$.∎

The proof is postponed to the Appendix -E. Theorem 3 shows that watermarking signals improve replay attack detection on MTCs. However, choosing the right watermarking level is vital to balancing control performance and detection accuracy. These insights will aid in characterizing the feedback from the detector and creating a dynamic and intelligent watermarking framework to address this trade-off.

III DynaMark Methodology

This section outlines the main components of the Markov Decision Process (MDP) model for the DWM problem in MTCs and the method used to solve it. The DynaMark architecture is denoted in Fig. 2. A typical MTC and its sensors form the physical environment. At each sampling interval, the agent observes the system state, including the latest measurement signal and the detector’s belief, and chooses the covariance of a zero-mean Gaussian watermark as an action. The watermark generator draws a signal from the distribution, integrates it with the control input, and transmits it to the machine tool. The $\chi^{2}$ detector assesses the gap between predicted and actual outputs, updates its belief regarding replay attacks, and provides an updated state and a reward balancing detection confidence, control performance, and energy consumption. An RL algorithm uses entities in the replay buffer to refine the policy, enabling the watermark to dynamically adapt to changing operational contexts.

III-A Markov Decision Process Formulation

Formally, an MDP problem [27, 28] is defined as a tuple $\mathcal{M}=(\mathcal{S},\mathcal{A},\mathcal{T},\mathcal{R},\rho_{0})$ where $\mathcal{S}$ denotes the state space, $\mathcal{A}$ is the action space, $\mathcal{T}:\mathcal{S}\times\mathcal{A}\to\mathcal{P}(\mathcal{S})$ represents the state transition function that is the probability distribution over states given being in state $s\in\mathcal{S}$ and taking action $a\in\mathcal{A}$; $\mathcal{R}:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\to\mathbb{R}$ is the reward function (or expected immediate reward) that is received after transitioning from state $s\in\mathcal{S}$ to state $s^{\prime}\in\mathcal{S}$ by taking action $a\in\mathcal{A}$; and $\rho_{0}:\mathcal{S}\to[0,1]$ is the starting state distribution. Next, we detail each component in the DynaMark context.

State Space: Let $s_{t}\in\mathcal{S}$ be the MDP state at time $t$. We define $s_{t}=(\mathbf{y}_{t},d_{t})$ where $\mathbf{y}_{t}$ is the measurement signal, and $d_{t}$ denotes the $\chi^{2}$ detector’s confidence at time $t$. The detection confidence, $d_{t}$, indicates the detector’s belief if the system is under attack, using measurement signal $\mathbf{y}_{t}$. In particular,$d_{t}$ indicates the detector’s confidence in an attack when an alarm is triggered. On the other hand, if no alarm sounds, $d_{t}$ illustrates the certainty that there is no attack. In MTCs’ closed architecture, sensor measurements are observable indicators of system operations, irrespective of the linearity assumption in Eq. (1). However, this assumption facilitates characterizing the belief. The belief is updated sequentially based on the results of the statistical tests. Define the random variables as follows: let $\sigma=1$ represent the occurrence of an actual ongoing replay attack, with a value of 0 otherwise; $I_{t}=1$ if the detector raises an alarm and 0 otherwise. Subsequently, $d_{t}:=\mathbb{P}(\sigma=1|I_{1:t})$.

Action Space: Using watermarking requires careful design of the covariance of the watermarking distribution $\mathcal{N}(0,U_{t})$ since the controller performance degrades due to the need for high accuracy in cyberattack detection. Specifically, if $U_{t}$ produces $\phi_{t}$ with high intensity, the control performance deteriorates, although this results in a marked improvement in the accuracy of detection. Conversely, if $U_{t}$ yields $\phi_{t}$ with low intensity, there is a reduction in detection accuracy, but the controller’s performance remains intact. We define the MDP action space $\mathcal{A}$ as the set of $n\times n$ PSD matrices. Observing $s_{t}\in\mathcal{S}$, the action at time $t$, denoted by $a_{t}\in\mathcal{A}$, is the covariance matrix of the watermark distribution ($U_{t}$).

State Transition Function: At time $t$, observing $s_{t}=(\mathbf{y}_{t},d_{t})$ and taking action $a_{t}=U_{t}$, a watermarking signal is sampled from $\mathbf{\phi}_{t}\sim\mathcal{N}(0,U_{t})$ and added to the control signal, resulting in $\mathbf{u}^{\prime}_{t}=\mathbf{u}_{t}+\mathbf{\phi}_{t}$. Then, the state $\mathbf{y}_{t}$ transitions according to the system dynamics to become $\mathbf{y}_{t+1}$. The detector updates detection belief from $d_{t}$ to $d_{t+1}$.

Theorem 4**.**

The detection confidence, $d_{t}$, evolves using Bayesian rule,

[TABLE]

∎

The proof of Theorem 4 is deferred to the Appendix -F. Eq.(16) updates the posterior attack probability based on the detector’s Type-I and Type-II errors. By the linear dynamic system described in Eq. 1, the state transition within the MDP is influenced by random Gaussian noise $\mathbf{w}_{t}$, and the state evolution occurs as follows:

[TABLE]

Lemma 3**.**

For the linear dynamic system described by Eq. 1, and its $\chi^{2}$ detector defined in Eq. (12), the Type-II error is

[TABLE]

where $\tau\in\mathbb{N}_{0}$ denotes the replay attack onset, $\mathcal{F}_{t|\tau=k}(\cdot)$ denotes the CDF of the generalized $\chi^{2}$ distribution, and $\mathcal{H}(\cdot)$ is the CDF of the $\chi^{2}$ distribution. ∎

The proof of Lemma 3 is deferred to Appendix -G. Eq.(17) calculates the $\chi^{2}$ detector’s Type-II error at time $t$ given threshold $\tilde{g}$ and onset prior $p(\tau|\sigma=1)$, which is then used in Eq. (20) to update the detector’s belief.

Remark 2**.**

DynaMark is not restricted to linear system dynamics. The linear-Gaussian model described in Eq. (1) is utilized to derive a closed-form expression for the residual distribution, which is subsequently used for characterizing the Type-II error, $\beta_{t}$, employed in the state and reward computation. If a system is nonlinear, one can obtain a real-time estimator of the test power (or Type-II error), such as through a data-driven surrogate or bootstrap calibration of a model-free statistic. In this case, the MDP formulation remains valid, and DynaMark can adjust its watermark covariance with the alternative estimator, instead of relying on $\chi^{2}$ detector.

Remark 3**.**

Computing $\beta_{t}$ at each time $t$ is essential. As the time horizon grows, calculating $\beta_{t}$ becomes more complex. Without a recursive relationship, a time-window of $w_{\beta}$ helps in computing the error, termed $\beta_{t}=\sum_{k=t-w_{\beta}}^{t}\mathcal{F}_{t|\tau=k}(\tilde{g})p(\tau=k|\sigma=1)+\mathcal{H}(\tilde{g})p(\tau>t|\sigma=1)$.

Reward Function: Recall the controller performance degrades due to the need for high accuracy in a replay attack detection that requires a careful choice of the watermarking covariance ($U_{t}$). The goal of adaptively deciding on $U_{t}$ is to establish a trade-off between detection accuracy and control performance while reacting to the changes in the environment. Let $r_{t}(s,a)$ denote the random reward of being in state $s$ and taking action $a$ at time $t$ that is defined as

[TABLE]

The reward function is designed to balance energy efficiency, control stability, and detection confidence, ensuring that watermarking effectively detects replay attacks without degrading system performance. The $l_{1}$-norm of $\mathbf{\phi}_{t}$ encourages sparse watermarking signals to conserve energy-overhead. The $l_{2}$-norm penalizes deviations from the un-watermarked system trajectory, maintaining control performance and stability. The term $|0.5-d_{t+1}|$ increases detection confidence by pushing $d_{t+1}$ toward either 0 or 1, avoiding uncertain detection outcomes. A confidence value close to 0.5 indicates high uncertainty, which weakens detection effectiveness.

III-B Policy Optimization Method

Let $\pi:\mathcal{S}\to\mathcal{P}(\mathcal{A})$ be a policy mapping each state $s$ to a probability distribution over the action space $\mathcal{A}$, including deterministic policies as a special case, where $\pi:\mathcal{S}\to\mathcal{A}$. Our goal is to establish a watermarking policy $\pi$ satisfying $a=\pi(s)\in\mathcal{A}$. Let $\Pi$ represent all feasible policies. Starting from initial state $s_{0}$, an optimal watermarking policy $\pi^{*}\in\Pi$ maximizes the expected total discounted reward with a discount factor $\gamma>0$, namely, \pi^{*}\in\arg\max_{\pi\in\Pi}\ \mathbb{E}\left[\sum\nolimits_{k=1}^{\infty}\gamma^{k-1}r_{k}(s,\pi(s))\big{|}\pi,s_{0}\right].

Policy Gradient algorithms directly learn a parameterized policy without computing a value function, making them effective for stochastic policies in noisy environments [29]. They optimize policy parameters $\theta$ to maximize the objective $J(\theta)=\mathbb{E}_{s,a}[R]$ via the gradient $\nabla_{\theta}J(\theta)=\mathbb{E}_{s,a}[\nabla_{\theta}\log\pi_{\theta}(a|s)Q^{\pi}(s,a)]$. Estimating the Q-functions $Q^{\pi}$ leads to actor-critic methods [30]. Deep Deterministic Policy Gradient (DDPG) extends PG to deterministic policies [31], combining deep neural networks to handle continuous action spaces [32]. It employs an actor-critic framework, where the critic estimates the Q-function and updates the policy via gradient descent. Thus, we use DDPG to derive the optimal DWM policy.

IV Evaluation Results

IV-A Numerical Experiment

We evaluated the DynaMark framework using a digital twin (DT) of the Siemens Sinumerik 828D MTC (see Fig. 3), which connects operators and machinery, handling tool changes and processes [33]. This DT replicates the MTC’s 2-axis motion control with high precision by tuning parameterized transfer functions based on real controller data. In our experiments, the motion was analyzed along the $y-$axis. The parameters defining the system dynamics in Eq. (1) and the replay attack are summarized in Appendix -H. Also, refer to Appendix -J for DDPG implementation details.

We evaluate DynaMark performance over 40 replications with $\alpha=0.005$ under normal conditions, to maintain an in-control average run-length (ARL0) of about 200. ARL generally measures the expected time to trigger an alarm, while ARL0 indicates the false-alarm rate. The attack scenario involves the attacker collecting full-system measurements, launching a replay attack $\tau=200$ by replacing true sensor data with recorded data $\mathbf{y}_{t}^{old}$, and manipulating the controller input $\mathbf{u}_{t}^{A}=-\mathbf{u}_{t}$.

Figures 4 and 5 present the evaluation results of DynaMark under normal and attack conditions, respectively. These results confirm DynaMark’s capacity to balance detection accuracy and control performance in MTCs. Using RL, DynaMark effectively modulates the strength of watermarking adaptively in response to system observations, thereby augmenting system security without necessitating system knowledge. This approach maintains system stability. DynaMark allows for the amplification of watermarking during periods of uncertainty, while reducing it when normalcy is observed within the system, thereby enabling efficient replay attack detection.

To benchmark DynaMark we also test two fixed-variance watermarks – “low” $(U_{t}=10^{-9})$ and “high” $(U_{t}=2.5{\times}10^{-3})$ – that bracket the RL policy’s operating range. Under normal conditions we compare energy consumption and control performance, while during a replay attack, we track detection speed via ARL1 (detection delay) that is computed as $\text{ARL}1=1/n\sum_{i=1}^{n}(T_{d,i}-\tau)$, where $T_{d}=\min\{t\geq\tau:I_{t}=1\}$. Lower $\text{ARL}_{1}$ indicates faster, and therefore better, detection.

Figure 6 illustrates the comparison results under normal and attack conditions.

1. (i)

DynaMark modulates watermark intensity just enough to stay well-below the high-variance scheme’s energy cost while avoiding the near-zero effort of the low-variance baseline; as a result, control performance remains essentially nominal, unlike the high-variance watermark that noticeably degrades motion accuracy. 2. (ii)

The RL policy raises alarm with ARL${}_{1}=1$ in all replications and maintains the detector belief at 1. The low-variance watermark is inconsistent, and no-watermark case never triggers. The high-variance watermark is fast but costly in energy/performance tradeoff. This shows DynaMark provides fast an reliable detection efficiently and stably.

To map the security-performance frontier, we swept nine fixed variances $U_{t}\in\{10^{-8},10^{-7},\dots,10^{-1},1.0\},$ averaging the true detection belief under attack ($1-d_{t}$, $t\geq\tau$) and control performance degradation under normal operation ($\|\mathbf{y}_{t+1}^{wom}-\mathbf{y}_{t+1}\|_{2}$). Fig.7 depicts the classic knee: detection improves rapidly up to $U_{t}\approx 10^{-4}$ with minimal performance loss, after which gains level-off and penalties increase sharply. DynaMark’s adaptive policy (star) aligns near this knee, balancing the trade-off and constantly adjusting $U_{t}$ to remain optimal under changing conditions.

IV-B Physical Testbed Implementation

This section analyzes DynaMark’s performance on a physical testbed, implemented using smart stepper-motors controlled by MTCs. Due to their crucial role in processes like drilling, milling, laser cutting, and 3D printing, MTC-controlled motors are prime targets for cyberattacks. Stepper-motors operate by transmitting electric pulses through coils, causing the shaft to rotate. When combined with a lead screw, linear motion results from the total pulses. Linear motion speed in the motor depends on constant current application.

IV-B1 Experiment Setup

The experimental platform integrates a $\rm NEMA17$ closed-loop stepper-motor with magnetic encoder, driven by an $\rm MKS$ $\rm Gen$ $\rm L$ $\rm V2.1$ control board and an $\rm MKS$ $\rm Servo$ $\rm 42A$ smart driver as illustrated in Fig 8. The $\rm MKS$ $\rm Servo$ $\rm 42A$’s firmware allows for servo-like control logic, which was then modified to accept $U_{t}$ updates via serial commands during motion using a command “watermark $U_{t}$”, while ensuring synchronization with the encoder’s closed-loop feedback. This is accomplished via serial communication with a PC running $\rm MATLAB$ at $115200$ baudrate with $\rm CR+LF$ termination. Additional firmware modifications enabled scripted cyberattack scenarios triggered via their respective commands. For every control tick, the driver emits a timestamped state vector

[TABLE]

where $\boldsymbol{\tau}_{t}$ are time steps, $\mathbf{y}_{t}$ are position measurements (converted from encoder counts to $\rm mm$), $\mathbf{u}_{t}$ is the control signal, $\boldsymbol{\phi}_{t}$ is the watermark signal, and $\rm attack_{flag}$ is a $\rm 1/0$ variable that flags the presence of an active attack.

DynaMark deployment for the motor involves three fundamental tasks: system identification, constructing a DT of the motor for RL training, and attack monitoring on the actual testbed. The system identification focuses on determining the parameters of the dynamic equation governing the motor’s operation as modeled by Eq. (1). We conceptualize the motor’s operation as a piecewise linear system evaluated at specific operating points. These operating points are defined by the speed and direction of the motor. Parameters $A$, $B$, $Q$, and $\overline{\mathbf{y}}$ are determined by fitting an ARX(1,1) model to data $\{\mathbf{y}_{t},\mathbf{u}_{t}\}$ at each operating point, then integrated to depict the system’s global non-linear behavior through localized linear models.

For parameter estimation, a watermark-free motion cycle is completed, collecting position measurements $\mathbf{y}_{t}$ and control signals $\mathbf{u}_{t}$ at the firmware’s native acquisition rate $\rm 1\ kHz$. The ARX(1,1) model parameters (see Table I) are estimated for four segments matching the operational commands. Figure 9a illustrates the continuous motion profile with four numbered segments. Red circles show segment indices (1-4), with vertical dashed lines marking segment boundaries.

During online decision-making or attack monitoring, trajectory updates are issued in discrete batches rather than as a fully continuous stream. This discretization leads to stepwise profiles in the recorded motion, even when the commanded trajectory is smooth. Occasional NaNs may appear in the streamed measurements during parsing or serial timing jitter, and these are removed during preprocessing. Minor communication and processing delays between the controller and the data acquisition pipeline can also introduce slight timing offsets, resulting in small variations when repeating experiments. Figure 9b illustrates the stepwise behavior from batchwise collection and processing. Nevertheless, the stepper-motor maintains accurate tracking across the entire motion profile under these normal operating conditions. For tested online batchwise collection settings, see Appendix -I.

To mitigate the cost of training the RL agent on the physical testbed, we construct the DT of the motor using these same stepwise trajectories. This ensures the DT accurately reflects the operational characteristics observed during online decision-making. For a detailed overview of the DT modeling workflow pertaining to the stepper motor, refer to Appendix -K, and for an exposition of the DDPG implementation, see Appendix -J.

The last task is performing attack monitoring on the actual testbed. While $\rm MATLAB$ serves as the primary environment for executing motion commands and operating the motor in real time, Python modules asynchronously update beliefs and select RL-based watermarks for dynamic watermark variance computation $U_{t}$. The RL policy is computationally intensive, so it is exported to the Open Neural Network Exchange ($\rm ONNX$) format, enabling Python-trained models (e.g., $\rm PyTorch$) to run efficiently in the $\rm MATLAB$-controlled environment. Leveraging $\rm ONNX$ Runtime optimizes RL policy inference, separating heavy computation from firmware and allowing flexible platform deployment. The overall online decision-making pipeline is illustrated in Algorithm 1 with detailed timing and multi-rate coordination, provided in Appendix -L.

IV-B2 Performance Evaluation

We first evaluate the DynaMark’s effectiveness in the stepper-motor DT environment under normal and attack conditions. This analysis serves as a controlled benchmark, similar to the numerical study for the MTC in Section IV-A, incorporating the motor’s piece-wise dynamics and processing latency. We considered a replay attack scenario on the stepper-motor starting at fast-time index $4000$, replaying measurements and flipping the control signal (see Appendix -K for DT time scales). Figure 10 shows the results, where the attack begins at decision epoch 8 in Fig. 10c. The event recurs at observation index $\approx 800$, where the trajectory in Fig. 10a diverges.

The stepper-motor DT verifies DynaMark’s dynamic adaptability (Fig. 10): during a replay attack, the physical path shifts sharply while spoofed data stays nominal. The detector’s belief surges to 1 in roughly two decision epochs with a six-sample detection delay. Post-alarm, DynaMark adjusts $U_{t}$ to maintain a strong residual gap. Results demonstrate DynaMark’s ability to (i) differentiate real from replayed dynamics, (ii) reduce detection delay, and (iii) adjust watermark intensity to maintain detection power.

Next, we validate the efficacy of DynaMark on the physical implementation. A similar attack scenario is considered wherein the adversary replays the measurements, while flipping the control signal. To ensure stable performance under normal conditions, we tuned the $\chi^{2}$ threshold to an empirical value of $16$ by analyzing residual distributions, addressing stochastic fluctuations and sensor noise, to meet the desired $\mathrm{ARL}_{0}\approx 1/\alpha$ with $\alpha=0.005$. As shown in Fig. 11, the trained RL policy successfully transfers to the physical implementation, exhibiting behavior consistent with the DT and validating the proposed DynaMark framework.

Under the replay attack beginning at decision epoch $7$ (iteration $\sim 600$), the physical trajectory in Fig. 11a diverges from the spoofed measurements, while the state estimate $\widehat{\mathbf{y}}_{t}$ follows the replayed input. This triggers the detector belief $d_{t}$ in Fig. 11b, which rises to $1$ within five samples. Figure 11c shows DynaMark adapting $U_{t}$: under nominal motion, it peaks near $\rm 0.009\ mV^{2}$ during acceleration and tapers on descent; post-alarm, it is reduced to conserve energy without affecting detection. These results confirm DynaMark’s real-time online applicability on the physical implementation.

IV-B3 Comparative Analysis

Section I-A discusses optimization-based watermarking paradigms, which offer a closed-form expression for watermarking covariance. To assess these paradigms on non-LTI plants, we conducted a controlled study on the stepper motor’s DT, characterized by linear time-variant dynamics. We approximated the dynamics (see Fig. 9a) as LTI with parameters $A=1.0$, $B=0.00372$, and $Q=3.1911\times 10^{-5}$. Using these parameters, we solved the optimization problem in [11] for five control performance loss budgets ($\delta$): 100%, 50%, 20%, 10%, and 5% of the optimal LQG cost with no watermark ($J_{0}$) considering the LQG cost weight as $X=1.0$. Table II lists the optimal variance for each budget. Each baseline was integrated, and its performance evaluated under normal and the same attack conditions as in Section IV-B2. Figure 12 shows that constant-variance baselines are unsuitable for non-LTI plants due to time-varying sensitivity to watermark excitation.

1. (i)

Constant-variance baselines show a monotone decrease in median energy and control performance degradation as the LQG-loss budget tightens. However, their inter-quartile range is significantly larger than DynaMark’s. DynaMark’s adaptive policy focuses energy near the lower tail of all baselines, achieving the lowest median control performance degradation with a notably tighter spread. This shows that a single variance cannot ensure low energy and adequate excitation when the true plant deviates from the LTI assumption. 2. (ii)

The inter-alarm interval measures the processed observations between alarms during a replay attack. Shorter intervals indicate higher detector sensitivity. DynaMark inter-alarm samples range from 1 to 4 (approximately, median of $3$ and mean of $3.2$), ensuring high confidence. Conversely, all five constant-variance baselines show slower, more variable alarm responses, despite higher energy use and control performance decline. A tighter LQG budget leads to longer silent intervals between alarms. This further shows that constant-variances fall short on a time-varying plant: they are either too aggressive in normal operation or too weak during an attack.

V Conclusion

This paper introduces DynaMark, an adaptive framework to generate dynamic physical watermarks for replay-attack detection in MTCs. A rigorous residual analysis showed that the Gaussian watermark maintains the detector’s $\chi^{2}$ structure during normal operations. It results in a tractable generalized $\chi^{2}$ distribution with replay attacks and forms a non-central $\chi^{2}$ distribution when subjected to flip attacks under specified conditions. These results support a reward that balances detection power, control performance, and energy consumption. Unlike current methods that depend on stationary LTI and Gaussian assumptions with full or partial system knowledge, DynaMark (i) uses an MDP for watermark design, (ii) directly learns an adaptive covariance policy from environmental interactions, bypassing the need for LTI conditions or detailed system knowledge, and (iii) adjusts watermark intensity based on changing dynamics and threats.

Experiments confirm DynaMark’s advantages. On a DT of a Siemens Sinumerik 828D, DynaMark maintained control performance with $70\%$ less energy than a high-variance baseline. A physical stepper-motor testbed further validated the policy; the detector’s confidence jumped to 1 almost instantly after a replay attack onset, while DynaMark dynamically adapts $U_{t}$ when confidence saturated. Compared to optimization-based baselines relying on the LTI assumption, these methods couldn’t match DynaMark’s low energy consumption, control performance degradation, and alarm precision, revealing their inadequacy for non-LTI systems.

This study focused on zero-mean independent Gaussian watermark signals. Exploring state- and frequency-shaped distributions can lower detectability by advanced adversaries and reduce excitation energy for effective monitoring. While evidence shows DynaMark stays stable, adding safe-RL constraints like control-barrier-function penalties would ensure all watermark adaptations are within actuator limits and maintain closed-loop performance during learning. Finally, an important practical extension is to integrate DynaMark with an online watermark-recovery module. Upon detecting an attack and halting production, the same RL framework can be used to generate an energy-efficient authentic control signal that safely returns the MTC to a certified state, reducing downtime and enabling the autonomous restart of normal operations.

Acknowledgments

This work was supported by the U.S. National Science Foundation under Grant 2501479.

-A Proof of Lemma 1

Since the watermark signals are known, under normal conditions, the conditional distribution of $\mathbf{y}_{t}$ follows a Gaussian distribution with mean $\mu_{t}+\theta_{t}$ and covariance $\mathcal{W}_{t}$. In addition,

[TABLE]

that implies $\widehat{\mathbf{y}}_{t}$ follows a Gaussian distribution with covariance $\mathcal{Y}_{t}$ and mean

[TABLE]

Accordingly, $\mathbf{r}_{t}\sim\mathcal{N}(B\phi_{t-1},Q)$ since $\mathbf{Cov}(\mathbf{y}_{t},\widehat{\mathbf{y}}_{t})=\mathcal{Y}_{t}$. Therefore, $g_{t}$ follows a noncentral $\chi^{2}$ distribution with parameter $\lambda=\phi_{t-1}^{\top}B^{\top}B\phi_{t-1}$ and $n$ degrees of freedom. ∎

-B Proof of Theorem 1

Under the second scenario, when the system is under flip attack, $\mathbf{u}_{t}^{A}=-\mathbf{u}_{t}-\phi_{t}$, and, by assumption,

[TABLE]

Then, the residuals under attack are

[TABLE]

Then, $\mathbf{r}_{t}^{A}\sim\mathcal{N}(-2B\phi_{t-1},Q)$ that implies $g_{t}^{A}$ follows a noncentral $\chi^{2}$ distribution with noncentrality parameter $\lambda=4\phi_{t-1}^{\top}B^{\top}B\phi_{t-1}$ and $n$ degrees of freedom. Consequently, injecting watermarking signals enhances the detection capabilities of the $\chi^{2}$ detector. ∎

-C Proof of Theorem 2

When the system is under flip attack (both scenarios),

[TABLE]

Then, the residuals

• •

under first flip attack scenario are,

[TABLE]

• •

under second flip attack scenario are,

[TABLE]

Under the first flip attack scenario

[TABLE]

and $g_{t}^{A}\sim\chi^{2}(\lambda)$ with noncentrality parameter

[TABLE]

and $n$ degrees of freedom.

Similarly, under the second flip attack scenario

[TABLE]

and $g_{t}^{A}\sim\chi^{2}(\lambda)$ with noncentrality parameter

[TABLE]

and $n$ degrees of freedom. Consequently, injecting watermarking signals enhances the detection capabilities of the noncentral $\chi^{2}$ detector introduced in Lemma 1. ∎

-D Proof of Lemma 2

On the onset of a replay attack, when $t=\tau$, the measurements signal is $\mathbf{y}_{\tau}^{A}=\mathbf{y}_{t_{1}}$, and the state estimation will not be affected by the attack as it is estimated by the previous measurement signal, i.e., $\widehat{\mathbf{y}}_{\tau}^{A}=\widehat{\mathbf{y}}_{\tau}$. Then, the residuals under attack follow

[TABLE]

Since $T_{r}$ is large enough to guarantee that the attacker can replay the sequence for an extended period of time during the attack, $\mathbf{Cov}(\mathbf{y}_{t_{1}},\widehat{\mathbf{y}}_{\tau})\to 0$. Therefore, by the distribution of $\mathbf{y}_{t_{1}}$ and $\widehat{\mathbf{y}}_{\tau}$, $\mathbf{r}_{\tau|\tau}$ follows a Gaussian distribution with mean $\mathbf{m}_{\tau|\tau}=\mu_{t1}-\mu_{\tau}$ and covariance $\mathcal{S}_{\tau|\tau}=\mathcal{W}_{t_{1}}+\mathcal{Z}_{t1}+\mathcal{Y}_{\tau}+\mathcal{Z}_{\tau}$.

After the onset of the attack ($t>\tau$), the state estimation follows $\widehat{\mathbf{y}}_{t}^{A}=A\mathbf{y}_{t-1}^{A}+B\mathbf{u}_{t-1}^{A}+B\phi_{t-1}$ that implies

[TABLE]

Since $\mathbf{u}_{t-1}^{A}=f(\mathbf{y}_{t-1}^{A};\eta)$ and $\mathbf{y}_{t-1}^{A}=\mathbf{y}_{t-\Delta t-1}$, it is assumed that $\mathbf{u}_{t-1}^{A}=\mathbf{u}_{t-\Delta t-1}$. Therefore, it holds that $\mathbb{E}(\widehat{\mathbf{y}}_{t}^{A})=\mu_{t-\Delta t}$. Furthermore, we have that

[TABLE]

The covariance between $\widehat{\mathbf{y}}_{t}^{A}$ and $\mathbf{y}_{t}^{A}$ follows that

[TABLE]

Then, the residuals under attack follow

[TABLE]

Therefore, by the distribution of $\mathbf{y}_{t-\Delta t}$ and $\widehat{\mathbf{y}}_{t}^{A}$, $\mathbf{r}_{t|\tau}$ follows a Gaussian distribution with mean $\mathbf{m}_{t|\tau}$ and covariance $\mathcal{S}_{t|\tau}$, where

[TABLE]

and

[TABLE]

∎

-E Proof of Theorem 3

Under a replay attack, $\mathbf{r}_{t|\tau}\sim\mathcal{N}(\mathbf{m}_{t|\tau},\mathcal{S}_{t|\tau})$. Using standard normal random variables, we can rewrite $\mathbf{r}_{t|\tau}=\mathbf{m}_{t|\tau}+\mathcal{S}_{t|\tau}^{1/2}\mathbf{Z}$ where $\mathbf{Z}\sim\mathcal{N}(0,I)$. Then, the test statistics under attack follow

[TABLE]

Apply the spectral theorem and write

[TABLE]

where $P_{t|\tau}$ is an orthogonal matrix, i.e., $P_{t|\tau}^{\top}P_{t|\tau}=P_{t|\tau}P_{t|\tau}^{\top}=I$; and, $\Lambda_{t|\tau}$ is a diagonal matrix with positive diagonal elements. Let $\mathbf{Y}_{t|\tau}=P_{t|\tau}\mathbf{Z}$ and $\mathbf{b}_{t|\tau}=P_{t|\tau}\mathcal{S}_{t|\tau}^{-1/2}\mathbf{m}_{t|\tau}$. Since $\mathbf{Z}$ is a multivariate standard normal distribution, $\mathbf{Y}_{t|\tau}$ also follows the same distribution because $P_{t|\tau}P_{t|\tau}^{\top}=I$. Then,

[TABLE]

that implies, test statistics follow the generalized $\chi^{2}$ distribution (weighted sum of noncentral chi-square variables)

[TABLE]

where $s=0$, $m=0$, and

[TABLE]

∎

-F Proof of Theorem 4

By the definition of the random variables $\sigma$, $\tau$, and $I_{t}$, the detector’s belief at time $t$ is defined as the probability of the existence of a replay attack given the sequence of the outcome of the detector $\{I_{t}\}$, i.e., $d_{t}=\mathbb{P}(\sigma=1|I_{1:t})$.

To compute this probability, notice that observing $\mathbf{y}_{t}$ and the outcome of the detector, $I_{t}$, and using the Bayesian rule,

[TABLE]

When a replay attack is present in the system, each outcome is independent of previous outcomes. Consequently, the sequence of outcomes, $\{I_{t}\}$, are conditionally independent given the existence of a replay attack that implies the conditional likelihood is computed as $p(I_{t}|I_{1:t-1},\sigma)=p(I_{t}|\sigma)$. Then, we have the following cases at time $t$:

• •

If there does not exists a replay attack in the system, i.e., $\sigma=0$, with probability one, the onset of the replay attack is in the future, i.e., $\mathbb{P}(\tau>t|\sigma=0)=1$. Therefore, the probability of raising an alarm or not depend only on the Type-I error, i.e.,

[TABLE]

• •

If there does exists a replay attack in the system, i.e., $\sigma=1$, depending the onset of the replay attack, probability of raising an alarm or not depends on both Type-I and Type-II error. In other words, if the replay attack has not begun, an alarm is raised with probability $\alpha$, and if it has started an alarm is raised with probability $1-\beta_{t}$, i.e.,

[TABLE]

Then, by conditioning on the onset of the replay attack, we have that

[TABLE]

In addition, conditioning on the existence of a replay attack and by the conditional independence of $\{I_{t}\}$, the conditional evidence is computed as

[TABLE]

Therefore, for $j\in\{0,1\}$, the detector’s belief at time $t$ can be computed as follows

[TABLE]

Assuming a prior distribution on the random variable $\sigma$, which by its definition is a Bernoulli distribution with some parameter $q$, the detector’s belief updates at each time using the relation in Eq. (20). ∎

-G Proof of Lemma 3

The Type-I error for detection is defined as the probability that the detector raises a false alarm when there is no replay attack in the system, i.e., $\alpha=\mathbb{P}(I_{t}=1|\sigma=0)$. Conversely, the Type-II error represents the probability of not raising an alarm when a replay attack is actually present, i.e., $\beta_{t}=\mathbb{P}(I_{t}=0|\sigma=1)$. Note that the Type-I error is time-invariant and can be controlled by carefully selecting the test statistic threshold, $\tilde{g}$. However, an operator could choose to vary $\tilde{g}$ over time, which would make the Type-I error time-dependent. For this study, we assume the threshold remains fixed. The Type-II error, on the other hand, can vary depending on the occurrence of a replay attack. At time $t$, if a replay attack has begun, then by Theorem 3, the probability of not raising an alarm depends on the generalized $\chi^{2}$ distribution of the test statistic under attack. If the attack has not begun, the probability of not raising an alarm is based on the $\chi^{2}$ distribution of the test statistic under normal conditions. This relationship is given by

[TABLE]

where $\mathcal{F}_{t|\tau=k}(\tilde{g})$ is the CDF of the generalized $\chi^{2}$ distribution with $s=0$, $m=0$, $\kappa=\vec{\boldsymbol{1}}$, $\omega_{t|\tau=k}=\left(\Lambda_{t|\tau=k}(1),\dots,\Lambda_{t|\tau=k}(n)\right)^{\top}$, $\lambda_{t|\tau=k}=\left(\mathbf{b}_{t|\tau=k}(1)^{2},\dots,\mathbf{b}_{t|\tau=k}(n)^{2}\right)^{\top}$, and $\mathcal{H}(\tilde{g})$ is the CDF of the $\chi^{2}$ distribution. ∎

-H Setup for numerical study DT (Section IV-A)

Table III lists parameters held constant throughout the numerical study. (Training–related hyperparameters appear separately in Appendix -J.)

-I Online Batchwise Collection Settings

Several configurations were tested to determine the optimal online collection settings, including the discrete batch size, the use of short pauses between transmissions, and whether to flush the serial buffer. We experimented with pause() durations of $10$, $20$, and $\rm 50\ ms$. A batch size of $100$ points was ultimately selected, as it minimized the occurrence of NaNs in the streamed data. A $\rm 10\ ms$ pause proved sufficient for stable operation, and explicit buffer flushing via flush() was necessary to ensure that remaining data from the previous iteration did not propagate into the current batch. The results are summarized in Table IV. The final selected configuration is highlighted in blue.

-J DDPG learning summary

To deploy the DynaMark framework, we implemented a DDPG agent using a PyTorch-based actor-critic setup. While DDPG is widely used among RL algorithms, it is known for its instability, often exhibiting sensitivity to hyperparameters and a tendency to converge to suboptimal solutions or diverge entirely [34]. To address these challenges, our implementation includes two critic networks to improve stability and reduce overestimation bias, with each network independently estimating the Q-value for the current state-action pair, $Q^{1}(s_{t},a_{t};\psi^{1})$ and $Q^{2}(s_{t},a_{t};\psi^{2})$, where $\psi^{1}$ and $\psi^{2}$ represent their respective parameters and $a_{t}=\mu(s_{t};\theta)$. The target Q-value used for training is computed as

[TABLE]

Target networks for the actor and critics were updated using a soft-update mechanism, $\theta_{\text{target}}\leftarrow\tau\theta+(1-\tau)\theta_{\text{target}}$ and $\psi_{\text{target}}^{i}\leftarrow\tau\psi^{i}+(1-\tau)\psi_{\text{target}}^{i}$. A replay buffer stored transitions $(s,a,r,s^{\prime},\text{done})$. To encourage exploration, an Ornstein-Uhlenbeck (OU) process [35] was used to introduce temporally correlated noise into the action space during training. Table V lists the DDPG hyperparameters for both numerical study (Section IV-A) and physical testbed (Section IV-B).

-K Digital-Twin Modeling Workflow for the Stepper-Motor Testbed

We construct the DT of the motor on the stepwise trajectory through a series of five conceptual steps:

1. (i)

segment the previously determined motion profile into the identical four sequential operational blocks utilized in the testbed, and allocate the linear parameters $\{A_{i},B_{i},Q_{i},\overline{\mathbf{y}}_{i}\}$ derived during the system identification phase to each segment; 2. (ii)

for each block, extract the respective control signal trace $\{\mathbf{u}_{t}\}$ and fit a Gaussian Mixture Model (GMM) to this one-dimensional signal. This results in a non-parametric surrogate $p_{i}(\mathbf{u})$, which replicates the empirical distribution of control actions without the necessity to explicitly model the closed-loop firmware; 3. (iii)

during simulation, iterate through fast time steps, at each step sampling $\mathbf{u}_{t}\sim p_{i}(\mathbf{u})$, adding the RL‑chosen watermark $\phi_{t}\sim\mathcal{N}(0,U_{t})$ to obtain the commanded input $\mathbf{u}^{\prime}_{t}=\mathbf{u}_{t}+\phi_{t}$, propagating the state $\mathbf{y}_{t+1}=A_{t}\mathbf{y}_{t}+B_{t}\mathbf{u}^{\prime}_{t}+\mathbf{w}_{t}$ with process noise $\mathbf{w}_{t}\sim\mathcal{N}(0,Q_{i})$, and switching to the next block when the simulated output first crosses its corresponding set‑point $\overline{\mathbf{y}}_{i}$; 4. (iv)

to mirror the real data‑collection routine, let the plant evolve for 500 fast samples under a held watermark variance, forward only the first 100 samples to the $\chi^{2}$ detector and updating the belief, accumulate the per‑sample reward over this 100‑point window, and return the last processed output together with the updated attack probability as the observation for the next RL decision; 5. (v)

if the fast‑time index is inside the chosen attack window, override the plant input by an adversarial increment $\mathbf{u}_{t}^{A}$ (a “flip attack” $\mathbf{u}_{t}^{A}=-\mathbf{u}_{t}$ in our experiments) while feeding the detector a stale measurement $\mathbf{y}_{t}^{A}=\mathbf{y}_{t-\Delta t}$ captured from a normal operation.

The GMM-driven, block-aware simulator is therefore capable of capturing both the stochastic control behavior and the batching latency of the physical motor, thereby offering a high-fidelity digital environment in which the RL agent can undergo training prior to deployment. We hold the same assumption on the prior belief in a replay attack as articulated in the numerical study, with $T=4.1\times 10^{4}$ delineating the maximum possible duration of the simulation. This upper boundary is established due to the stochastic nature of pulse timing inherent in the GMM-based controller surrogate, leading to the simulated trajectory concluding within the interval $[3.95\times 10^{4},4.1\times 10^{4})$. This setting ensures the completion of each episode while preventing unnecessary roll-out that extends beyond the physical motion profile. Additionally, the time-window for computing $\beta_{t}$ has been set to $w_{\beta}=500$ fast-time steps. Moreover, we assume $\Delta t=0$, which grants the attacker comprehensive access and the capability to replay all operational measurements at their respective original timestamps.

The DT uses two nested temporal resolutions: Fast time for the stepper-motor’s physical sampling rate with $T_{\text{fast}}\leq 4.1\times 10^{4}$ samples per episode, and plant blocks of length $L_{\text{plant}}=500$. At each decision time block boundary, DynaMark chooses a watermark covariance $U_{t}$, held fixed for $500$ fast-time steps, resulting in an $U_{t}$ trace with $T_{\text{fast}}/L_{\text{plant}}\approx 82$ points. Only the first $L_{\text{proc}}=100$ samples in each block are sent to the $\chi^{2}$ detector, whose belief $d_{t}$ is updated at one-fifth the rate of the underlying dynamics. Processed windows yield $T_{\text{fast}}/L_{\text{plant}}\times L_{\text{proc}}\leq 8200$ detection samples. This design separates high-fidelity simulation from intensive inference and learning, enabling the RL policy to respond in line with the motor’s linear segments.

-L Multi-rate Online Decision-making by DynaMark

The $\rm ONNX$-based online pipeline operates across three coordinated strobes, each mapped to its own timescale $\Delta T_{i}$ and hold duration $\tau_{i}$. The $\Delta T_{i}$ parameter defines the interval between consecutive executions of that strobe, while the hold duration specifies how long the most recently computed output remains active until it is updated in the next cycle.

Strobe 1 ($T_{1}$): High-frequency serial reads from the controller buffer at $\rm 1000~Hz$ capture the state vector $\mathbf{y}_{t}$ as in Eq. (19). Every $\Delta T_{1}$ interval, the latest $100$-sample batch is retrieved and retained for downstream processing. A buffer flush follows to remove stale data. The output of Strobe 1 is held constant for $\tau_{1}$ until the next read cycle.

Strobe 2 ($T_{2}$): On a slower timescale $\Delta T_{2}$, the retained sample batch is processed by the $\chi^{2}$ detector and Python-based belief update module to estimate the current system state and attack likelihood $d_{t}$. The updated belief is held for $\tau_{2}$ until the next invocation.

Strobe 3 ($T_{3}$): Operating at the slowest rate $\Delta T_{3}$, the $\rm ONNX$-based watermark generator uses the latest belief $d_{t}$ to compute the watermark covariance $U_{t}$. The watermark $\phi_{t}\sim\mathcal{N}(0,U_{t})$ is superimposed on the nominal control $\mathbf{u}_{t}$ to produce the watermarked signal $\mathbf{u}^{\prime}_{t}=\mathbf{u}_{t}+\phi_{t}$. This watermarked control is held for $\tau_{3}$ until the next update, ensuring consistent actuation between refresh cycles.

By co-designing $\Delta T_{i}$ and $\tau_{i}$ for all three strobes, along with batch size, buffer management, and telemetry bandwidth, the system achieves deterministic timing in this multi-rate, cross-platform compute pipeline. This design enables real-time watermark adaptation without interrupting motor motion.