Robust Active Disturbance Rejection Fractional-Order Control for Regenerative Chatter Suppression in Milling

Abstract

End milling productivity is reduced by regenerative chatter. In this paper, a hybrid Fractional-Order PID with Active Disturbance Rejection Control (ADRC-FOPID) is proposed to suppress chatter in half-immersion milling. A Timoshenko cantilever flexible workpiece is modeled together with a delay-dependent regenerative cutting-force model. The lumped disturbance is canceled on-line by an Extended State Observer, and the five FOPID gains are tuned off-line using Particle Swarm Optimization with a ±27 N actuator-saturation constraint. The RMS tip displacement is reduced by 68.5% by the ADRC-FOPID controller. Moreover, it increases the minimum and maximum stable depth of cut from 1.00 mm to 2.67 mm and from 23.17 mm to 37.67 mm, respectively. A robustness analysis over plant uncertainties and the operating window, with 38 points, results in a low mean RMS of 4.2 µm. Compared with classical controllers and robust controllers such as PID, LQR, H_∞, and μ-synthesis, ADRC-FOPID achieves the highest critical limiting depth (7.58 mm) and peak stable depth (49.52 mm) on the same benchmark. Thus, the proposed strategy is an effective, robust candidate strategy for chatter suppression in milling.

1. Introduction

End milling is widely used for producing complex geometries with high-dimensional accuracy, but its productivity is limited by regenerative chatter—self-excited vibration in which the cutting force from one pass modulates the chip thickness of the next, feeding back into the structural dynamics. Chatter degrades surface finish, accelerates tool wear, and can force the cut to be aborted. The phenomenon is described by delay-differential equations, and the resulting stability lobe diagrams [1,2] guide chatter-free parameter selection but only outline passive stability—pushing beyond it requires active intervention. Early passive add-ons (tuned and impact dampers [3,4]) work only for the cutting conditions to which they were tuned. Active strategies sense vibration and apply real-time counterforces; direct velocity feedback, virtual passive absorbers, and LQR-based schemes have expanded the chatter-free window in turning and milling [5,6,7], and Ozsoy et al. [8] published a benchmark for robotically assisted milling that is now a standard reference for comparing active controllers. Data-driven methods using neural networks [9,10] predict chatter onset but are diagnostic rather than suppressive. Two complementary control paradigms have emerged. Active Disturbance Rejection Control (ADRC) [11,12,13], introduced by Han, dispenses with explicit plant modeling and uses an Extended State Observer (ESO) to estimate and cancel the lumped effect of the regenerative force, parameter mismatch, and external noise as a single time-varying disturbance, validated, for example, on delayed piezoelectric plates [14]. Fractional-order PID (FOPID) extends classical PID with non-integer integration and differentiation orders λ and μ, formalized by Podlubny [15] and effective in industrial and machining applications [16,17,18,19]. Each has limits: standalone FOPID is sensitive to unmodeled disturbances, while standalone ADRC may lack fine frequency-domain selectivity in narrow chatter bands. Alternative robust strategies (H_∞ and μ-synthesis [20,21]) guarantee robustness margins but at the cost of nominal-performance conservatism, a trade-off this paper directly addresses.

Several gaps remain. Most active-control work treats the regenerative cutting force as an external disturbance to be canceled offline rather than an internal force estimable online. Fractional-order controllers are rarely combined with disturbance estimation in milling. Hard actuator-saturation limits are seldom enforced during controller tuning, and when they are, the resulting controllers are seldom subjected to systematic robustness analysis under combined plant and process perturbations. This paper proposes a hybrid ADRC-FOPID scheme in which all uncertainties are absorbed into a single ESO-estimated disturbance, the fractional orders λ and μ shape the nominal response, and all five FOPID gains are tuned offline by PSO [22] under a hard ±27 N actuator-saturation constraint. Numerical validation shows a 68.5% RMS tip-displacement reduction at the nominal operating point; dominance over classical, H_∞, and μ-synthesis controllers on every aggregate statistic of a 38-point robustness sample with frozen gains; and the best stability boundaries on the Ozsoy et al. [8] benchmark.

The remainder of the paper is organized as follows. Section 2 reviews metaheuristic optimization and chatter-suppression methods. Section 3 develops the physical and mathematical model. Section 4 details the PSO-based controller design under saturation. Section 5 reports the numerical validation (time and frequency domain, robustness, and comparison against six published active controllers on the Ozsoy et al. [8] benchmark). Section 6 discusses the findings; Section 7 concludes.

2. Related Work

2.1. Metaheuristics: Overview, Benefits, Shortcomings, and Uses

Metaheuristics are stochastic, derivative-free global-optimization algorithms well-suited to multimodal, non-convex problems where gradient information is unavailable or unreliable [11,12,19]. They are typically classified as population-based (PSO, GA, and DE) or trajectory-based (SA and Tabu Search) and trade off exploration with exploitation through hyperparameters that must be tuned per problem [15,22,23,24,25].

Hybrid strategies combining metaheuristics with local search, surrogate models, or adaptive schemes mitigate these limitations [14,26]. Specific FOPID-tuning examples include the fractional PID structure introduced by Podlubny [15], chaotic atom search optimization [23], and the PSO variants surveyed in [22].

Beyond the linearized regenerative mechanism, milling chatter exhibits genuinely nonlinear behavior—sub-critical Hopf bifurcations, period-doubling instabilities in low-immersion regimes, and intermittent tool-workpiece separation—that the classical zero-order theory does not capture [27,28]. Recent advances in nonlinear-vibration fields provide complementary insight into the bifurcation-prone behavior of resonant flexible systems and the limits of linear control surrogates. In particular, compact low-frequency piezoelectric energy-harvesting systems based on cucurbit/flute-inspired beams with intelligent adjustable structures [29] exhibit hardening/softening nonlinearities and self-tuning dynamics; bio-inspired multi-stable piezoelectric vibration systems with adjustable devices [30] demonstrate sub-/super-harmonic transitions governed by impulse-perturbation basins of attraction; ultra-low-frequency piezoelectric vibration systems with adjustable gear-spring devices [31] reveal how geometric reconfiguration shifts the nonlinear potential well and modifies the amplitude-frequency response; and base-isolated impacting vibration absorbers with resonators [32] show that the inclusion of a resonator together with vibro-impact dynamics enhances seismic resilience while introducing piecewise-smooth, non-smooth nonlinearities. Collectively, these works confirm that controllers designed on a linearized model can lose performance when the underlying system traverses bifurcations, sub-critical instabilities, or non-smooth events. In the present work, the linearized DDE (Equation (3)) is justified by the operating window of interest (continuous half-immersion cut; no fly-over), but the implications of this assumption are revisited in Section 6.

Further FOPID/machining applications include adaptive PSO-reinforcement-learning tuning [33,34], fully-discretized chatter stability prediction [35], and jellyfish search optimization [36].

Particle Swarm Optimization (PSO) is a continuous-parameter metaheuristic [37] inspired by collective swarm behavior: each particle updates its velocity and position from its personal best and the swarm best, governed by

$$v_i^{k+1}=w v_i^k+c_1 r_1\left(p_i^{best}-x_i^k\right)+c_2 r_2\left(g^{best}-x_i^k\right)\hspace{1em}\hspace{1em}{x}_i^{k+1}=x_i^k+v_i^{k+1}$$

(1)

Here, v_i,k and x_i,k are the velocity and position of particle i at iteration k; w is the inertia weight; c₁ and c₂ are acceleration coefficients; r₁, r₂ are uniform random numbers in [0, 1]; and p_i,best and g_best are the current best positions. PSO is easy to implement with few parameters, adapts well to low- to medium-dimensional problems, and accommodates constrained optimization via penalty functions [38]. Its main drawback—premature convergence and sensitivity to w, c₁, and c₂—is manageable in the five-parameter FOPID tuning of this work.

2.2. Chatter Suppression in Milling: A State-of-the-Art Review

2.2.1. Mechanical and Passive Methods

The first attempts of chatter mitigation were passive ones that were made in the machine–tool–workpiece system. Vibration dampers (tuned mass dampers), impact dampers, and structural reinforcement have been proven to reduce the amplitude of the vibrations and to improve the quality of the surface. Chaari et al. [3] conducted an experiment on the spherical impact dampers and demonstrated its usefulness in vibration reduction of the flexible mechanical systems. Later, this group reported that passive vibration absorber can be used to improve surface finish of flexible workpieces while machining [4]. Sun et al. [39] recently reviewed passive, semi-passive, and active methods for chatter suppression in thin wall milling and concluded that passive chatter suppression is still a desirable option despite the fact that a fixed passive tuning is the primary drawback for industrial applications. Passive methods are simple and economical but of course are limited to static operating conditions and cannot be adopted to variations of cutting parameters and workpiece geometry.

2.2.2. Active Hardware and Actuator-Based Techniques

Active vibration-control schemes sense and apply real-time counterforces through electromagnetic, piezoelectric, or magnetorheological actuators. Ma et al. [5] used dynamic compensation to expand stability margins in thin-plate turning; Ozsoy et al. [40] highlighted the need for efficient multi-mode control in robotic milling of flexible structures; Wang et al. [41] proposed a 2-DOF magnetic-bearing tool-holder with delayed output feedback for low-immersion milling; Guo et al. [42] developed an active contact-force actuator to expand the chatter-free window in robotic milling. Most active methods assume ideal knowledge of the system dynamics and do not account for unmodeled perturbations or parameter variation.

2.2.3. Classical and Frequency-Domain Control

PID controllers are widely used for their simplicity. Acceleration or displacement signals can be fed to an actuator for chatter mitigation, but classical PID tuning typically relies on trial-and-error or linear models that capture neither the nonlinearity nor the delay of regenerative chatter. Frequency-domain tools such as stability lobe diagrams [1,2] complement closed-loop suppression by identifying chatter-free operating conditions.

2.2.4. Robust and Optimal Control

Machining processes face significant uncertainty (cutting-coefficient variation, tool wear, and fixture compliance), motivating robust formulations. Bahari Kordabad and Gros [20] proposed a Lyapunov-based optimal control framework for time-delay milling systems. H_∞ and H₂ designs maintain stability against bounded disturbances; LQR/LQG provide systematic effort-versus-performance trade-offs (Li et al. [43] combined LQR with ANFIS and showed it outperforms fixed-gain controllers). Yang et al. [44] integrated a Kalman estimator with model-predictive control on an adjustable piezoelectric actuator. Du et al. [45] combined μ-synthesis with an active time-delay term to address non-smooth dynamics, multi-mode participation, and time-varying coefficients in flexible-workpiece milling.

2.2.5. Fractional-Order Control

In the classical PID, the fractional-order PID (FOPID) adds fractional orders λ and μ to the integral and derivative actions, respectively, resulting in the following transfer function:

$$C\left(s\right)=K_p+K_i s^{-\lambda }+K_d s^{\mu }$$

(2)

Equation (28) gives the time-domain definition used in this work. The extra orders λ and μ provide finer phase- and gain-shaping than integer-order PID, particularly for resonant or complex dynamics systems. Frikh et al. [18] used analytical and computational FOPID tuning in wind turbines; Saxena et al. [17] compared ANFIS, FOPID-PSO, and fuzzy tuning on high-performance drilling, reporting superior fractional-order performance; Jin et al. [16] proposed a self-tuning FOPID with torque-observer compensation for PMSMs. Zheng et al. [46] combined a stable fractional-order PID with an ADRC inner loop for first-order-plus-time-delay systems, showing that disturbance rejection eases the fractional tuning conditions while preserving robustness under modeling error. Such hybrid FOPID + disturbance-estimation schemes have not been studied for regenerative chatter suppression in milling under hard actuator-saturation constraints.

2.2.6. Adaptive and Intelligent Control

Adaptive control methods (Self-Tuning Regulators; MRAC) and intelligent techniques (neural networks, fuzzy logic, and neuro-fuzzy) have also been applied to machining: Ghoshal [10] used a neural network to forecast chatter stability for variable-pitch cutters, and Basit et al. [9] reviewed machine-learning chatter detection. These methods are primarily detection-oriented, and real-time suppression remains limited by their computational cost.

2.2.7. Metaheuristic-Tuned Machining Controllers

The dynamics of machining and the non-linearity of chatter have motivated use of metaheuristics for optimizing the controller parameters. Different machines operations have been considered, and PID and FOPID gain tuning has been performed using PSO, GA, and DE. To get a glimpse of the same, Saxena et al. [17] demonstrated the effectiveness of PSO-tuned FOPID in the field of drilling machines. Karahan and Karci [11] proposed a hybrid swarm intelligence algorithm-based strong fractional-order fuzzy PID sliding mode controller for a 6-DOF robotic manipulator as an example of how the metaheuristics can be applied to a complex mechatronic system.

2.2.8. Positioning of the Present Work

Three gaps motivate the present work: (i) most active-control studies treat the regenerative cutting force as a known disturbance rather than as a lumped uncertainty estimated and rejected on-line; (ii) fractional-order controllers are rarely combined with disturbance estimation in milling; and (iii) physical search constraints (actuator saturation) and statistical validation are seldom enforced in metaheuristic tuning.

The combination of ADRC and FOPID as a generic control architecture is not, in itself, new: the hybrid has been examined for first-order-plus-dead-time systems [46] and tuned analytically for power-plant control [33]. The specific contributions of the present work are four-fold. First, the regenerative delay-differential term $-K_t\ast a_p\ast [x_{w\left(t\right)}-x_{w(t-\tau )}]$ is absorbed into the lumped-disturbance estimate of the ESO, leading to the bandwidth-selection rule ${\omega }_{\mathrm{o}}\ge \sqrt[3]{20}\ast \omega \mathrm{c}$ given in Equation (26a). Second, the PSO tuning is constrained by a hard ±27 N actuator envelope (Equation (31)) so that the optimized gains remain physically realizable. Third, robustness is assessed on a 38-point combined parametric-disturbance sample with frozen gains, including a worst-case combined perturbation in which the three uncertain parameters (modal mass, modal stiffness, and tangential cutting coefficient) are varied simultaneously at their most unfavorable values (Section 5.6.2). Fourth, a like-for-like benchmark is performed against the six controllers reported in [8], including $H_{\infty }$ and μ-synthesis, on the same plant and under the same saturation limit. To the authors’ knowledge, these four elements have not previously been addressed jointly for regenerative chatter in milling.

To clarify the originality further, we emphasize three points that, taken together, separate the present scheme from prior ADRC–FOPID hybrids. Prior hybrids (Zheng et al. [46] for first-order-plus-time-delay processes; Sun et al. [33] for power-plant FOPTD models) target single-input single-output plants without an intrinsic time delay in the closed-loop transfer function—the delay enters only as an input lag, and the ESO is asked to cancel a smooth, slowly-varying disturbance. In contrast, milling chatter is governed by a delay-differential equation (DDE) of neutral type in which the delay couples the present and past states of the workpiece directly through the regenerative kernel [x_w(t) − x_w(t − τ)]. Treating this state-dependent delayed feedback as part of the ESO’s lumped disturbance—rather than as an external disturbance—is, to our knowledge, new and is the source of the bandwidth-selection rule of Equation (26a). The PSO objective itself is also distinct: it maximizes the worst-case stability lobe $b_{crit}$ (N;p) under a hard ±27 N actuator-saturation penalty rather than a smooth time-domain integral criterion (ITAE, IAE)—a chatter-specific cost function that has not been used to tune ADRC–FOPID elsewhere. Finally, the validation protocol (Section 5.6) imposes the same gains on a 38-point combined parametric/disturbance sample and benchmarks against six published controllers on a third-party plant [8] under the same saturation budget; the previously cited ADRC–FOPID hybrids are validated only on their nominal plant. The architecture itself reuses well-established blocks; the novelty lies in their problem-specific assembly, tuning objective, and validation envelope rather than in the construction of a brand-new control law.

3. Physical System Description and Mathematical Modeling

The active milling configuration being explored is comprised of three key subsystems: a flexible cantilevered workpiece, an electromagnetic actuator for active force injection, and a self-excitation source in the form of a regenerative cutting mechanism. In addition, to simulate the realistic machining conditions, the model also involves the structural drift and the broadband noise, which denote the long-term compliance variation and high-frequency environmental disturbances, respectively.

The physical setup is shown in Figure 1, which shows a beam clamped at one end and half immersion end milling using a rotating tool. An electromagnetic actuator is attached orthogonally to the beam to apply active control forces, and a displacement sensor measures the lateral deflection at the free end. The equivalent lumped-parameter model is shown on the right part of the schematic, in which the actuator and beam are modeled as two second-order mass-spring-damper systems. The actuator subsystem, which consists of mass $M_a$, stiffness $K_a$, and damping $C_a$, produces a lateral control force $F_a\left(t\right)$ when the voltage u(t) is applied to it. This force is passed on directly to the beam with modal parameters $M_w$, $K_w$, and $C_w$ and subjected to the regenerative cutting force $F_m\left(t\right)$.

3.1. Beam Dynamics

A structural-steel beam ($E$ = 210 GPa; $\rho$ = 7850 kg/m³; $\nu$ = 0.30) is clamped at one end and then cantilevered. The beam is modeled as a reduced-order Timoshenko beam model of only the first bending mode. The lateral displacement at the free end $x_w\left(t\right)$ satisfies the second-order differential equation:

$$M_w{\ddot{x}}_w\left(t\right)+C_w{\dot{x}}_w\left(t\right)+K_w{x}_w\left(t\right)=F_m\left(t\right)+F_a\left(t\right)+d_{\mathrm{drift} }\left(t\right)+d_{\mathrm{noise} }\left(t\right)$$

(3)

Parameters of the dominant first bending mode are given in

Table 1. Dynamic system and actuator parameters used in simulation.

Parameter	Value/Matrix
Modal mass	$30 \mathrm{k}\mathrm{g}$
Modal stiffness	$2.40\ast {10}^7 \mathrm{N}/\mathrm{m}$
Modal damping	$\zeta =0.0088,$ $C=2\zeta \sqrt{Mk}$
Actuator ${\mathit{A}}_{\mathit{a}}$ matrix	$\left[-15.834-2785.6 1 0 \right]$
Actuator ${\mathit{B}}_{\mathit{a}}$ matrix	$\left[1 0 \right]$
Actuator ${\mathit{C}}_{\mathit{a}}$ matrix	$\left[3 0 \right]$
Feedthrough gain	0
Saturation limit	$\pm 27 \mathrm{N}$
Drift amplitude	$0.1\ast K_w$
Disturbance RMS	$50 \mathrm{N}$

.

The receptance calculated at the free tip of the cantilever over the band 0–2700 Hz, based on a Timoshenko-beam finite-element model, is shown in Figure 2a. The dominant mode is the first bending mode at f₁ = 143.2 Hz. The spindle-speed range studied in this work (1000–3000 rpm with Z = 4) is chosen in such a way that the corresponding tooth-passing frequency is $Z . N/60$. The chatter-excitation band of ($N/60$) is between 67 and 200 Hz. The range explicitly covers the first natural frequency, which is the dominant frequency in the system and is chosen to put the system in the worst-case dynamic situation for the purpose of testing the chatter-suppression controller. In addition, the higher-order resonances at 895.8 Hz and 2499.8 Hz are approximately 32 dB and 50 dB below, respectively, and well outside the frequency range explored in the study. The use of the single-mode reduction in the rest of the paper is thus physically appropriate as well as being computationally justified.

To formalize this argument, the error transfer function E(s) = G(s) − G₁(s) was evaluated analytically using the modal parameters of Table 1 and the published peak amplitudes of the second and third resonances (Figure 2). The modal parameters back-calculated under the cantilever-mode assumption of uniform tip-effective modal mass $m_{eff}$ = 30 kg are mode 1 (f₁ = 143.2 Hz, ζ₁ = 0.00878, ${\left|G_1\right|}_{peak}$ = 2.344 µm/N), mode 2 (f₂ = 895.8 Hz, ζ₂ = 0.00936, ${\left|G_2\right|}_{peak}$ = 0.0562 µm/N), and mode 3 (f₃ = 2499.8 Hz, ζ₃ = 0.00954, ${\left|G_3\right|}_{peak}$ = 0.00708 µm/N). Across the chatter excitation band 67–200 Hz, the relative error satisfies $\left|E\right(j_{\omega }\left)\right|/\left|G\right(j_{\omega }\left)\right|$ < 2.96%, dropping to 0.052% at f₁ that governs regenerative stability. The static residual flexibility $R_{stat}=1/k_2+1/k_3$ = 1.187 × 10⁻³ $\mathsf{\mu }\mathrm{m}/\mathrm{N}$—representing the quasi-static contribution of the truncated modes—amounts to 2.88% of the first-mode static compliance; adding it to G₁(s) as a Craig–Bampton correction reduces the band-wide error below 0.1%.

To rule out control- and observation-spillover, the spillover norm ${\left|\right|T_{spill}\left|\right|}_{\infty }=sup \left|G_{act}\right(j_{\omega }) K(j\omega ) E(j_{\omega }\left)\right|$ is computed with the PSO-tuned ADRC-FOPID gains of

Table 2. Optimal PID and FOPID controller gains obtained via PSO.

Controller	${\mathit{K}}_{\mathit{p}}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{d}}$	$\mathit{\lambda }$	$\mathit{\mu }$
PID	$2.3 \ast {10}^3$	$0.012$	$180$	–	–
FOPID	1.8 ∗ 103	0.45	240	0.85	1.15

and the actuator transfer function from Table 1. The resulting norm is 1.5 × 10⁻⁴ at f₂ = 895.8 Hz and 2.2 × 10⁻⁵ at f₃ = 2499.8 Hz, at least four orders of magnitude below the destabilizing threshold of unity. The observer bandwidth ${\omega }_{\mathrm{o}}=3\ast {\omega }_1=2699 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ satisfies the bandwidth-selection rule derived in Section 4.2 (Equation (26a)) while remaining safely below the second mode (${\omega }_{\mathrm{o}}/{\omega }_2$ = 0.48), so the ESO does not couple into the truncated higher-order dynamics. The single-mode reduction is therefore retained with a documented quantitative margin. A formal modal truncation error analysis is provided below: the analytical band-wide error norm ‖E(jω)‖_∞ over 67–200 Hz is below 2.96% (falling to 0.052% at the dominant f₁ = 143.2 Hz), the static residual flexibility correction R_stat brings the band-wide error below 0.1%, and the closed-loop spillover norm ‖T_spnk‖∞ is at most 1.5 × 10⁻⁴ at the second mode and 2.2 × 10⁻⁵ at the third—four to five orders of magnitude below the destabilizing threshold of unity. The 895.8 Hz and 2499.8 Hz higher-order bending modes are therefore not ignored: they are quantitatively shown to contribute negligibly to the regenerative-loop dynamics in the operating window of interest.

3.2. Regenerative Cutting Force Model

3.2.1. System Configuration and Physical Mechanisms

The whole milling configuration is shown in Figure 3, where both structural dynamics and cutting mechanics and regenerative feedback are coupled. The model includes orthogonal structural compliance, multi-tooth cutting kinematics, and the resolution of the cutting forces in the rotating tool and stationary workpiece frames.

Figure 3a is the physical model of the workpiece with orthogonal modal parameters $k_x,C_x$ (feed direction) and $k_y,C_y$ (normal direction) under cutting forces $F_{tj}$ (tangential) and $F_{rj}$ (radial) of the tooth $j$ at angle position ${\phi }_j$ in the engagement zone $\left[{\phi }_{st},{\phi }_{ex}\right]$. Figure 3b shows the regenerative mechanism that the dynamic chip thickness modulation displays between $P_t$ (current cutting point) and $P_w$ (surface waviness in previous tooth pass).

3.2.2. Mechanistic Cutting Force Formulation

Following the oblique cutting theory extended by Altintas and Budak [2], the instantaneous cutting forces are expressed through the mechanistic model. For tooth $j$ at angular position ${\phi }_j(t)=\Omega t+(j-1)2\pi /Z$, the force components shown in Figure 3 are

$$F_{tj}\left({\varphi }_j\right)=K_t{a}_p{h}_j\left({\varphi }_j\right)g\left({\varphi }_j\right)and{F}_{rj}\left({\varphi }_j\right)=K_r{a}_p{h}_j\left({\varphi }_j\right)g\left({\varphi }_j\right)$$

(4)

The cutting coefficients $K_t$ and $K_r$ encapsulate the material properties, tool geometry, and cutting conditions. The window function $g\left({\phi }_{-}j\right)$ ensures force contribution only during active engagement:

$$g\left({\varphi }_j\right)=\left\{\begin{array}{c}1, {\varphi }_{st}\le {\varphi }_j\le {\varphi }_{ex}\\ 0,otherwise\end{array}\right.$$

(5)

3.2.3. Coordinate Transformation and Force Projection

The transformation from the rotating tool frame to the stationary workpiece coordinates $(\mathit{x},\mathit{y})$ requires consideration of the instantaneous angular position:

$$T\left({\varphi }_j\right)=\left[-cos{\varphi }_j-sin{\varphi }_j sin{\varphi }_j-cos{\varphi }_j\right]$$

(6)

Applied to the force components,

$$\left[F_x\left({\varphi }_j\right)F_y\left({\varphi }_j\right)\right]=T\left({\varphi }_j\right)\left[F_{tj} F_{rj}\right]=\left[-F_{tj}cos{\varphi }_j-F_{rj}sin{\varphi }_j F_{tj}sin{\varphi }_j-F_{rj}cos{\varphi }_j\right]$$

(7)

3.2.4. Dynamic Chip Thickness and Regenerative Mechanism

The regenerative effect illustrated in Figure 3b manifests through the interaction between consecutive tooth passes. The instantaneous chip thickness comprises static, dynamic, and regenerative components:

$$h_j\left({\varphi }_j,t\right)=h_{static}+h_{dynamic}$$

(8)

Expanding

$$h_j\left({\varphi }_j,t\right)=f_z\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_j+\underset{\mathrm{feed} \mathrm{direction} \mathrm{modulation} }{\underbrace{\left[x_w\left(t\right)-x_w\left(t-\tau \right)\right]\sin {\varphi }_j}}+\underset{\mathrm{normal} \mathrm{direction} \mathrm{modulation}}{\underbrace{\left[y_w\left(t\right)-y_w\left(t-\tau \right)\right]\cos {\varphi }_j}}$$

(9)

The time delay $\tau =60/\left(ZN\right)$ represents the temporal separation between successive teeth at the same angular location, creating the characteristic delay-differential equation structure governing stability.

3.2.5. Altintas–Budak Directional Factors

The time-averaged directional cutting coefficients, derived through Fourier series expansion and retaining the zero-order terms:

Feed direction coefficients ($\mathit{x}$-axis with $k_x,C_x$):

$$\begin{array}{ll}{\alpha }_{xx}& ={\displaystyle \frac{N_t}{2\pi }}{\int }_{{\varphi }_{st}}^{{\varphi }_{ex}}\left[-K_t\mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm{c}\mathrm{o}\mathrm{s}\varphi -K_r{\mathit{sin}}^2\varphi \right]d\varphi \\ {\alpha }_{xy}& ={\displaystyle \frac{N_t}{2\pi }}{\int }_{{\varphi }_{st}}^{{\varphi }_{ex}}\left[K_t{\mathit{sin}}^2\varphi -K_r\mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm{c}\mathrm{o}\mathrm{s}\varphi \right]d\varphi \end{array}$$

(10)

Normal direction coefficients (y-axis with $k_y$, $C_y$):

$$\begin{array}{ll}{\alpha }_{yx}& ={\displaystyle \frac{N_t}{2\pi }}{\int }_{{\varphi }_{st}}^{{\varphi }_{ex}}\left[-K_t{cos}^2\varphi -K_{r}sin\varphi cos\varphi \right]d\varphi \\ {\alpha }_{yy}& ={\displaystyle \frac{N_t}{2\pi }}{\int }_{{\varphi }_{st}}^{{\varphi }_{ex}}\left[K_{t}sin\varphi cos\varphi -K_r{cos}^2\varphi \right]d\varphi \end{array}$$

(11)

For half-immersion up-milling (${\phi }_{st}=0,{\phi }_{ex}=\pi$), analytical evaluation yields

$${\alpha }_{xx}=-{\displaystyle \frac{N_t{K}_r}{2}},{\alpha }_{yy}={\displaystyle \frac{N_t\left(K_t-K_r\right)}{2}}$$

(12)

3.2.6. Spatially Distributed Force Integration

The total regenerative cutting force, accounting for continuous chip removal over the engagement arc:

$$F_m\left(t\right)=-{\displaystyle \frac{a_p}{2{\varphi }_h}}{\int }_{-{\varphi }_h}^{{\varphi }_h}\left[K_t\mathit{sin}\varphi \mathit{cos}\varphi \left(x_w\left(t\right)-x_w\left(t-\tau \right)\right)+K_r{\mathit{sin}}^2\varphi f_z\right]d\varphi$$

(13)

where ${\phi }_h=arccos\left(1-a_p/R\right)$ is the half-immersion angle. This integral formulation captures the spatially distributed nature of the cutting process and is more accurate than the lumped models commonly used in stability analysis.

3.2.7. Linearized Directional Coefficients

Through perturbation analysis about the steady-state cutting condition, two fundamental coefficients emerge:

Dynamic directional coefficient (regenerative feedback strength):

$${\Gamma }_{dyn}={\displaystyle \frac{1}{2{\varphi }_h}}{\int }_{-{\varphi }_h}^{{\varphi }_h}\left(K_{t}sin\varphi cos\varphi -K_r{cos}^2\varphi \right)d\varphi$$

(14)

Nominal coefficient (steady-state force):

$${\Gamma }_{nom}={\displaystyle \frac{1}{2{\varphi }_h}}{\int }_{-{\varphi }_h}^{{\varphi }_h}\left(K_{t}sin\varphi cos\varphi +K_r{sin}^2\varphi \right)d\varphi$$

(15)

The sign reversal in the $K_r$ term between Equations (10) and (11) reflects the dual role of the radial cutting force: it has a stabilising effect on the regenerative component while still contributing to the nominal steady-state cutting load.

3.3. Actuator Dynamics

The actuator is modeled as a second-order electromagnetic inertial actuator, expressed in state-space form as

$${z\dot{}}_a\left(t\right)=A_a{z}_a\left(t\right)+B_{a}u\left(t\right);F_a\left(t\right)=C_a{z}_a\left(t\right)+D_{a}u\left(t\right)$$

(16)

To ensure physical feasibility, a hard saturation constraint is applied to the output force [20]:

$$\left|F_a\left(t\right)\right|\le F_{max}$$

(17)

This upper bound is enforced during both time-domain simulation and controller tuning processes.

3.4. Disturbance Modeling

Two forms of disturbance are superimposed on the beam dynamics:

Structural drift simulates low-frequency thermal or fixturing effects using a sinusoidal modulation:

$$d_{drift}\left(t\right)=A_{drift}sin\left({\omega }_{drift}t\right),A_{drift}=0.1K_w$$

(18)

Broadband noise is introduced as zero-mean Gaussian excitation with an RMS amplitude of 50 N, affecting a frequency range up to 2 kHz.

These disturbances reproduce realistic operating conditions and are used to evaluate the closed-loop performance of the proposed control architecture.

4. Controller Design and PSO-Based Tuning

The proposed control architecture consists of three functional blocks that are tightly coupled together: a Tracking Differentiator (TD), which generates smooth reference signals; an Extended State Observer (ESO), which reconstructs the workpiece state and the lumped disturbance; and a Fractional-Order PID (FOPID) controller, which implements the nominal feedback action.

4.1. Closed-Loop Control Architecture

The ADRC-FOPID closed-loop architecture is shown in Figure 4. The Tracking Differentiator (TD) processes the reference v into a smooth trajectory $v_1$ and its derivative $v_2$. The Extended State Observer (ESO) receives the measured workpiece response and the controllable input $b_0. u$, reconstructing the internal plant states and the total lumped disturbance. The errors fed to the FOPID are the differences between the estimated states $z_1$, $z_2$ and the references $v_1$, $v_2$:

$$e_1=v_1-z_1,e_2=v_2-z_2$$

(19)

The FOPID produces the nominal control $u_0$. In parallel, the ESO provides a total-disturbance estimate $z_3$ that lumps the regenerative cutting force, structural uncertainty, parameter variation, and external perturbations; subtracting it in the 1/$b_0$ branch (Figure 4) yields the actuator command:

$$u=u_0-{\displaystyle \frac{z_3}{b_0}},$$

(20)

where $b_0$ is the nominal input gain. The actuator applies the resulting force, the delay element e^−sT reproduces the regenerative mechanism, and the response feeds the ESO to close the loop. An offline PSO layer (Figure 4) tunes the controller gains against the cost function subject to the actuator-saturation constraint.

4.2. Extended State Observer Design

The ESO extends the plant state to include the total disturbance as an estimated state. The plant (Equation (3)) is rewritten as

$$\ddot{x}=f_0(x,\dot{x},w,t)+bu$$

(21)

where $f_0(x,\dot{x},w,t)$ lumps all internal and external disturbances (regenerative force, drift, and noise). Treating any mismatch between the actual and nominal input gain as part of the disturbance gives

$$f=f_0(x,\dot{x},w,t)+\left(b-b_0\right)u$$

(22)

where $b$ is the actual input gain, and $b_0$ is its nominal value used in the control law. The term $\left(b-b_0\right)u$ accounts for input gain uncertainty. Then, the plant can be expressed in the canonical form:

$$\ddot{x}=f+b_{0}u$$

(23)

Assuming that $f$ is differentiable and its derivative $h=\dot{f}$ is bounded, the extended statespace model is constructed by defining the states as displacement $x_1=x$, velocity $x_2=\dot{x}$, and total disturbance $x_3=f$:

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=x_3+b_{0}u\\ {\dot{x}}_3=h\\ y=x_1\end{array}\right.$$

(24)

A linear ESO for this system is designed as

$$\left\{\begin{array}{l}{\dot{\hat{x}}}_1={\hat{x}}_2+{\beta }_1\left(x_1-{\hat{x}}_1\right)\\ {\dot{\hat{x}}}_2={\hat{x}}_3+{\beta }_2\left(x_1-{\hat{x}}_1\right)+b_{0}u\\ {\dot{\hat{x}}}_3={\beta }_3\left(x_1-{\hat{x}}_1\right)\end{array}\right.$$

(25)

where ${\hat{x}}_1,{\hat{x}}_2$, and ${\hat{x}}_3$ are estimates of the displacement, velocity, and total disturbance, respectively. The observer gains ${\beta }_1,{\beta }_2$, and ${\beta }_3$ are chosen to place the poles of the estimation error dynamics at desired locations. It is common to parameterize them as

$${\beta }_1=3{\omega }_o,{\beta }_2=3{\omega }_o^2,{\beta }_3={\omega }_o^3$$

(26)

where ${\omega }_o$ is the observer bandwidth [13,47,48], chosen well above the chatter frequency for accurate estimation but limited by noise and sampling [24,25]. ${\omega }_0$ is selected from the system’s natural frequency and disturbance spectrum, with the regenerative-DDE rule derived below.

The disturbance estimate ${\hat{x}}_3$ is used in the control law to cancel the total disturbance, effectively reducing the plant to a double integrator (in the nominal case) that can be easily controlled by the FOPID.

The bandwidth-selection rule for the regenerative DDE is derived as follows. The estimation-error transfer function of the linear ESO is E(s) = s³/(s + ω_o)³ times F(s), giving an asymptotic amplitude error of (${\omega }_c$/ω_o)³ at the dominant chatter frequency ${\omega }_c$. Requiring less than 5% residual error yields

$${\omega }_o\ge \sqrt[3]{20}\ast {\omega }_c\approx 2.71·{\omega }_c$$

(26a)

With ${\omega }_c$= ${\omega }_1$ = 900 rad/s, the rule mandates ω_o ≥ 2442 rad/s; the chosen value ${{\mathsf{\omega }}_o=3. \omega }_1$ = 2699 rad/s satisfies this with an 11% margin while remaining safely below the second resonance (ω_o/ω₂ = 0.48; see Section 3.1). The nominal input gain $b_0=1/M_w=0.0333 {\mathrm{kg}}^{-1}$ follows directly from the canonical ADRC form $x_{ddot}=f\left(t\right)+b_0\ast u$ applied to Equation (3).

4.3. Control Law Formulation

The classical PID controller is defined by the equation

$${u\left(t\right)=K}_{p}e\left(t\right)+K_i{\int }_0^{t}e\left(\tau \right)d\tau +K_d\dot{e}\left(t\right)$$

(27)

The FOPID controller generalizes this structure by including fractional-order dynamics:

$${u\left(t\right)=K}_{p}e\left(t\right)+K_i{D}_t^{-\lambda }\left[e\left(t\right)\right]+K_d{D}_t^{\mu }\left[e\left(t\right)\right]$$

(28)

where $D_t^{-\lambda }$ and $D_t^{\mu }$ denote the fractional integral and derivative, respectively, implemented using the Grünwald–Letnikov approximation [49]. The parameters $\lambda$ and $\mu$ offer additional flexibility in tuning gain and phase, enabling robust damping across a wider frequency band [18].

4.4. PSO-Based Gain Optimization

Let the state of particle i on iteration k be a vector ${\mathrm{x}}_i^k={\left[K_p,K_i,K_d,\lambda ,\mu \right]}^T$; lambda, mu are defined parameters that are bounded (see

Table 3. Parameter bounds used in PSO-based tuning.

Parameter	PID Bounds	FOPID Bounds
${\mathit{K}}_{\mathit{p}}$	$[{10}^2,{10}^5]$	$[{10}^2,{10}^5]$
${\mathit{K}}_{\mathit{i}}$	$[{10}^{-3},{10}^2]$	$[{10}^{-2},{10}^4]$
${\mathit{K}}_{\mathit{d}}$	$[{10}^1,{10}^3]$	$[{10}^1,{10}^3]$
$\mathit{\lambda }$	–	$[0.1,2.0]$
$\mathit{\mu }$	–	$[0.1,2.0]$

). The velocity of each particle is given as ${\mathrm{v}}_i^k$, updated as

$$\begin{array}{c}{\mathrm{v}}_i^{k+1}=w^k{\mathrm{v}}_i^k+c_1{r}_1\left({\mathrm{p}}_i^{\mathrm{best} }-{\mathrm{x}}_i^k\right)+c_2{r}_2\left({\mathrm{g}}^{\mathrm{best} }-{\mathrm{x}}_i^k\right),\\ {\mathrm{x}}_i^{k+1}={\mathrm{x}}_i^k+{\mathrm{v}}_i^{k+1},\end{array}$$

(29)

where

• The weight of inertia is denoted by $w^k$ and reduces in a linear manner $({\mathrm{w}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.9;{\mathrm{w}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=0.4)$ as the optimization progresses to explore and exploit [50];
• $c_1,c_2=2$ are cognitive acceleration and social acceleration coefficients, the usual values that favor convergence [51];
• The random numbers $r_1,r_2$ are $\cup \left(\mathrm{0,1}\right)$ and independent;
• ${\mathrm{p}}_i^{\mathrm{best}}$ is the most optimal position of particle $\dot{i}$ that has been spotted;
• $g^{\mathrm{best}}$ is the best position of all the particles in the world.

Velocity clamping is used to ensure that there is no movement of the particles off the search space: each element of the velocity of particle $v_i^{k+1}$ is confined to a range of $\pm 30\mathrm{\%}$ of the parameter ranges. This converts convergence to become stable rather than unduly confining exploration.

4.4.1. Population Size and Iterations

The PSO algorithm is run for a population of 40 particles and $K_{max}=60$ iterations.

These values follow from a preliminary convergence study (Figure 5): the algorithm converges well before the iteration cap, with no further objective improvement from larger swarm sizes.

The cost function and constraint handling are now defined. The objective is to maximize the minimum critical depth of cut across a discrete set of spindle speeds N = {N₁, N₂, …, N_m} while keeping the actuator within saturation. Stability-lobe theory of delay-differential equations gives the critical depth as a function of spindle speed and controller parameters p; for a single-mode regenerative-force model, the closed-loop characteristic equation is

$$1+{\mathsf{\Gamma }}_{dyn}{G}_{cl}(s,p)\left(1-e^{-s\tau }\right)=0$$

(30)

where G_cl(s,p) is the closed-loop transfer function from regenerative force to displacement, including actuator, controller, and ESO dynamics. To maximize the critical depth across N, the controller parameters p are optimized by PSO with cost function

$$\left(p\right)=-\underset{N\in N}{min} b_{crit}\left(N;p\right)+\alpha I_{sat}\left(p\right)$$

(31)

where $I_{sat}=1$ if $\left|F_a\left(t\right)\right|>F_{max}$ at any time, and 0 otherwise. The penalty weight $\alpha$ is sufficiently large to eliminate infeasible solutions. The critical depth of cut is computed via

$$b_{crit}\left(N;p\right)=\underset{\omega >0}{min} {\displaystyle \frac{\sqrt{{\left(M_w{\omega }^2-K_w\right)}^2+{\left[C_w+C_{ctrl}\left(p\right)\right]}^2{\omega }^2}}{{\Gamma }_{dyn}\sqrt{2-2coscos\left(\omega \ast \tau \left(N\right)\right)}}}$$

(32)

where $\tau \left(N\right)=60/\left(ZN\right)$, and $C_{ctrl}\left(p\right)$ represents the effective damping induced by the controller.

The range of spindle speeds of interest (1000–3000 rpm, in 50 rpm increments) is chosen as the set of spindle speeds, typically the first few lobes. Computation Critical depth is calculated numerically by sweeping on a grid of fine resolution (e.g., by 0.1 Hz steps) around the structural natural frequency.

4.4.2. Search Space Bounds

Parameter bounds (Table 3) follow from physical and stability criteria: the fractional orders are restricted to (0, 2) to preserve their integrator/differentiator character.

The ranges for the parameters used in the search are summarized in Table 3, with a distinction between the PID ranges and the FOPID ranges.

These constraints provide a reasonable and large set of feasible physically and numerically control strategies, ranging from conservative to aggressive, to be explored by the optimizer.

Note: The upper bound on K_i is set to 10⁴ in FOPID (versus 10² for integer-order PID) for two reasons. First, the fractional integral D^−λ (0 < λ < 1) rolls off at −20λ dB/decade rather than −20 dB/decade, so the same numerical coefficient yields a lower low-frequency gain—the coefficient must be larger to deliver equivalent integral action. Second, the bound only delimits the search space, not the operating value: across all PSO runs, the best K_i lay well below the bound (Table 2 reports 0.45, ≈4 orders of magnitude below). As a direct sensitivity check requested in review, the PSO tuning of the FOPID was repeated with the same upper bound on K_i as used for the integer-order PID (K_i ∈ [10⁻³, 10²]); the best run converged to K_i = 0.42 with b_crit changed by less than 1.5% and the closed-loop RMS by less than 2%. The wider upper bound is therefore an exploration safeguard rather than a source of unfair advantage: removing it does not materially alter the reported performance, and the PSO consistently selects a small operating value of K_i for FOPID regardless of the bound because the fractional integrator already provides the required low-frequency action through the order λ.

4.4.3. Statistical Validation

Since PSO is stochastic, the optimization is repeated R = 10 times with independent random initializations of the swarm. The best solution across all runs is retained, and the mean and standard deviation of the objective are reported to quantify run-to-run consistency.

5. Numerical Validation and Results

Four control cases are compared: open-loop, classical PID, FOPID, and the proposed ADRC–FOPID. Simulations include actuator saturation, structural drift, and broadband disturbances $b_{crit}$, under physical constraints.

Time domain responses, frequency domain analysis, and stability lobe diagrams are used to measure the performance.

5.1. Simulation Setup and System Parameters

All numerical simulations were performed in MATLAB R2025b (The MathWorks, Inc., Natick, MA, USA). The simulations were conducted with a spindle speed of 2000 rpm (close to the first natural frequency of the cantilevered workpiece, ≈143 Hz), corresponding to a critical condition for the onset of chatter. The simulations were carried out with a fixed time step of $10 \mathsf{\mu }\mathrm{s}$ for 30 s, which allows for a high-resolution capture of the effects of regenerative vibrations and closed-loop dynamics.

Two types of disturbances were taken into consideration to simulate industrial conditions: Low-frequency structural drift: sinusoidal variation of stiffness at 0.5 Hz with a ±10% amplitude.

Broadband stochastic disturbance: 50 N RMS Gaussian white noise as a representation of unbalanced spindle motion, coolant flow variability, and tool runout. The dynamic parameters of the workpiece and actuator, such as modal parameters, control matrices, and disturbance models, are presented in Table 1.

Table 4. Cutting parameters for half-immersion milling.

Parameter	Value
Axial depth	$1 \mathrm{m}\mathrm{m}$
Feed per tooth	$0.05 \mathrm{m}\mathrm{m}$
Number of teeth	$4$
Tool radius	$8 \mathrm{m}\mathrm{m}$
Tangential stiffness	$760 \mathrm{N}/{\mathrm{mm}}^2$
Radial stiffness	$210 \mathrm{N}/{\mathrm{mm}}^2$

presents the half-immersion milling process cutting conditions like feed rate, depth of cut, and tool geometry.

5.2. Controller Tuning and Displacement Analysis

PSO was used to tune the controller gains under the actuator-saturation constraint, with the objective of maximizing the minimum critical depth across the spindle-speed range. The tuned parameters are listed in Table 2.

The ADRC-FOPID controller uses the same five PSO-tuned FOPID gains as standalone FOPID ($K_p=1.8\times {10}^3, K_i=0.45, K_d=240, \lambda =0.85,\mathrm{a}\mathrm{n}\mathrm{d} \mu =1.15$), augmented by the two ADRC design parameters fixed at the $values {\omega }_{\mathrm{o}}=3\ast {\omega }_1=2699.3 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ (observer bandwidth; Section 4.2) and b₀ = 1/$M_w$ = 0.0333 kg⁻¹ (nominal input gain; canonical ADRC form). The corresponding ESO gains computed from Equation (26) $\mathrm{a}\mathrm{r}\mathrm{e} {\beta }_1=3·{\omega }_{\mathrm{o}}=8.10\times {10}^3 {\mathrm{s}}^{-1}, {\beta }_2=3·{{\omega }_{\mathrm{o}}}^2=2.19\times {10}^7 {\mathrm{s}}^{-2}, \mathrm{a}\mathrm{n}\mathrm{d} {\beta }_3={{\omega }_{\mathrm{o}}}^3=1.97\times {10}^{10} {\mathrm{s}}^{-3}.$ These design parameters are chosen at the lower edge of the Gao [13] envelope ω_o in $[3\ast {\omega }_c, 10\ast {\omega }_c]$ to minimize sensor-noise amplification (which scales as ${{\omega }_{\mathrm{o}}}^3$) while satisfying the regenerative-DDE rule (Equation (26a)) with 11% margin.

5.3. Time-Domain Response

Figure 6 shows the tip displacement at 2000 rpm over 30 s. The open-loop signal exhibits continuous amplitude-modulated chatter with bursts reaching ≈20 µm. PID provides only marginal envelope attenuation, FOPID reduces amplitude with residual bursts, and the proposed ADRC–FOPID yields the smallest and smoothest response. RMS displacement averaged over ten independent noise realizations (

Table 5. RMS displacement for each control scenario.

Scenario	RMS Displacement (µm)	Improvement vs. Open-Loop
Open-Loop	3.78 ± 0.21	–
PID	3.66 ± 0.19	3.2% ± 0.5%
FOPID	2.71 ± 0.14	28.3% ± 1.2%
ADRC–FOPID	1.19 ± 0.11	68.5% ± 1.5%

) drops from 3.78 ± 0.21 µm (open-loop) to 1.19 ± 0.11 µm under ADRC–FOPID—a 68.5% reduction—against 3.2% for PID and 28.3% for FOPID.

5.4. Frequency Domain Analysis

The single-sided FFT amplitude spectrum |P₁(f)| at ap = 1.0 mm and N = 2000 rpm is shown in Figure 7 for the four cases up to 400 Hz, with three reference frequencies marked: the injected disturbance ≈50 Hz (●), the tooth-pass frequency Z·N/60 = 133.3 Hz (■), and the first natural frequency ≈143 Hz (◆). Open-loop peaks reach 3.7–3.8 µm with a 160 Hz sidelobe. PID only marginally lowers the chatter peak; FOPID attenuates more uniformly through fractional phase/gain shaping. ADRC–FOPID reduces the tooth-pass peak to ≈1.2 µm (≈70% attenuation), eliminates the 160 Hz sidelobe, and drops the disturbance peak from 3.7 to ≈2.9 µm—the smaller margin reflects the narrow-band disturbance falling on the shallower edge of the ESO roll off. The spectral ranking matches the time-domain RMS results.

5.5. Stability Lobe Diagrams

Stability lobe diagrams (SLDs) for the closed loop were computed from the distributed-delay formulation [1,2] over 1000–3000 rpm. Figure 8 shows the SLDs for the four cases.

Two typical spindle speeds are indicated:

The minimum value of the critical depth $b_{\mathrm{m}\mathrm{i}\mathrm{n}}$ (worst-case stability) of the system is 1424 rpm, where the most effective destabilizer of the process is the alignment of the regenerative phase.

The maximum stability $b_{\mathrm{m}\mathrm{a}\mathrm{x}}$ of the system is at 2133 rpm, which is the optimum regenerative interference.

ADRC–FOPID raises the worst-case critical depth from 1.00 to 2.67 mm (+167%) and the peak stable depth from 23.17 to 37.67 mm (+62.6%). Per-scenario minimum and maximum stable-depth values appear in

Table 6. Critical depth bounds at representative spindle speeds.

Scenario	${\mathit{b}}_{\mathit{m}\mathit{i}\mathit{n}}\left(\mathbf{mm}\right)$	Improvement vs. Open-Loop	${\mathit{b}}_{\mathbf{max}}$(mm)	Improvement vs. Open-Loop
Open-Loop	1.00	—	23.17	—
PID	1.50	50.0%	27.50	18.7%
FOPID	2.00	100.0%	30.83	33.1%
ADRC–FOPID	2.67	167.0%	37.67	62.6%

.

5.6. Robustness Analysis

In Section 5.6, controller gains are held at the nominal values (Table 2) while plant parameters, disturbance levels, and cutting conditions are varied; what is measured is the robustness of the closed loop without operating-point retuning.

5.6.1. Spectral Suppression of the Chatter Resonance Peak

Time-domain RMS can obscure the residual peak in the chatter band that governs surface finish and the practical onset of chatter. Figure 9 reports the peak amplitude of the displacement spectrum over the chatter band (100–200 Hz, centered on the 143 Hz first bending mode) for the four scenarios at axial depths of 1.0, 1.5, and 2.0 mm. The open-loop peak rises monotonically (7.29 → 14.56 µm) as the regenerative loop gain grows with depth. PID barely attenuates the peak (3–4%): its fixed bandwidth lacks a phase margin near the structural mode. FOPID reduces the peak by 1.8–2.5× through fractional phase shaping (2.86, 5.18, and 8.18 µm). ADRC–FOPID compresses it to 1.17, 2.41, and 3.60 µm—53–59% below FOPID and 4.0–6.2× below open-loop, with the relative gain growing with depth since the ESO absorbs the regenerative force into the lumped disturbance and leaves the loop seen by the FOPID near a double integrator even under heavy cuts.

Figure 10 shows the steady-state response over a 200 ms window (1.70–1.90 s) at the three depths, with in-window RMS, and actuator-saturation duty cycle. Three points emerge. (i) In-window RMS decreases monotonically from PID to FOPID to ADRC–FOPID at every depth; at 2.0 mm, the reduction is 46% (PID→FOPID) and 26% (FOPID→ADRC–FOPID). (ii) ADRC–FOPID operates near the ±27 N saturation envelope across all depths (duty cycle 95–98%) versus 76–94% for FOPID and only 1–66% for PID—the classical PID under-uses the saturation budget while ADRC–FOPID consistently spends it to cancel the lumped disturbance. (iii) Despite near-saturation actuation, the ADRC–FOPID displacement remains well-damped and free of high-frequency limit cycles, confirming that the PSO-tuned gains place the closed loop in the saturated-but-stable region.

5.6.2. Plant Parameter Uncertainty

Real machines exhibit several-percent deviations in modal mass m, stiffness k, and the tangential cutting coefficient from temperature drift, fixturing variation, tool wear, and workpiece dispersion. Each parameter was varied by ±15% from nominal, together with a worst-case combined perturbation. Figure 11 shows the resulting RMS displacement with controller gains frozen at the nominal values. Two regimes appear. Variations that reduce the regenerative loop gain have a modest effect (8.1 µm at m −15%, 6.9 µm at k +15%). Variations that increase the loop gain—particularly m +15% and k −15%—degrade the open-loop badly (32.9 and 75.8 µm, up to 8× nominal). PID and standalone FOPID only partially recover (k −15%: 36.6 and 34.7 µm, respectively). ADRC–FOPID stays below 5.0 µm at k −15% and below 4.2 µm at m +15% (1.1–1.3× the 3.8 µm nominal). Under combined worst-case perturbation, it reaches 4.8 µm versus 12.4 (open-loop), 11.1 (PID), and 6.5 µm (FOPID). The margin reflects the ESO-absorbing parameter mismatch into the lumped disturbance, keeping the loop seen by the FOPID close to its nominal double integrator regardless of plant deviation.

5.6.3. Disturbance-Amplitude Robustness and Combined Stress

The second robustness axis is broadband-disturbance amplitude (combining spindle imbalance, coolant flow, and tool runout). Figure 12 sweeps a multiplicative gain on the nominal disturbance from 0.5× to 3×—a sixfold power range. Open-loop RMS rises from 9.6 to 13.6 µm (+42%), PID from 9.2 to 10.2 µm, and FOPID from 5.1 to 6.3 µm, while ADRC–FOPID stays flat at 3.8–3.9 µm. Its slope below 0.03 µm per unit gain (under 2% RMS change over a 3× sweep) indicates high disturbance-rejection bandwidth and no ESO saturation across the range.

A stiffer test combines the worst plant perturbation (k − 15%) with a full spindle-speed sweep. Figure 13 shows RMS displacement from 1000 to 2800 rpm. The open-loop exhibits a sharp resonance at ≈2000 rpm, where regenerative phase alignment coincides with the destabilized mode (75.8 µm, ≈8× nominal). PID and FOPID also peak there (36.6 and 34.7 µm) with only marginal off-resonance improvement. ADRC–FOPID stays nearly flat at 3.5–4.9 µm with no resonance peak—within a 40% window of nominal even under combined structural destabilization and worst-case excitation.

5.6.4. Aggregate Robustness Statistics

Combining the eight plant scenarios (Section 5.6.2), six disturbance amplitudes (Section 5.6.3) and a five-point rpm sweep yields a 38-point robustness sample per controller. Four statistics summaries it: mean RMS (typical performance), 90th-percentile RMS (less sensitive than the strict maximum and particularly relevant since the worst few percent of conditions dominate surface roughness and tool wear), CV = σ/mean (dispersion normalized across amplitude scales), and maximum RMS (worst-case bound). Figure 14 and

Table 7. Aggregate robustness statistics over the 38-point uncertainty sample (Section 5.6.2 and Section 5.6.3 combined). CV denotes the coefficient of variation, defined as the ratio of standard deviation to mean.

Controller	Mean RMS (µm)	p90 RMS (µm)	CV (%)	Max RMS (µm)
Open-loop	14.8	31.0	114.8	75.8
PID	10.7	16.0	72.0	36.6
FOPID	7.2	7.9	101.3	34.7
ADRC–FOPID	4.1	5.0	16.0	6.1

report all four. ADRC–FOPID minimizes every statistic simultaneously: mean 4.1 µm versus 7.2, 10.7, and 14.8 µm (FOPID, PID, and open-loop); p90 5.0 µm versus 7.9, 16.0, and 31.0 µm; CV 16.0%—a quarter of PID, a sixth of FOPID, and a seventh of open-loop; and maximum 6.1 µm, six to twelve times smaller than the other cases. Mean, tail, and dispersion all collapse together: ADRC–FOPID is not only better on average; it is far less variable across operating conditions.

5.6.5. Sensitivity to Process Parameters

Two process parameters of direct practical relevance—feed per tooth and radial engagement—were swept independently. Figure 15 (left) varies feed from 0.025 to 0.12 mm at fixed depth 1 mm and 2000 rpm. RMS rises monotonically with feed (as the cutting force scales linearly), but slopes differ: open-loop and PID ≈ 39 µm/mm, FOPID ≈ 34 µm/mm, and ADRC–FOPID ≈ 32 µm/mm, with ADRC–FOPID lowest across the range. Figure 15 (right) varies radial engagement from 4 to 14 mm. Open-loop and PID show a strongly non-monotonic peak around 8 mm (half-immersion maximizes regenerative phase coupling), with recovery toward full immersion. ADRC–FOPID has a milder peak at 8 mm and stays bounded between 3.5 and 5.2 µm across the sweep versus an open-loop peak of 9.8 µm. FOPID alone slightly outperforms ADRC–FOPID at very small engagements (≤6 mm), where the regenerative force is too small for the ESO disturbance-rejection action to compensate for its noise amplification; at higher engagement, ADRC–FOPID resumes its margin.

5.6.6. Two-Dimensional Operating Window ($a_p$ × rpm)

The preceding sweeps fix one operating point and vary one parameter at a time. Figure 16 maps RMS displacement over the full two-dimensional operating window (axial depth 0.5–2.5 mm × rpm 1000–2800) for all four controllers on a common color scale. The open-loop map (left) has an unstable region with RMS > 35 µm, peaking at 74.9 µm at the worst corner (2.5 mm, 1000 rpm), plus a second hot spot near 2000 rpm. PID and FOPID cool the map (peaks 15.2 and 12.0 µm) but leave a residual hot zone at high depth and 2000–2500 rpm. The ADRC–FOPID map (right) is the most uniformly cool: RMS exceeds 6.0 µm only at the most demanding corner (2.5 mm, 2500 rpm). With nominal gains, performance is bounded across a 5× axial-depth range and a 2.5× spindle-speed range without operating-point retuning.

Across the six analyses, the ADRC-FOPID retains its margin without operating-point retuning: the ESO absorbs uncertainties into a lumped disturbance estimate that is canceled on-line, while the FOPID shapes the nominal residual response.

5.7. Comparative Analysis Against Published Active-Control Benchmarks

The proposed controller was benchmarked against state-of-the-art active chatter-suppression strategies using the experimental setup of Ozsoy, Sims, and Ozturk [8]: a flexible workpiece on a 2-DOF flexible robot arm (first bending mode 142.1 Hz, modal damping 0.88%, and modal stiffness 2.43 × 10⁷ N/m) with a proof-mass inertial actuator (±27 N). All six controllers from [8], together with the proposed FOPID and ADRC–FOPID, were re-implemented and PSO-tuned on the same plant under the same saturation budget. Cutting conditions (16 mm 4-flute end mill; half-immersion down-milling on Al-7075-T6) and the Altintas–Budak zero-order stability method were identical, giving a like-for-like comparison. Figure 17 shows the closed-loop receptance around the chatter band. The uncontrolled peak is 2.34 µm/N; the six benchmark controllers reduce it to 0.9–1.0 µm/N (≈60% attenuation), the standalone FOPID to 0.69 µm/N (≈70%), and the proposed ADRC–FOPID to 0.40 µm/N (83% attenuation)—less than half of any other controller, with the stiffness asymptote preserved at low frequency and the inertia asymptote at high frequency because the active disturbance rejection damps the resonance without adding low-frequency loop gain.

To ensure a like-for-like comparison, every controller in

Table 8. Verification of the analytical implementation against the Ozsoy et al. [8] benchmark and comparison of the proposed FOPID and ADRC-FOPID controllers. Columns “$b_{\mathrm{crit}}$ ours/$b_{\max }$ ours” are the values produced by the present SLD computation; columns “$b_{\mathrm{crit}}$ paper/$b_{\max }$ paper” are taken from [8]; Δ columns give the relative error in percent. “(novel)” indicates a controller introduced in this paper for which no reference value is available.

Controller	${\mathit{b}}_{\mathbf{crit}}$ Ours (mm)	${\mathit{b}}_{\mathbf{crit}}$ [8] (mm)	Δ ${\mathit{b}}_{\mathbf{crit}}$(%)	${\mathit{b}}_{\mathbf{max}}$ Ours (mm)	${\mathit{b}}_{\mathbf{max}}$ [8] (mm)	Δ ${\mathit{b}}_{\mathbf{max}}$(%)
Uncontrolled	1.24	1.2	+3.3	43.94	45.1	−2.6
DVF	3.20	3.2	+0.0	46.59	47.2	−1.3
VPA	2.49	2.5	−0.4	45.33	46.5	−2.5
PID	3.19	3.2	−0.3	46.54	47.3	−1.6
LQR	3.17	3.2	−0.9	46.47	47.2	−1.5
H∞	3.11	3.1	+0.3	47.06	46.2	+1.9
μ-synthesis	3.11	3.1	+0.3	47.06	46.7	+0.8
FOPID	4.37	—	novel	47.71	—	novel
ADRC-FOPID	7.58	—	novel	50.92	—	novel

was re-tuned by the same PSO algorithm under the same cost function (Equation (31)), the same actuator-saturation penalty, and the same plant. The structural form of each controller was preserved as in [8], but its free parameters were treated as PSO decision variables: for PID, the three classical gains; for LQR, the entries of $Q=diag(q_1, q_2)$ and R; for $H_{\mathrm{\infty }}$ and μ-synthesis, the weighting filters $W_{\mathrm{S}\left(\mathrm{s}\right)}$ and $W_{\mathrm{T}\left(\mathrm{s}\right)}$ parametrised by their DC gain and crossover frequency. Ten independent PSO runs were performed for each controller; the reported gains correspond to the best run, and the mean ± standard deviation of the best objective is given in Appendix A. As a separate code-verification sanity check, the SLD computation was confirmed to reproduce the published $b_{crit}$ values of [8] within 1%; this verification is independent of the comparison protocol above. The fairness of this comparison is supported by two quantitative indicators: (i) the run-to-run coefficient of variation of the best b_crit is below 0.72% for every one of the eight controllers (

Table A1. Statistical replication of PSO tuning over ten independent runs for each controller. CV = coefficient of variation = standard deviation divided by mean.

Controller	${\mathit{n}}_{\mathit{p}}$	${\overline{\mathit{J}}}_{60}$	$\mathit{\sigma }$	$\mathit{C}\mathit{V} \left[\mathit{\%}\right]$	${\mathit{k}}_{\mathbf{conv}}$
DVF	1	0.660	0.0001	0.02	35
VPA	2	0.550	0.0002	0.04	38
PID	3	0.451	0.0002	0.05	44
LQR	4	0.403	0.0005	0.13	47
H∞	5	0.304	0.0004	0.12	52
μ-synthesis	6	0.275	0.0009	0.31	53
FOPID	7	0.218	0.0016	0.72	58
ADRC-FOPID	8	0.188	0.0011	0.59	59

; Figure A1), demonstrating that each PSO run has converged to a tight neighborhood of its global optimum; (ii) the reproduction error of the six benchmark controllers against [8] is below 1% on b_crit and below 2.6% on b_max, indicating that the PSO has rediscovered the literature-reported gains for the simpler baselines and has not under-tuned them. Together, these two checks address the concern that classical controllers (PID and LQR) might be artificially handicapped relative to the proposed ADRC–FOPID: every controller was independently optimized under identical conditions, with the same swarm size, iteration cap, cost function, and saturation penalty.

Stability lobe diagrams for all controllers (Altintas–Budak zero-order method) are shown in Figure 18 and summarized in Table 8. The verification reproduces all six benchmark values within 1% on the critical depth and within 2.6% on the peak stable depth. The proposed ADRC–FOPID reaches a critical depth of 7.58 mm—6.1× the uncontrolled case (1.24 mm), 2.4× DVF/PID (≈3.2 mm), and 1.7× standalone FOPID (4.37 mm)—and the highest peak stable depth (50.92 mm) of all controllers tested, without exceeding the ±27 N saturation budget and without controller-specific resources such as the additional shaper required by μ-synthesis. The ADRC–FOPID curve dominates the others uniformly across 500–3000 rpm rather than only at isolated lobe peaks.

ADRC–FOPID outperforms both the classical strategies (DVF, VPA, and PID) and the H_∞/μ-synthesis robust controllers tuned in [8] under the same saturation constraint. Without an explicit uncertainty model, the lumped disturbance estimate captures regenerative dynamics and parametric variation online, whereas H_∞ and μ-synthesis must trade nominal performance against design-time robustness margins.

6. Discussion

The proposed ADRC–FOPID combines ADRC-based lumped-disturbance estimation with the frequency-domain flexibility of FOPID, with the goal of avoiding the nominal-versus-robust trade-off inherent in H_∞ and μ-synthesis designs. The numerical results support this premise: a single set of gains tuned at one operating point delivers the best nominal performance (Section 5.5), the best aggregate robustness on a 38-point uncertainty sample (Section 5.6), and the best stability boundaries on the Ozsoy et al. [8] benchmark (Section 5.7).

Against the six controllers in [8] (critical depth ≈ 3.1–3.2 mm), the present scheme increases the critical depth by 1.7× over pure FOPID and 2.4× over PID, quantifying for regenerative milling the ADRC benefits previously reported in non-machining applications [11,12,15,16,17,18,19]. Section 5.6 additionally sweeps a six-fold disturbance amplitude alongside combined ±15% structural uncertainty in m, k, and the cutting coefficient—a joint sweep rarely reported in the machining-control literature.

The two stages of the scheme are complementary: the Extended State Observer estimates the lumped disturbance (regenerative cutting force, model uncertainties, and unmodeled friction) and cancels it through the inner loop, leaving the outer controller to see a near-nominal double integrator; the FOPID then uses its fractional orders to shape the closed-loop frequency response in ways unattainable with integer-order PID. ADRC achieves robustness through online disturbance rejection rather than worst-case design margin, which explains the advantage over the H_∞ and μ-synthesis controllers in [8] designed under the same actuator-saturation budget.

Two practical implications follow. First, robustness holds across a 5× change in an axial depth of cut and a 1.8× change in spindle speed (Figure 16), so a single controller covers a wide operating window without online re-tuning. Second, the ±27 N actuator-saturation budget is well below the bandwidth of typical proof-mass actuators, so the strategy does not require the very high-bandwidth hardware often assumed by active-control designs. Because the regenerative force enters the ESO as part of the lumped disturbance rather than as an external input, the framework extends naturally to other regenerative processes such as thin-wall milling and slender-tool turning.

Limitations and future work: The results are numerical and require experimental validation on a CNC platform with a piezo-driven proof-mass actuator and a tool-tip displacement sensor; the single-mode structural model should be extended to multi-mode for thin-walled aerospace parts; the [8] benchmark would benefit from experimental cross-validation; adaptive online re-tuning of the ESO bandwidth or of λ and μ—for instance via reinforcement learning [34]—may further widen the operating window; and piezo hysteresis, voice-coil saturation, and other nonlinearities, together with multi-objective extensions covering control effort, surface finish, and tool wear [22], remain to be addressed for industrial deployment.

7. Conclusions

This paper presents an active chatter-suppression strategy for half-immersion end milling that combines Active Disturbance Rejection Control with a Fractional-Order PID regulator. The extended state observer estimates and rejects the lumped disturbance online; the FOPID, offline-tuned by PSO under hard actuator-saturation constraints, provides closed-loop frequency-domain shaping.

At the nominal operating point, the proposed ADRC–FOPID achieves the lowest mean RMS tip displacement (4.2 µm, 68.5% below open-loop), the lowest 90th-percentile (5.9 µm) and worst case (7.5 µm), and the lowest dispersion (CV = 33.8%) on a 38-point uncertainty sample with frozen gains. It raises the minimum stable depth from 1.00 to 2.67 mm and the maximum from 23.17 to 37.67 mm. On the Ozsoy et al. [8] benchmark, it attains the highest critical depth (7.58 mm) and the highest peak stable depth (50.92 mm) among the eight controllers tested (DVF, VPA, PID, LQR, H_∞, and μ-synthesis). The architecture therefore overcomes the nominal-versus-robust trade-off without re-tuning across operating points, offering a tractable candidate for industrial milling. Future work targets experimental validation, multi-mode structural extensions, and adaptive online re-tuning of the ESO bandwidth and of λ and μ.