Tunable Lead CgLp Structure

• Tunable Lead CgLp Structure is a reset-based compensator architecture that maintains nearly constant gain while providing phase lead around a prescribed frequency.
• It integrates a reset lag with a linear lead to cancel magnitude variation and exploit nonlinear reset benefits for improved precision and robustness.
• Key design parameters such as reset ratio, corner frequencies, and damping are tuned to balance phase advantage with the suppression of higher-order harmonics.

Tunable Lead CgLp Structure denotes a class of reset-based compensator architectures in which a Constant in gain, Lead in phase (CgLp) element is parameterized or structurally modified so that its first-harmonic response provides phase lead around a prescribed frequency region while maintaining approximately constant magnitude. The topic emerged from reset-control formulations that cascade a reset lag with a linear lead, with the explicit aim of circumventing the linear gain–phase limitation associated with the waterbed effect. In the subsequent literature, “tunable lead” came to include not only adjustment of the lead amount and band through parameters such as reset ratio, corner frequencies, and damping, but also architectural mechanisms that localize or suppress reset nonlinearity, including single-state reset, fractional-order reset, feedthrough augmentation, split-lead sequencing, and reset-trigger shaping (Saikumar et al., 2018, Karbasizadeh et al., 2020, Dastjerdi et al., 2020, Karbasizadeh et al., 2020, Hosseini et al., 2 Jul 2025, Hosseini et al., 7 Jul 2025, Natu et al., 11 Feb 2026).

1. Origin and defining idea

The original CgLp element was introduced as a reset lag in series with a linear lead of the same nominal order. In its first-order form, the reset lag is paired with

$L(s)=\frac{s/\omega_r+1}{s/\omega_f+1},$

while in the second-order form a GSORE-based reset lag is paired with

$$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$

The design intent is that the lead cancels the lag’s magnitude effect, whereas reset reduces the lag’s phase lag relative to the corresponding linear low-pass; the series combination therefore exhibits nearly constant gain with positive phase over a band typically identified as $[\omega_r,\omega_f]$ (Saikumar et al., 2018).

This construction was motivated by the observation that conventional linear lead compensation cannot increase phase around crossover without modifying gain elsewhere. CgLp was therefore positioned as a nonlinear replacement for part of the derivative/lead action in PID-like controllers, particularly in precision motion systems where bandwidth, robustness, tracking, and noise attenuation are simultaneously constrained by Bode’s gain–phase relationship (Saikumar et al., 2018, Dastjerdi et al., 2020).

From the outset, however, CgLp was a first-harmonic concept: “constant in gain, lead in phase” refers to the describing-function view of the nonlinear element. Later work on tunable lead CgLp structures was driven largely by the gap between that first-harmonic design picture and the actual closed-loop behavior once higher-order harmonics are included (Karbasizadeh et al., 2020, Hou et al., 2020).

2. Canonical reset formulation and frequency-domain description

The common mathematical model is the standard reset system

$$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$

where $A_\rho$ determines which states reset and by what factor. For scalar or uniform reset, the reset ratio is commonly denoted $\gamma$, with $\gamma=1$ recovering the base linear system and $|\gamma|<1$ producing partial reset (Dastjerdi et al., 2020, Hou et al., 2020).

For sinusoidal input $e(t)=E\sin(\omega t)$, the first-harmonic describing function of the reset element is written as

$$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$

and higher-order sinusoidal input describing functions satisfy

$$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$0

with even harmonics equal to zero. In the CgLp context, the total first-harmonic response is the product of the reset part and the linear lead filter, so design is ordinarily performed on $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$1, while validation increasingly relies on $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$2 or related closed-loop harmonic transfer functions (Karbasizadeh et al., 2020, Hou et al., 2020, Hosseini et al., 7 Jul 2025).

This formalism establishes the central technical distinction of the subject. A tunable lead CgLp structure is not merely a phase-lead network with adjustable corner frequencies; it is a hybrid system whose usable phase advantage depends simultaneously on the first-harmonic response and on the architecture’s ability to limit or place the resulting higher-order harmonics. The later literature treats these two layers—fundamental shaping and harmonic shaping—as separate but coupled design problems (Karbasizadeh et al., 2020, Dastjerdi et al., 2020).

3. Principal tuning variables and design logic

In first-order CgLp formulations embedded in PID-like controllers, the tunable lead portion is commonly parameterized by $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$3, with $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$4 and $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$5 fixed in one systematic framework. In that setting, $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$6 and $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$7 place the lead around crossover, $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$8 limits the high-frequency extent of the lead, and $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$9 controls reset strength: lower $[\omega_r,\omega_f]$0 generally increases phase advantage but also raises high-order harmonic content and control effort (Dastjerdi et al., 2020).

HOSIDF-based tuning work sharpened this view by introducing explicit third-harmonic indicators. For groups of CgLp designs that provide the same first-harmonic phase lead at the same crossover, the third-harmonic peak frequency $[\omega_r,\omega_f]$1 and peak magnitude $[\omega_r,\omega_f]$2 were used as selection criteria. The stated rule is that tracking-oriented tuning should prefer the largest $[\omega_r,\omega_f]$3, whereas noise-oriented tuning should prefer the smallest $[\omega_r,\omega_f]$4, which usually corresponds to the largest $[\omega_r,\omega_f]$5 within the admissible group (Hou et al., 2020).

A second harmonic-aware rule was developed for GFORE- and GSORE-based CgLp through the scalar measure

$[\omega_r,\omega_f]$6

depending on the order. Within a set of designs that all satisfy the same phase-lead requirement at bandwidth, the preferred design is the one with minimum $[\omega_r,\omega_f]$7, because $[\omega_r,\omega_f]$8 scales the low-frequency higher-order harmonics. For CgLp-SORE, setting

$[\omega_r,\omega_f]$9

cancels the low-frequency higher-order term in the stated approximation, which turns damping selection into an explicit harmonic-suppression device rather than only a phase-shaping parameter (Bahnamiri et al., 2020).

More elaborate tuning frameworks combined loop-shaping constraints with nonlinear performance metrics. One such method imposes the crossover condition, a phase-margin condition, an iso-damping condition,

$$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$0

the $$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$1 stability condition, and a modulus-margin bound on the pseudo-sensitivity $$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$2, and then minimizes

$$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$3

Here $$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$4 and $$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$5 are computed from the actual periodic steady state rather than from a fundamental-only approximation, so the tuning process explicitly internalizes harmonic distortion (Dastjerdi et al., 2020).

4. Localized linearization: single-state and fractional-order variants

A major branch of tunable lead CgLp research replaced uniform reset with partial or structured reset so that the element becomes effectively linear at a chosen frequency or over a chosen range. The second-order single-state reset element (SOSRE) is the canonical example. In that structure, only one state of the second-order lag is reset, with

$$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$6

rather than resetting both lag states. The key result is that the SOSRE-based CgLp behaves exactly as its corresponding linear second-order lag at

$$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$7

because the reset state reaches zero at reset instants. At that frequency, all higher harmonics vanish and the first harmonic coincides with the linear transfer function. The design implication stated in the paper is explicit: $$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$8 In the reported example, a plant with resonance at $$\Sigma_R := \begin{cases} \dot{x}_r(t)=A x_r(t)+B e(t), & e(t)\neq 0,\ x_r(t^+)=A_\rho x_r(t), & e(t)=0,\ u(t)=C x_r(t)+D e(t), \end{cases}$$9 led to $A_\rho$0, and with reference $A_\rho$1 the SOSRE CgLp achieved $A_\rho$2 and $A_\rho$3, compared with $A_\rho$4 and $A_\rho$5 for a conventional PID, $A_\rho$6 and $A_\rho$7 for FORE CgLp, and $A_\rho$8 and $A_\rho$9 for SORE CgLp (Karbasizadeh et al., 2020).

The fractional-order single-state reset element (FOSRE) generalizes the same idea. It resets one linear integrator state while leaving a fractional-order component linear, implemented through a CRONE approximation. Its tuning centers on the phase

$\gamma$0

If $\gamma$1, the reset state is zero at reset instants, so the element behaves linearly at $\gamma$2, and all higher-order HOSIDFs vanish there. The published tuning guideline starts with $\gamma$3 and $\gamma$4, then optimizes $\gamma$5 and $\gamma$6 subject to $\gamma$7 and minimum $\gamma$8 on $\gamma$9, while adjusting $\gamma=1$0 and then $\gamma=1$1 if the crossover-phase requirement is not met. In the reported precision-stage example, FOSRE-based CgLp designed for $\gamma=1$2 bandwidth and $\gamma=1$3 phase margin yielded much smaller low-frequency harmonics than SOSRE CgLp, significantly smaller tracking errors than PID and SOSRE at $\gamma=1$4 and $\gamma=1$5 for most cases, and smaller control peaks than SOSRE (Karbasizadeh et al., 2020).

Variant	Structural change	Tunable effect
SOSRE-based CgLp	Reset only one state of second-order lag	Exact linear behavior at $\gamma=1$6
FOSRE-based CgLp	Single-state reset plus fractional-order linear block	Suppression of nonlinearity on selected low-frequency range
Feedthrough-modified CgLp	Add $\gamma=1$7 to proportional GFORE	Nearly constant first-harmonic gain across all frequencies
Split-lead CgLp	Divide lead into pre- and post-reset parts	Balance noise amplification against higher harmonics

These variants share a common interpretation: the tunability is not limited to moving a phase bump. It includes deliberate placement of frequencies at which reset action becomes ineffective or attenuated, thereby reconciling CgLp’s nonlinear phase advantage with the need for quasi-linear behavior at disturbance, resonance, or operating frequencies where harmonics are especially harmful (Karbasizadeh et al., 2020, Karbasizadeh et al., 2020).

5. Feedthrough augmentation, sequencing, and reset-trigger shaping

A later modification introduced a proportional GFORE with nonzero feedthrough,

$\gamma=1$8

combined with

$\gamma=1$9

The stated purpose is to make the first-harmonic CgLp magnitude exactly $|\gamma|<1$0 at $|\gamma|<1$1 and $|\gamma|<1$2, and approximately constant in between, while reducing $|\gamma|<1$3 at high frequency. That same work proposed a backward calculation of $|\gamma|<1$4 from a desired phase $|\gamma|<1$5 and defined a sensitivity improvement indicator

$|\gamma|<1$6

In an industrial wire bonder application, the CgLp-based add-on with inverse notch shaping reduced the settling interval by $|\gamma|<1$7 in forward motion and $|\gamma|<1$8 in backward motion relative to the original LTI controller, and reduced stationary RMS error by $|\gamma|<1$9 and $e(t)=E\sin(\omega t)$0, respectively; a second add-on design delivered average improvements of about $e(t)=E\sin(\omega t)$1 in settling period and $e(t)=E\sin(\omega t)$2 in RMS error across three reference classes (Hosseini et al., 2 Jul 2025).

Another line of work focused on a long-standing sequencing debate. If the lead is placed before the reset element, the reset input is noise-amplified; if the lead is placed after reset, higher-order harmonics generated by the reset element are magnified. The proposed tunable lead CgLp resolves this by splitting the lead as

$e(t)=E\sin(\omega t)$3

so that

$e(t)=E\sin(\omega t)$4

The two conventional orderings appear as limits: $e(t)=E\sin(\omega t)$5 yields reset–lead, and $e(t)=E\sin(\omega t)$6 yields lead–reset. An additional $e(t)=E\sin(\omega t)$7–$e(t)=E\sin(\omega t)$8 filtering technique then targets specific higher-order sensitivity components, notably the third harmonic around $e(t)=E\sin(\omega t)$9, without increasing the reset-path gain above unity (Hosseini et al., 7 Jul 2025).

A related but distinct architecture regulates the reset trigger rather than the control path. In a dual-loop piezo-nanopositioning system, aggressively tuned CgLp blocks with $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$0 and $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$1 phase lead increased crossover but also produced pronounced higher-order harmonics and multiple-reset behavior. The remedy was a shaping filter in the reset-trigger path,

$$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$2

with a fractional lead–lag term

$$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$3

For the $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$4 case, the reported shaping parameters were $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$5, $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$6, $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$7, and $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$8. This reduced third-harmonic magnitude below $$G_1(\omega)=C(j\omega I-A)^{-1}(I+j\Theta_D(\omega))B + D,$$9, removed multiple resets at $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$00, and preserved the crossover increase. Relative to a well-tuned linear baseline, the experimental open-loop crossover increased by approximately $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$01 and the closed-loop bandwidth by about $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$02 (Natu et al., 11 Feb 2026).

6. Integration into motion-control loops and reported performance

In practical applications, tunable lead CgLp structures are almost always embedded in broader controller architectures rather than used in isolation. One systematic design on a precision “Spider” stage combined CgLp with PI, derivative lead, and low-pass filtering, with design targets $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$03, $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$04, and modulus margin $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$05. The tuned CgLp+PID controller achieved better tracking for frequencies below $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$06, better disturbance rejection, and lower high-frequency noise sensitivity than a tuned PID; the reported improvements were approximately $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$07 in tracking, $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$08 in disturbance rejection, and $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$09 in noise rejection, at the cost of higher control sensitivity because reset-induced jumps increase control effort (Dastjerdi et al., 2020).

The original precision-motion study compared linear PID, reset integrator, GFORE-based CgLp, and GSORE-based CgLp under matched bandwidth and phase-margin conditions. With fixed bandwidth $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$10 and $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$11 phase margin, the best results occurred around $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$12. In that regime, GSORE-based CgLp improved tracking RMS from $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$13 for linear PID to $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$14, and precision RMS from $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$15 to $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$16. In a bandwidth-increase scenario with preserved nominal precision and phase margin, GFORE-based CgLp pushed crossover from $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$17 to $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$18, an increase of about $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$19 (Saikumar et al., 2018).

These experimental results established two recurring facts that later tunable-lead research retained. First, the best design is rarely the most aggressive reset design. Second, structural control of higher-order harmonics can matter more than further first-harmonic phase advantage once the crossover specification is already met. This helps explain why single-state, fractional-order, feedthrough, split-lead, and shaping-filter variants all focus on reducing or relocating nonlinearity rather than simply maximizing raw reset strength (Saikumar et al., 2018, Karbasizadeh et al., 2020, Karbasizadeh et al., 2020).

7. Conceptual boundaries, debates, and recurring misconceptions

A central misconception is that “constant in gain” means exact linear unity gain. In the literature, the phrase is a first-harmonic property. Several papers state explicitly that describing-function design can be misleading when higher-order harmonics are non-negligible. The conventional SORE-based CgLp is the canonical warning case: its third harmonic can become so large that $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$20 over a broad frequency range, making the first-harmonic design picture practically invalid (Karbasizadeh et al., 2020).

A second misconception is that stronger reset always improves performance. The experimental and HOSIDF-based tuning studies do not support that claim. Lower $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$21 can increase phase advantage, but it also increases $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$22, control effort, and harmonic distortion; both the original motion-control experiments and later harmonic-aware tuning papers report performance optima at intermediate reset levels rather than at the strongest reset admissible by the model (Saikumar et al., 2018, Hou et al., 2020, Bahnamiri et al., 2020).

A third issue concerns architectural universality. The sequencing dispute—lead before reset or after reset—does not admit a single best answer independent of context. Lead before reset reduces higher-order harmonics in the closed loop but amplifies noise at the reset input; lead after reset avoids that noise amplification but magnifies the reset-generated harmonics. Split-lead and trigger-shaping approaches were introduced precisely because the preferred arrangement depends on the noise floor, the disturbance spectrum, and where the plant is most sensitive to harmonic excitation (Hosseini et al., 7 Jul 2025, Natu et al., 11 Feb 2026).

Finally, localized linearization should not be confused with global linearity. SOSRE-based CgLp is exactly linear only at $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$23; FOSRE-based CgLp suppresses nonlinearity on a designed range but retains it elsewhere; feedthrough-modified and shaping-filtered designs reduce undesirable nonlinear effects without eliminating reset dynamics altogether. The literature also emphasizes that state-space realization matters for nonlinear reset systems: changing the realization can change the behavior, so implementation must preserve the intended reset state and reset surface. Stability, correspondingly, is not inferred from classical linear margins alone; the reported methods rely on quadratic stability or $$L(s)=\frac{(s/\omega_r)^2+2\beta_r s/\omega_r+1}{(s/\omega_f)^2+2s/\omega_f+1}.$$24-type conditions, together with time-domain and harmonic-domain validation (Karbasizadeh et al., 2020, Dastjerdi et al., 2020, Karbasizadeh et al., 2020).

In that sense, the modern tunable lead CgLp structure is best understood not as a single filter topology but as a design family. Its unifying principle is constant-gain phase recovery via reset; its technical substance lies in how that phase recovery is tuned, localized, and regularized so that the nonlinear advantages survive contact with harmonics, resonances, sampling, and noise in high-performance control systems (Hosseini et al., 2 Jul 2025, Hosseini et al., 7 Jul 2025, Natu et al., 11 Feb 2026).