Robust PI-Lead Control Design

• Robust PI-Lead Control is a strategy that combines a proportional-integral compensator with a lead element to enhance bandwidth, phase margin, and robustness.
• The method utilizes model-free tuning based on ultimate sensitivity and sustained oscillation detection, offering practical loop-shaping without explicit plant models.
• It finds applications in precision mechatronics, flight control, and robust PID design, effectively reducing overshoot and improving disturbance rejection.

Searching arXiv for the cited papers to ground the article in current metadata. {"query":"id:(Ruderman, 26 Nov 2025) OR (Ruderman, 26 Nov 2025) robust PI-Lead control", "max_results": 5} {"query":"(Ruderman, 26 Nov 2025)", "max_results": 5} to=arxiv_search 招商总代ीjson {"query":"(Ruderman, 26 Nov 2025)", "max_results": 5} Robust PI-Lead control, in the literature summarized here, denotes a family of feedback designs that combine a proportional-integral compensator with a lead element in order to obtain adequate bandwidth, phase margin and robustness under model uncertainty or limited plant knowledge. In the linear model-free formulation, the target controller is $C(s)=K_p\,\frac{T_i s+1}{T_i s}$ together with a standard lead element $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$, $0<\alpha<1$, chosen from experimental observations only (Ruderman, 26 Nov 2025). Related work treats the lead stage as a nonlinear reset element for precision mechatronics, derives discrete PI-Lead laws from incremental nonlinear dynamic inversion and time-delay control for flight-control systems, and embeds PI-Lead synthesis in robust PID parameter-space constructions (Palanikumar et al., 2018).

1. Canonical form and operating assumptions

In the model-free formulation, the plant is assumed to be type–one integrating, open-loop stable, minimum-phase, and possibly subject to an unknown pure delay $e^{-s\tau}$. The measured signals are a real-time measurable input $u(t)$ and a noisy output $x(t)$, and the plant is written as

$G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$

with $\tilde G(s)$ stable and minimum-phase (Ruderman, 26 Nov 2025). This restriction is substantive: a common misconception is that model-free PI-Lead tuning is assumption-free, whereas the stated procedure is explicitly developed for this plant class.

The same source frames the controller design around the loop $L(s)\,C(s)\,G(s)$, where the PI part provides the single-integrator structure and the lead part supplies additional phase margin. The design objective is not stated as exact pole placement but as a practical loop-shaping goal: adequate bandwidth, phase margin and robustness obtained experimentally, without internal model or identification. A plausible implication is that the method is intended for settings in which plant experiments are easier to obtain than a reliable parametric model, particularly motion-control systems.

A broader interpretation of robust PI-Lead control appears in related literature. In precision mechatronics, the lead stage can be replaced by a nonlinear reset element rather than a linear differentiator, while in flight control an incremental law can be mapped into a discrete PI-Lead increment. In robust PID theory, PI-Lead is treated as a constrained subfamily of PID controllers. This suggests that “robust PI-Lead control” is not a single synthesis doctrine but a structural class realized through several design paradigms.

2. Ultimate-sensitivity tuning without modeling

The model-free design in “Model-free practical PI-Lead control design by ultimate sensitivity principle” follows the ultimate sensitivity principle in Ziegler–Nichols style: close the loop with a pure gain $K$, increase $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$0 until the closed-loop output exhibits sustained oscillations, denote the gain by $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$1, and denote the oscillation frequency by $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$2, or equivalently measure $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$3 (Ruderman, 26 Nov 2025). The proposed PI-Lead synthesis then proceeds in three steps.

The first step determines the integrator time constant $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$4. One sets $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$5, together with any pre-gain $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$6 needed to stabilize the initial loop, applies a step reference, and gradually decreases $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$7. At the first occurrence of sustained oscillations, $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$8, one records

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$9

and then chooses

$0<\alpha<1$0

The stated interpretation is that the factor $0<\alpha<1$1 shifts the PI corner one decade below the critical phase-drop region.

The second step determines $0<\alpha<1$2 from transient overshoot and therefore from phase margin. With $0<\alpha<1$3 fixed, one varies $0<\alpha<1$4 until the closed loop exhibits a transient overshoot

$0<\alpha<1$5

in the range $0<\alpha<1$6. The overshoot is related to the damping ratio $0<\alpha<1$7 by

$0<\alpha<1$8

and the corresponding loop phase margin is

$0<\alpha<1$9

Ensuring $e^{-s\tau}$0 therefore guarantees $e^{-s\tau}$1.

The third step introduces the lead element. The method fixes $e^{-s\tau}$2, corresponding to approximately $e^{-s\tau}$3 maximum lead, and places the lead’s maximum-phase frequency at half to one decade above the PI corner, with the convenient choice

$e^{-s\tau}$4

This yields

$e^{-s\tau}$5

and the unity-DC-gain lead element

$e^{-s\tau}$6

The illustrative experiment uses a noise-perturbed voice-coil actuator with gravity and friction, sampled at $e^{-s\tau}$7. The pre-gain is $e^{-s\tau}$8, the step amplitude is $e^{-s\tau}$9, sustained oscillations appear at $u(t)$0, and the measured values are $u(t)$1 and $u(t)$2, giving

$u(t)$3

Overshoot tuning gives $u(t)$4 and $u(t)$5, so that

$u(t)$6

and the lead element becomes

$u(t)$7

3. Loop-shaping criteria, robustness margins, and empirical behavior

The loop-shaping formulation defines the loop transfer $u(t)$8 and the standard sensitivity functions

$u(t)$9

The stated robustness targets are phase margin $x(t)$0, gain margin at least $x(t)$1, and sensitivity peak

$x(t)$2

with the latter linked to disturbance rejection and robust stability (Ruderman, 26 Nov 2025).

The same work gives practical notes that clarify the intended experimental workflow. If necessary, a static pre-gain $x(t)$3 may be used, for example to counteract gravity in a motion system. In Step 1, the reference is stepped and self-oscillations are observed at low frequencies, where they stand well above sensor noise. In Step 2, percent overshoot $x(t)$4 is measured, and the paper states that noise has little effect on peak detection. In Step 3, the lead element is implemented directly from (2.5). No internal model or identification is used at any stage.

The experimental comparison reported for the voice-coil actuator states that PI-Lead, relative to plain PI, reduced overshoot by approximately $x(t)$5 and improved settling under external perturbations. Against a classic Ziegler–Nichols PID tuned via $x(t)$6, $x(t)$7, and $x(t)$8, the PI-Lead gave a smoother, less aggressive action and superior robustness at low frequencies. A plausible implication is that the added lead is being used primarily to recover phase margin that would otherwise be traded away by aggressive integral action.

4. Reset-lead realizations and mitigation of the waterbed effect

A different line of work develops a nonlinear reset-lead element for precision mechatronics, starting from the observation that industrial PID consists of Lag, Lead, and Low Pass Filters, and that linear PID is inherently bounded by the waterbed effect, which creates a trade-off between precision tracking on one side and stability robustness on the other (Palanikumar et al., 2018). The reset controller is partitioned into a resetting part $x(t)$9 and a non-resetting part $G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$0 in series. For the reset-lead, the resetting block is chosen as

$G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$1

with state-space realization

$G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$2

and reset law

$G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$3

where $G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$4 tunes the degree of nonlinearity. The non-resetting block is

$G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$5

The overall open loop is $G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$6, and the first-harmonic describing function of the reset element is

$G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$7

The design procedure is explicit. One identifies the plant $G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$8; for the Lorentz stage,

$G(s)=\frac{x(s)}{u(s)}=\frac{\tilde G(s)}{s},$9

Then one selects desired gain crossover $\tilde G(s)$0 and phase margin $\tilde G(s)$1. Two cases are given: Case A with $\tilde G(s)$2 and $\tilde G(s)$3, and Case B with $\tilde G(s)$4 and $\tilde G(s)$5. The remaining placements are $\tilde G(s)$6, $\tilde G(s)$7, and $\tilde G(s)$8. The reset-lead replaces the linear lead, $\tilde G(s)$9 is chosen to give the same $L(s)\,C(s)\,G(s)$0 from describing-function analysis, with $L(s)\,C(s)\,G(s)$1 as an example, and $L(s)\,C(s)\,G(s)$2 compensates the reset-induced corner shift. Finally, $L(s)\,C(s)\,G(s)$3 is scaled so that $L(s)\,C(s)\,G(s)$4.

The robustness rationale is stated in contrast to the linear “tamed differentiator”

$L(s)\,C(s)\,G(s)$5

In the linear case, adding phase near $L(s)\,C(s)\,G(s)$6 requires $L(s)\,C(s)\,G(s)$7, which also raises high-frequency gain and hurts precision. By contrast, the reset-lead contributes $L(s)\,C(s)\,G(s)$8 of phase for $L(s)\,C(s)\,G(s)$9 without boosting gain at high $K$0, permits $K$1, and recovers low-frequency gain for tracking. The measured improvements are reported in two ways. For Case A, with the same $K$2 and $K$3, RMS tracking error is reduced by $K$4, low-frequency $K$5 sensitivity improves, and high-frequency $K$6 is approximately $K$7 lower than for linear PID. For Case B, with the same $K$8 and the same high-frequency $K$9 as PID, $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$00 increases from $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$01 to $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$02, a $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$03 bandwidth increase.

The time-domain results use a 4th-order triangular reference with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$04 amplitude and velocity, acceleration, and jerk limits. Case A reports PID RMS error $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$05 versus Reset-PID A $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$06, and Case B reports PID RMS $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$07 versus Reset-PID B $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$08. With $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$09 white noise injected, PID maximum error is approximately $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$10, while Reset-PID A is approximately $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$11. Closed-loop identification shows that $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$12 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$13 closely match describing-function predictions. A common misconception is that the lead element in PID must be implemented as a linear differentiator; the reset-lead results show a distinct nonlinear alternative.

5. Incremental PI-Lead synthesis via INDI and time-delay control

For flight-control systems, PI-Lead appears through the equivalence between incremental nonlinear dynamic inversion (INDI) and time-delay control (TDC). The plant is written as

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$14

where $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$15 are body rates and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$16 are control deflections (B. et al., 2017). The one-step nonlinear dynamic inversion law is

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$17

with virtual input $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$18 chosen from desired error dynamics. Taking a small time delay $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$19 and performing a first-order Taylor increment around $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$20 gives

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$21

and therefore the standard INDI law

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$22

In the alternative extended form $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$23, time-delay estimation of $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$24 yields

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$25

so INDI and TDC deliver the same incremental law.

With sampled time $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$26, the discrete law is

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$27

Choosing second-order desired error dynamics with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$28 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$29 leads to the standard discrete PI-Lead increment

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$30

Term-by-term matching gives

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$31

For closed-loop tuning in terms of natural frequency and damping ratio, one chooses

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$32

so that

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$33

and therefore

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$34

The damping ratio may be mapped to phase margin through

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$35

The paper’s longitudinal pitch-rate example controls $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$36 to follow $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$37 and uses desired one-pole error dynamics $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$38 with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$39. The incremental law becomes

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$40

The equivalent PI parameters are $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$41 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$42. For $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$43 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$44, the mapping yields $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$45; with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$46 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$47, the simulation uses $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$48, $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$49, white-noise $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$50, and reports coincident INDI and PI-Lead closed-loop responses. The reported Bode plot shows a $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$51 bandwidth of approximately $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$52 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$53, consistent with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$54.

6. Robust stabilization regions and PI-Lead as a constrained PID family

A more formal robust-control treatment is provided by robust PID theory for finite sets of SISO LTI plants $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$55. The central result is that the set of all continuous-time PID controllers

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$56

that simultaneously stabilize all plants is exactly the union of convex polygonal slices in $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$57-space, each slice taken at fixed $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$58 (Bajcinca, 2013). If

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$59

then

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$60

and each $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$61 is a convex polygon whose edges arise from singular frequencies on the imaginary axis.

The continuous-time construction uses

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$62

On $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$63, with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$64 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$65, the condition $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$66 decouples into

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$67

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$68

At a singular frequency $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$69, one obtains the straight-line relation

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$70

in the $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$71-plane and the $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$72-plot relation

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$73

The algorithm computes common stabilizing $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$74-intervals, solves $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$75 for singular frequencies, forms boundary lines, computes transition signs, intersects the associated half-spaces, and returns the robustly stabilizing region as a union of polygonal slices.

Within this framework, a PI-Lead controller is written as

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$76

and embedded in standard PID form $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$77 through

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$78

The PI-Lead family therefore satisfies the linear constraints

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$79

The implementation proposed in the source is to compute the robust PID region $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$80, impose these linear equations to reduce dimensionality to a one-parameter curve, and intersect that curve with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$81 to obtain admissible PI-Lead gains. A common misconception is that robust PI-Lead tuning is necessarily heuristic; this parameter-space construction shows a complete picture of all PI-Lead controllers that robustly stabilize the given plant family.

The illustrative example uses two second-order plants,

$L(s)=\frac{\tau s+1}{\alpha \tau s+1}$82

with $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$83 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$84. The common stabilizing interval is reported as approximately $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$85. Choosing $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$86 and $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$87 gives $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$88, and a permissible interval such as $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$89 is obtained by enforcing the polygon inequalities. The Robsin toolbox is reported to implement singular-frequency detection, polygon construction, $L(s)=\frac{\tau s+1}{\alpha \tau s+1}$90-interval gridding, and robust region visualization.

Robust PI-Lead control therefore spans purely experimental tuning for type-one integrating plants, nonlinear reset realization for precision mechatronics, incremental discrete synthesis for flight control, and exact robust-stabilization region computation for plant families. This suggests a common structural theme—PI action combined with lead phase advance—while the notion of robustness varies across the literature, ranging from empirical phase-margin targets and low-frequency disturbance rejection to Lyapunov-based verification and simultaneous stabilization of multiple LTI plants.