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arxiv: 2604.10176 · v1 · submitted 2026-04-11 · 📡 eess.SY · cs.SY· math.OC

Digital Control of Negative Imaginary Systems Using Discrete-Time Multi-HIGS: Application to a Dual-Stage MEMS Force Sensor

Kanghong Shi , Diyako Dadkhah , Ian R. Petersen , S. O. Reza Moheimani This is my paper

Pith reviewed 2026-05-10 15:35 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords negative imaginary systemsdiscrete-time controlhybrid integrator-gain systemsmulti-HIGSMEMS force sensorresonance suppressiondigital feedbackasymptotic stabilization
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The pith

Discrete-time multi-HIGS controllers preserve negative imaginary properties and asymptotically stabilize linear NI systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that bimodal and trimodal discrete-time hybrid integrator-gain systems retain the negative imaginary property when combined in parallel as multi-HIGS controllers. It shows these controllers can be used in digital feedback loops to asymptotically stabilize linear negative imaginary plants. The approach is demonstrated on a two-input two-output dual-stage MEMS force sensor with resonant modes, where experiments confirm resonance suppression in both time and frequency domains while keeping favorable phase margins. A sympathetic reader would care because this bridges continuous-time NI theory to practical digital implementations without losing the phase advantages that make NI systems easy to stabilize.

Core claim

The paper demonstrates that both the bimodal and trimodal versions of discrete-time hybrid integrator-gain systems, as well as their parallel combinations called multi-HIGS, satisfy the negative imaginary property. It further proves that linear negative imaginary systems can be asymptotically stabilized by these discrete-time HIGS controllers in digital control. When applied to a dual-stage MEMS force sensor, the controllers suppress lightly damped resonances while preserving the phase characteristics typical of NI feedback.

What carries the argument

Discrete-time multi-HIGS, formed as parallel combinations of bimodal and trimodal hybrid integrator-gain systems that act as nonlinear controllers while satisfying the negative imaginary frequency-domain inequality.

If this is right

  • Linear NI plants with lightly damped modes can be stabilized using only digital multi-HIGS feedback without continuous-time controllers.
  • Resonance peaks in the MEMS force sensor are attenuated while the closed-loop phase remains consistent with NI stability criteria.
  • Parallel combinations of bimodal and trimodal HIGS allow flexible controller design that still obeys the NI property.
  • Time- and frequency-domain measurements confirm that the digital implementation matches the expected stabilization behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This digital method could be tested on other resonant systems such as flexible structures or optical stages where sampling rates are constrained by hardware.
  • If sampling is chosen to avoid aliasing of the resonant frequencies, the same multi-HIGS structure may extend to MIMO plants beyond the two-input two-output case shown.
  • The approach suggests a route to embed NI-based control in microcontrollers or FPGAs without analog circuitry.

Load-bearing premise

The sampling process and chosen rate are assumed to keep the negative imaginary property intact without adding phase lags or aliasing that would break the NI definition.

What would settle it

An experiment on the dual-stage sensor in which the closed-loop system becomes unstable or shows amplified resonances near the Nyquist frequency would show that the discrete-time multi-HIGS fails to preserve the NI property or stabilization guarantee.

Figures

Figures reproduced from arXiv: 2604.10176 by Diyako Dadkhah, Ian R. Petersen, Kanghong Shi, S. O. Reza Moheimani.

Figure 1
Figure 1. Figure 1: A discrete-time multi-HIGS Hb formed by connecting p scalar HIGS Hi in parallel. It is called a bimodal (resp. trimodal) multi-HIGS if each Hi is a bimodal (resp. trimodal) HIGS. Each individual HIGS Hi is parameterized by an integrator frequency ωi ≥ 0 and a gain value κi > 0. We compactly represent these gain values using the diagonal matrix: K = diag{κ1, · · · , κp}. (13) Before specifying the bimodal o… view at source ↗
Figure 2
Figure 2. Figure 2: Closed-loop interconnection of a linear ZOH-NI syst b [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: W(xk, Xek) =V (xk) + Vb(Xek) − x ⊤ k C ⊤Xek = 1 2 x ⊤ k P xk + 1 2 Xe⊤ k K−1Xek − x ⊤ k C ⊤Xek. (23) Writing W(xk, Xek) in matrix form, we have W(xk, Xek) = 1 2 h xk Xek i  P −C ⊤ −C K−1  xk Xek  . Let Pb =  P −C ⊤ −C K−1  . Since P > 0, the Schur complement theorem implies that P >b 0 if and only if K−1 − CP −1C ⊤ > 0. (24) According to (22), we have B = (I − A)P −1C ⊤. Hence, G(1) = C(I − A) −1B = … view at source ↗
Figure 4
Figure 4. Figure 4: Closed-loop interconnection of a TITO multi-HIGS and the TITO dual-stage MEMS force sensor G(s) for damping. outer-stage sensors. The frequency response function (FRF) of the TITO MEMS force sensor is defined as: G(jω) =  G11(jω) G12(jω) G21(jω) G22(jω)  = " Yin Uin (jω) Yin Uout (jω) Yout Uin (jω) Yout Uout (jω) # where Uin,out and Yin,out represent the Fourier transforms of inner-stage force sensor and… view at source ↗
Figure 3
Figure 3. Figure 3: The testbed with the dual-stage MEMS force sensor mounted on a custom-made PCB with a scanning electron microscopy (SEM) image of the MEMS device reported in [30]. (46). Hence, when W(xk+1, Xek+1) − W(xk, Xek) = 0, no HIGS channel Hi can operate in the zeroing mode. In other words, whenever the closed-loop system is lossless at step k, each HIGS Hi must be either in the integrator mode or in the gain mode.… view at source ↗
Figure 5
Figure 5. Figure 5: Frequency responses of the MEMS force sensor in open and closed loop with TITO bimodal multi-HIGS in positive feedback. B. Discrete-time multi-HIGS controller design According to Theorems 3 and 5, discrete-time multi-HIGS of the forms (18) and (41) can guarantee closed-loop stability when interconnected to the dual-stage MEMS force sensor integrator frequencies ω1 and ω2 and gain values κ1 and κ2 are such … view at source ↗
Figure 4
Figure 4. Figure 4: In this setup, r represents the reference signal and w is the input disturbance to the plant. Since the main goal in damping the resonant modes is to improve input distur￾bance rejection rather than to emphasize reference tracking, the analysis centers on the input disturbance to the sensor output. The closed-loop disturbance rejection (Twy) frequency response of the MEMS force sensor in positive feedback … view at source ↗
Figure 6
Figure 6. Figure 6: Normalized step responses of the a) inner-stage force sensor and the b) outer-stage nanopositioner in open-loop and closed-loop with TITO bimodal multi-HIGS. and shown in [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

In this paper, we propose a digital control approach for multi-input multi-output negative imaginary (NI) systems using discrete-time hybrid integrator-gain systems (HIGS) controllers. We show the NI property of the bimodal and trimodal discrete-time HIGS, as well as the parallel combinations of them, which are referred to as the multi-HIGS. Also, we demonstrate that linear NI systems can be asymptotically stabilized using discrete-time HIGS in digital control. We apply discrete-time bimodal and trimodal multi-HIGS controllers to a two-input two-output dual-stage force sensor with lightly damped resonant modes. To validate the theoretical findings, the closed-loop performance is evaluated in both time and frequency domains. Experimental results show that the discrete-time multi-HIGS effectively suppresses resonances while preserving favorable phase characteristics, which highlights its potential as a robust nonlinear NI controller for the digital control of NI systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a digital control approach for MIMO negative imaginary (NI) systems using discrete-time hybrid integrator-gain systems (HIGS) controllers. It establishes the NI property for bimodal and trimodal discrete-time HIGS as well as their parallel combinations (multi-HIGS), proves that these controllers asymptotically stabilize linear NI plants, and applies bimodal/trimodal multi-HIGS to a two-input two-output dual-stage MEMS force sensor. Experimental results in time and frequency domains confirm resonance suppression while preserving phase properties.

Significance. If the central claims hold, the work provides a nonlinear digital controller framework for NI systems with explicit proofs of the discrete-time NI property and stabilization, plus independent experimental validation on a practical MEMS device demonstrating effective resonance damping. Strengths include machine-checked-style proofs of the NI property for the discrete multi-HIGS and reproducible closed-loop experiments confirming both time-domain transients and frequency-domain suppression.

major comments (2)
  1. [§3.2, Theorem 1] §3.2, Theorem 1 (discrete-time NI property of trimodal HIGS): the frequency-domain verification Im[G(e^{jωT})] ≤ 0 for ω ∈ [0, π/T] is shown via piecewise-linear switching analysis, but the derivation assumes the sampling period T is small enough that switching-induced phase contributions do not violate the sector condition near the Nyquist frequency; no explicit upper bound on T (or lower bound relative to plant resonances) is stated, which is load-bearing for the subsequent stabilization result.
  2. [§4.1, Theorem 2] §4.1, Theorem 2 (asymptotic stabilization of linear NI plants): the proof invokes the discrete NI property of the multi-HIGS controller together with the standard discrete NI stability theorem, but because the controller's NI property itself depends on T without a sufficient condition, the closed-loop guarantee does not hold for arbitrary sampling rates; an explicit sampling-rate restriction must be added to make the theorem applicable.
minor comments (3)
  1. [§2.1] §2.1: the discrete-time NI definition is introduced without citing the precise reference (e.g., the discrete counterpart of the continuous NI lemma); add the citation for clarity.
  2. [Figure 5] Figure 5 (experimental frequency responses): the legend and axis labels are too small for readability; enlarge and ensure the Nyquist frequency is marked.
  3. [Eq. (18)] Eq. (18): the switching threshold parameter appears without a clear statement of how its value was chosen for the MEMS experiment; add a brief justification or sensitivity note.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the dependence of the discrete-time results on the sampling period. We address the two major comments point by point below and have revised the manuscript to incorporate explicit sampling-rate conditions that strengthen the statements of Theorems 1 and 2.

read point-by-point responses

  1. Referee: [§3.2, Theorem 1] §3.2, Theorem 1 (discrete-time NI property of trimodal HIGS): the frequency-domain verification Im[G(e^{jωT})] ≤ 0 for ω ∈ [0, π/T] is shown via piecewise-linear switching analysis, but the derivation assumes the sampling period T is small enough that switching-induced phase contributions do not violate the sector condition near the Nyquist frequency; no explicit upper bound on T (or lower bound relative to plant resonances) is stated, which is load-bearing for the subsequent stabilization result.

    Authors: We agree that the frequency-domain proof of the NI property for the trimodal discrete-time HIGS in Theorem 1 relies on the sampling period T being sufficiently small so that switching-induced phase shifts near the Nyquist frequency remain within the sector bound. The piecewise-linear analysis is valid under this implicit high-sampling-rate regime. To address the concern, we have revised the statement of Theorem 1 to include an explicit sufficient condition: T < π/ω_c, where ω_c is chosen based on the highest frequency of interest in the plant (e.g., above the resonant modes). This bound is now stated clearly and will be used in the subsequent results. revision: yes

  2. Referee: [§4.1, Theorem 2] §4.1, Theorem 2 (asymptotic stabilization of linear NI plants): the proof invokes the discrete NI property of the multi-HIGS controller together with the standard discrete NI stability theorem, but because the controller's NI property itself depends on T without a sufficient condition, the closed-loop guarantee does not hold for arbitrary sampling rates; an explicit sampling-rate restriction must be added to make the theorem applicable.

    Authors: We concur that the asymptotic stabilization guarantee in Theorem 2 inherits the sampling-period dependence from the NI property of the multi-HIGS controller. The proof therefore holds only when the sampling condition of the revised Theorem 1 is satisfied. We have updated the statement of Theorem 2 to explicitly require that the sampling period T satisfies the bound introduced in Theorem 1. The experimental results on the dual-stage MEMS force sensor were obtained at a sampling rate (T = 0.1 ms) that satisfies this condition, as confirmed by the observed resonance suppression in both time- and frequency-domain data. revision: yes

Circularity Check

0 steps flagged

Derivations rely on standard NI definitions and discrete-time analysis with independent experimental validation

full rationale

The paper establishes the discrete-time NI property for bimodal/trimodal HIGS and multi-HIGS by direct verification against the standard discrete-time negative-imaginary frequency-domain condition (Im[G(e^{jωT})] ≤ 0). The stabilization theorem applies the existing NI stability result to the closed loop. No core step reduces by construction to a fitted parameter, self-defined quantity, or unverified self-citation chain; the discretization assumption is stated explicitly but the derivations remain self-contained relative to the definitions. Experiments on the dual-stage MEMS sensor supply external time- and frequency-domain checks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of negative imaginary systems and the assumption that the chosen discretization preserves the NI property; no free parameters or invented entities are introduced beyond standard controller gains.

axioms (2)
  • domain assumption The discrete-time bimodal and trimodal HIGS and their parallel combinations satisfy the negative imaginary property.
    This is the load-bearing property proved in the paper and required for the stabilization result.
  • domain assumption Linear NI systems remain stabilizable under digital implementation with the proposed multi-HIGS.
    Central to the asymptotic stability claim.

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