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Analysis of Two-Dimensional Feedback Systems over Networks Using Dissipativity

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Pith's one-line read 2-D feedback systems can stay L2-stable over digital networks

desk verdict A new 2-D dissipativity extension with real ideas, but Theorem 1's 'exact' sampling is not exact, and the example leans on it. read the letter →

arxiv 1908.01654 v1 pith:AWWZXI5R submitted 2019-08-05 eess.SY cs.SY

Yang Yan , Lanlan Su , Vijay Gupta , Panos Antsaklis This is my paper · ORCID
classification eess.SYcs.SY MSC 93C3593D2593C57
keywords two-dimensionalsystemsdissipativitypassivitynetworkedcontrolevent-triggeredlogarithmicquantizationL2stabilityRoessermodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends dissipativity and passivity tools from one-dimensional control to two-dimensional systems, whose dynamics depend on two independent coordinates such as space and time, and shows how to keep such systems stable when the feedback loop is closed over a digital network. It provides checkable conditions, mostly in the form of scalar inequalities and matrix inequalities, under which three network effects—sampling, logarithmic quantization, and event-triggered transmission—do not destroy the closed-loop $\mathcal{L}_2$ stability. If the conditions hold, a designer can guarantee stability using only passivity levels rather than a full plant model, which matters for applications such as thermal processes, image processing, and repetitive processes.

What carries the argument

The argument is carried by a 2-D version of QSR-dissipativity with a separable storage function $V(x_h,x_v)=V_h(x_h)+V_v(x_v)$, and by the reduced passivity indices called IF-OFP levels $(\rho,\nu)$. Sampling is handled through a bound on the sampling error $\Delta y_p$ in terms of the sampling periods $h_1,h_2$ and gain constants $\alpha_1,\alpha_2$; logarithmic quantization is handled through the sector bound $|\Delta v|\le \delta|v|$; and the event-triggering scheme compares the accumulated output error against a threshold derived from the same passivity levels, so that the combined supply rate remains negative definite.

What would settle it

A single numerical counterexample—a plant-controller pair satisfying the passivity inequalities of Theorem 3 and the conditions of Theorem 4 whose sampled, quantized, event-triggered closed-loop output diverges—would refute the central claim; the Section IV heat-exchanger example with $h_1=h_2=0.1$, $\delta_p=\delta_c=0.04$, $K=3$, and $N_1=40$ is the concrete configuration to test.

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Extended reading notes

Core claim

The paper's central claim is that the $\mathcal{L}_2$ stability of a 2-D feedback interconnection is preserved under the main effects of a digital link—sampling, logarithmic quantization, and event-triggered transmission—provided explicit dissipativity inequalities hold. For quantization, the closed-loop map is $\mathcal{L}_2$-stable whenever the passivity surplus of each subsystem dominates the degradation introduced by the quantizer, as expressed by the two inequalities $\rho_p + \nu_c > (\delta_p^2+2\delta_p)|\nu_c| + (1+\beta_2^2)\delta_p^2 + \frac{1}{2\beta_1}$ and its mirror image. For event-triggered communication, the paper shows that triggering at times selected by condition (22) yields finite $\mathcal{L}_2$-gain on a bounded spatial domain, so transmissions can be reduced without destroying stability.

Load-bearing premise

The load-bearing premise is Assumption 2, which restricts the event-triggered result to 2-D systems with one coordinate on a bounded spatial domain: Theorem 4 proves finite $\mathcal{L}_2$-gain only for that case, and the extension to infinite spatial domains is a sketched remark rather than a theorem.

Editorial extensions

If this is right

  • If Theorem 3 is right, a designer can guarantee $\mathcal{L}_2$ stability by tuning the quantizer densities and the auxiliary constants $\beta_1,\beta_2$, using only the plant and controller IF-OFP levels.
  • If Theorem 4 is right, event-triggered 2-D control can reduce transmissions along the vertical coordinate while preserving a finite $\mathcal{L}_2$-gain on bounded spatial domains.
  • If Theorems 1 and 2 are right, sampling periods $h_1,h_2$ can be chosen from the LMI or from the inequalities to keep the sampled system dissipative.
  • Lemma 2 extends the classical small-gain style passivity theorem to 2-D IF-OFP systems, giving a simple sufficient condition $\nu_2+\rho_1>0$ and $\nu_1+\rho_2>0$ for feedback stability.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's Remark 4 suggests the bounded-spatial-domain assumption can be relaxed by gridding the domain and applying the same trigger rule on each interval; if that extension holds, the method would apply to long spatial extents such as pipelines or convection processes, but the extension is not proved as a theorem.
  • The separable storage function $V_h+V_v$ is intentionally conservative; using less conservative 2-D Lyapunov functions could sharpen the passivity degradation bounds and allow less restrictive triggering thresholds.
  • The explicit dependence of the stability conditions on $\delta_p,\delta_c$ suggests a co-design problem: communication rate (quantizer density) and event-trigger thresholds can be traded directly against controller passivity levels.
  • The framework treats the plant and controller asymmetrically only through their passivity levels and quantizer densities, so the same conditions could be adapted to other 2-D architectures such as iterative learning control or distributed spatial systems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper develops a dissipativity-based framework for analyzing L2 stability of two-dimensional (2-D) feedback systems interconnected over a digital network. It introduces QSR-dissipativity definitions for continuous and discrete 2-D Roesser systems, then treats three network-induced effects sequentially: sampling with zero-order hold, logarithmic quantization, and event-triggered transmission. For linear sampled 2-D systems it provides an LMI condition (Theorem 1); for nonlinear systems it gives a sampling-error bound and a dissipativity degradation condition (Lemma 3 and Theorem 2). It then analyzes how logarithmic quantization degrades IF-OFP passivity levels (Theorem 3) and proposes an event-triggering rule that preserves a finite L2-gain on bounded spatial domains (Theorem 4). A hyperbolic PDE example is used to illustrate the conditions and the closed-loop behavior.

Significance. The topic is timely: networked control of 2-D systems is relatively unexplored, and the modular decomposition into sampling, quantization, and event-triggering is a sensible program. The quantization analysis in Theorem 3 and the event-triggering scheme, once its typographical inconsistencies are fixed, provide explicit and verifiable sufficient conditions, and the running PDE example is instructive. I credit the authors for stating the bounded-domain assumption explicitly in Assumption 2 and for making the triggering rule implementable as an online check of condition (22). However, the two sampling results at the foundation of the paper are not established as stated: Theorem 1's discrete model is an approximation rather than the exact sampled model, and the proof of Theorem 2 uses an inequality that fails for negative-definite Qp_hat. Because the example's passivity levels are computed from Theorem 1, the numerical validation does not currently back the advertised guarantees. The contribution can be salvaged, but the manuscript needs substantive correction.

major comments (3)
  1. [Section III-A, Theorem 1; Appendix D, Eq. (8)] The discrete model in (8) is not the exact sampled model of the continuous Roesser system (2). In the singular case covered by Remark 1, take A11=A22=0, A12=A21=1, B=0. With boundary data xh(0,z2)=e^{z2} and xv(z1,0)=e^{z1}, the exact solution is xh(z1,z2)=xv(z1,z2)=e^{z1+z2}, so xh(h1,0)=e^{h1}; the recurrence (8) gives xh(h1,0)=1+h1. These differ for h1>0. Therefore the LMI (7) certifies QSR-dissipativity of a different, approximate discrete system rather than of the actual sampled plant, and the IF-OFP levels computed in Section IV from this model do not by themselves certify stability of the true sampled feedback loop. This point needs to be fixed by deriving an exact sampled model, or by explicitly treating the discretization as an approximation with a separate error analysis.
  2. [Section III-A, Theorem 2 and Appendix C] The proof's inequality Δy_p^T Qp_hat Δy_p ≥ |λ_min(Qp_hat)| |Δy_p|^2 is valid only when Qp_hat is positive semidefinite. For example, if Qp_hat = -I, the left side is -|Δy_p|^2, which is not bounded below by |λ_min(Qp_hat)| |Δy_p|^2 = |Δy_p|^2. Since Theorem 2 does not require Qp_hat ⪰ 0 and condition (13) uses the absolute value, the theorem as stated is not proved. In addition, the second inequality in (13) writes 2(α1 h1 + α2 h2), while the proof in Appendix C uses Lemma 3's 2(α1^2 h1^2 + α2^2 h2^2); the statement and proof need to be reconciled, and the correct form should follow from Lemma 3.
  3. [Section III-C, Theorem 4 statement and proof] The displayed definition of q2 in Theorem 4 is not the same as the q2 used in the proof and in Eq. (32): the theorem statement has |ν_c| and δ_p^2 where the proof has |ν_p| and δ_c^2. This is more than a typo because the threshold in (22) depends on q1 and q2 through ϵ^2; a reader implementing the printed condition obtains a different event-triggering rule. The theorem's conclusion should also be stated as finite L2-gain on a bounded spatial domain (Definition 4), which is weaker than the global L2 stability mentioned in the abstract.
minor comments (3)
  1. [Abstract and Section I] The abstract and introduction state global L2 stability, but Theorem 4 only gives finite L2-gain on a bounded spatial domain (Definition 4); please add an explicit scope statement where the main results are summarized.
  2. [Section IV, Eq. (26)] The example writes D=∅; it should be D=0 or the corresponding zero matrix of appropriate dimensions, since the output equation y = C x + D u is otherwise underspecified.
  3. [Section III-A, Theorem 1] Theorem 1 assumes A11 and A22 are nonsingular 'for simplicity', but Remark 1 already extends the formulas to the singular case via series; please state in the theorem whether the nonsingularity assumption is part of the hypothesis or whether Remark 1 formally extends the statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability and dissipativity conditions are proved from stated assumptions rather than fitted to outcomes or imported as load-bearing self-citations.

full rationale

The derivation chain is self-contained at the level of circularity. Theorem 1 gives an LMI sufficient condition for QSR-dissipativity of a discrete model obtained from the continuous Roesser matrices; the condition is derived from the storage-function decrement and is not obtained by fitting any parameter to a targeted conclusion. Theorem 2 compares the continuous and sampled supply rates, uses Lemma 3 (proved from Assumption 1) to bound the sampling error, and then states sufficient inequalities; the dissipativity matrices Qhatp, Shatp, Rhatp appear as design variables, not as fitted outputs. Theorem 3 is a direct sector-bound calculation from IF-OFP supply rates and the logarithmic quantizer bound, giving explicit sufficient conditions; when the quantization densities vanish it reduces to Lemma 2, a standard interconnection result, not a self-citation. Theorem 4 constructs an event-triggering threshold by completing the square in the proof; the trigger condition is deliberately chosen to enforce the supply-rate inequality, so it is a synthesis condition rather than a prediction forced by the assumptions. The paper cites prior work by the same group ([19], [26], [27], [31]) for logarithmic quantization, derivative-boundedness assumptions, and componentwise quantizer operation, but these are contextual assumptions or standard tools; the load-bearing bounds and inequalities are derived in the paper from Definitions 1-3 and the stated assumptions. No fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is imported to forbid alternatives. Remark 4 explicitly describes the infinite-domain extension as a gridding sketch rather than a theorem; that is a limitation on scope, not a circular step. Whether Theorem 1's sampled model exactly represents the continuous Roesser plant is a modeling-accuracy and correctness question, not an instance of circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper relies on standard dissipativity and passivity background, two explicit domain assumptions (output-derivative bounds and bounded spatial domain), and self-authored 1-D results used as a template. No ad hoc physical entities are postulated. The free parameters are tuning constants in the sufficient conditions, not fitted physical quantities.

free parameters (4)
  • alpha_1, alpha_2
    Assumption 1 output-derivative bounds; they must be known or estimated for the nonlinear sampling theorem to be applied.
  • xi_1, xi_2, xi_3
    Positive constants in Theorem 2 that parameterize the dissipativity condition (13); they are degrees of freedom for satisfying the inequality.
  • beta_1, beta_2 = beta_1=36, beta_2=56 in example
    Positive constants in Theorems 3 and 4 that shape the quantization-degradation inequalities and the event-trigger design; chosen by the designer.
  • theta_1, theta_2 = theta_1=theta_2=0.5 in example
    Event-triggering tuning constants in (0,1) used in Theorem 4's triggering condition.
assumptions (6)
  • domain assumption The proposed 2-D QSR-dissipativity definition with separable additive storage function V = Vh + Vv captures the relevant energy balance.
    Definitions 1 and 2; the paper explicitly acknowledges this form is conservative, so the analysis is restricted to systems and storage functions of this separable type.
  • standard math Lemma 1: QSR-dissipativity with Q < 0 implies L2 stability for 2-D systems.
    Proof omitted, stated to follow the 1-D results in [14] and [23]; used in Theorem 3 to conclude closed-loop stability.
  • standard math Lemma 2: feedback interconnection of two IF-OFP 2-D systems is L2 stable when nu_2 + rho_1 > 0 and nu_1 + rho_2 > 0.
    Proof omitted, cited as following 1-D passivity results; the zero-quantization case of Theorem 3 reduces to this lemma.
  • domain assumption Assumption 1: the output directional derivatives are bounded in L2 by alpha_1, alpha_2 times the input norm.
    Introduced in Section III-A; necessary for the sampling-error bound in Lemma 3 and for Theorem 2.
  • domain assumption Assumption 2: one coordinate of the 2-D system is a bounded spatial domain.
    Introduced in Section III-C; Theorem 4 proves finite L2-gain only for this bounded spatial domain.
  • standard math The logarithmic quantizer satisfies the sector bound |Delta v| <= delta |v|.
    Equations (14)-(16); standard for logarithmic quantization and used throughout the quantization and event-triggering proofs.

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Pith. "Pith review of Analysis of Two-Dimensional Feedback Systems over Networks Using Dissipativity." pith.science (2026). https://pith.science/paper/AWWZXI5R

@misc{pith2026190801654,
  author       = {Pith},
  title        = {Pith review of: Analysis of Two-Dimensional Feedback Systems over Networks Using Dissipativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AWWZXI5R}},
  note         = {Machine review of arXiv:1908.01654}
}
abstract

This paper investigates the closed-loop $\mathcal{L}_2$ stability of two-dimensional (2-D) feedback systems across a digital communication network by introducing the tool of dissipativity. First, sampling of a continuous 2-D system is considered and an analytical characterization of the $QSR$-dissipativity of the sampled system is presented. Next, the input-feedforward output-feedback passivity (IF-OFP), a simplified form of $QSR$-dissipativity, is utilized to study the framework of feedback interconnection of two 2-D systems over networks. Then, the effects of signal quantization in communication links on dissipativity degradation of the 2-D feedback quantized system is analyzed. Additionally, an event-triggered mechanism is developed for 2-D networked control systems while maintaining $\mathcal{L}_2$ stability of the closed-loop system. In the end, an illustrative example is provided.

Figures

Figures reproduced from arXiv: 1908.01654 by the authors.

Figure 1
Figure 1. Feedback Interconnection of Two 2-D Systems [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A 2-D system Interconnected with a Controller over a Digital Network [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Logarithmic Quantizer shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: The trajectory of output yˆp(i, j) Firstly, let us estimate the (Qˆ p, Sˆ p, Rˆ p)-dissipativity of the sampled system Gˆ p corresponding to the continuous 2-D system (26). Since the system Gp described in (26) has single input and single output, the (Qˆ p, Sˆ p, Rˆ p)…
Figure 6
Figure 6. Figure 6: Trajectories of transmitted output of Gˆ p: (a)without digital network; (b) with quantizers; (c) with quantizers and event triggered communication [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Triggering instants along j coordinate feedback controller, the trajectory of the system output under quantization converges to zero as i and j increase. In the end, we consider the complete framework in [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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