pith. sign in

arxiv: 2411.01793 · v2 · pith:IPFDKHAKnew · submitted 2024-11-04 · 🧮 math.OC

H₂-Optimal Estimation of Linear Delayed and PDE Systems

Danio Braghini , Sachin Shivakumar , Matthew M. Peet This is my paper

Pith reviewed 2026-05-23 18:07 UTC · model grok-4.3

classification 🧮 math.OC
keywords H2-optimal estimationpartial integral equationsPDE systemsdelay systemsobserver synthesislinear partial integral inequalitiesconvex optimization
0
0 comments X

The pith

Re-characterizing the H2 norm as an initial-condition map and using PIE representations turns H2-optimal observer design for PDE and delay systems into a convex LPI problem.

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

The paper establishes a method to design H2-optimal observers for linear systems with delays and PDEs. These systems lack the transfer functions and state-space forms needed for standard H2 estimation techniques. The approach first redefines the H2 norm in terms of a map from initial conditions to outputs. It then uses the partial integral equation representation to convert the problem into convex optimization over linear partial integral inequalities. A reader would care because this offers a systematic way to synthesize estimators for distributed systems where conventional tools do not apply.

Core claim

By re-characterizing the H2-norm in terms of a map from initial condition to output and leveraging the Partial Integral Equation (PIE) state-space representation of systems of linear PDEs coupled with ODEs, this characterization of the H2-norm is recast as a convex optimization problem defined in terms of Linear Partial Integral (LPI) inequalities. A class of PIE-based observers is then parameterized, the observer synthesis problem is recast as an LPI, and the resulting observers are validated using numerical simulation.

What carries the argument

The PIE state-space representation, which converts the initial-condition-to-output H2-norm map into linear partial integral inequalities that support convex optimization for observer synthesis.

If this is right

  • The H2-optimal estimation problem for these systems becomes a convex LPI optimization.
  • A parameterized class of PIE-based observers can be synthesized directly from the LPI.
  • The method applies to systems of linear PDEs coupled with ODEs.
  • Resulting observers can be checked through numerical simulation.

Where Pith is reading between the lines

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

  • The same initial-condition map might allow other performance criteria to be handled via similar LPI problems if the underlying inequalities can be derived.
  • Practical deployment would depend on the availability of numerical solvers capable of handling the resulting LPIs at scale.
  • The representation could open routes to joint observer-controller design within the same convex framework.

Load-bearing premise

The H2-norm characterization via the initial-condition-to-output map combined with the PIE representation accurately captures estimation performance for linear delayed and PDE systems without transfer functions or standard state-space forms.

What would settle it

A numerical simulation of a specific PDE or delay system in which an LPI-synthesized observer fails to achieve the H2 performance level predicted by the optimization would settle the claim.

Figures

Figures reproduced from arXiv: 2411.01793 by Danio Braghini, Matthew M. Peet, Sachin Shivakumar.

Figure 1
Figure 1. Figure 1: Numerical estimation of an H2-optimal estimator for an unstable reaction-diffusion equation (Eq. (15)) using measurement at the boundary along with process and sensor disturbance w(t) = sin(100t) and PDE initial condition ξ(0, s) = −s 2/2. (a): Evolution of error in estimate of the PDE state T e(t) = T x˜(t) − ξ(t). (b): Evolution of the regulated output (z(t)) of both estimator and PDE. This PDE is entere… view at source ↗
read the original abstract

The $H_2$ norm is a commonly used performance metric in the design of estimators. However, $H_2$-optimal estimation of most PDEs is complicated by the lack of transfer function and state-space representations. To address this problem, we first re-characterize the $H_2$-norm in terms of a map from initial condition to output. We then leverage the Partial Integral Equation (PIE) state-space representation of systems of linear PDEs coupled with ODEs to recast this characterization of $H_2$-norm as a convex optimization problem defined in terms of Linear Partial Integral (LPI) inequalities. We then parameterize a class of PIE-based observers and solve the associated $H_2$-optimal estimation problem. The observer synthesis problem is then recast as an LPI, and the resulting observers are validated using numerical simulation.

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

1 major / 0 minor

Summary. The paper claims that the H2 norm for estimation can be re-characterized as a map from initial conditions to output for linear delayed and PDE systems. Using the Partial Integral Equation (PIE) representation, this is recast as a convex LPI optimization problem. A class of PIE-based observers is parameterized, the H2-optimal estimation problem is solved via LPI, and the resulting observers are validated numerically.

Significance. If the initial-condition re-characterization is shown to be equivalent to the standard input-to-output H2 norm (including disturbance effects), the approach would enable convex synthesis of optimal estimators for infinite-dimensional systems without requiring transfer functions or finite-dimensional state-space forms. The PIE-to-LPI conversion and numerical validation are noted strengths, but the equivalence is load-bearing for the central claim.

major comments (1)
  1. [Abstract] Abstract: The re-characterization of the H2-norm strictly via the initial-condition-to-output map for the autonomous error dynamics must be shown to incorporate the precise disturbance feedthrough operators from the PIE representation of the plant. If it does not, the resulting LPI solves a different problem than standard H2-optimal estimation under process/measurement noise.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need to confirm equivalence between our initial-condition re-characterization and the standard input-to-output H2 norm. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The re-characterization of the H2-norm strictly via the initial-condition-to-output map for the autonomous error dynamics must be shown to incorporate the precise disturbance feedthrough operators from the PIE representation of the plant. If it does not, the resulting LPI solves a different problem than standard H2-optimal estimation under process/measurement noise.

    Authors: We agree that explicit confirmation is required. In the manuscript the error system is itself expressed as a PIE whose operators already embed the original plant's disturbance feedthrough terms (see the construction of the error PIE in Section 3). The H2-norm re-characterization is then applied directly to this PIE: the initial-condition distribution is chosen according to the input operator of the error PIE, which includes all feedthrough contributions. Consequently the autonomous initial-condition-to-output map recovers the full input-to-output H2 norm, including any direct feedthrough. To make this equivalence transparent we will add a short lemma (or remark) immediately after the re-characterization statement and update the abstract to reference it. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reformulation is independent of fitted inputs

full rationale

The derivation begins with a re-characterization of the H2 norm as an initial-condition-to-output map for autonomous error dynamics, then applies the existing PIE representation to convert the resulting quadratic cost into an LPI feasibility problem. This step is a change of representation rather than a self-definition or fitted-parameter renaming. No load-bearing premise reduces to a self-citation chain, an ansatz smuggled via prior work, or a prediction that is forced by construction from the same data used to define the observer. The observer parameterization and LPI synthesis are presented as direct consequences of the PIE state-space form, which is treated as an external modeling tool. The paper therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the approach assumes PIE representations exist for the systems (likely drawn from prior literature by the authors) and that LPI inequalities can be solved numerically without additional free parameters specified here.

axioms (1)
  • domain assumption Systems of linear PDEs coupled with ODEs admit a PIE state-space representation.
    Invoked to recast the H2-norm and observer problem; location: abstract description of leveraging PIE.

pith-pipeline@v0.9.0 · 5681 in / 1143 out tokens · 25540 ms · 2026-05-23T18:07:06.978575+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Adap- tive and model-based control theory applied to convectively unstable flows,

    N. Fabbiane, O. Semeraro, S. Bagheri, and D. S. Henningson, “Adap- tive and model-based control theory applied to convectively unstable flows,” Applied Mechanics Reviews , vol. 66, no. 6, p. 060801, 2014

    work page 2014

  2. [2]

    H2 optimal actuator and sensor place- ment in the linearised complex Ginzburg–Landau system,

    K. K. Chen and C. W. Rowley, “ H2 optimal actuator and sensor place- ment in the linearised complex Ginzburg–Landau system,” Journal of Fluid Mechanics, vol. 681, pp. 241–260, 2011

    work page 2011

  3. [3]

    A computational scheme for the optimal sensor/actuator placement of flexible structures using spatial H2 measures,

    W. Liu, Z. Hou, and M. A. Demetriou, “A computational scheme for the optimal sensor/actuator placement of flexible structures using spatial H2 measures,” Mechanical systems and signal processing , vol. 20, no. 4, pp. 881–895, 2006

    work page 2006

  4. [4]

    An H2 norm approach for the actuator and sensor placement in vibration control of a smart structure,

    P. Ambrosio, F. Resta, and F. Ripamonti, “An H2 norm approach for the actuator and sensor placement in vibration control of a smart structure,” Smart Materials and Structures, vol. 21, no. 12, p. 125016, 2012

    work page 2012

  5. [5]

    The exploration of chemical reaction networks,

    J. P. Unsleber and M. Reiher, “The exploration of chemical reaction networks,” Annual review of physical chemistry , vol. 71, no. 1, pp. 121–142, 2020

    work page 2020

  6. [6]

    Time-delay systems: an overview of some recent advances and open problems,

    J.-P. Richard, “Time-delay systems: an overview of some recent advances and open problems,” automatica, vol. 39, no. 10, pp. 1667– 1694, 2003

    work page 2003

  7. [7]

    A convex Sum-of-Squares approach to analysis, state feedback and output feedback control of parabolic PDEs,

    A. Gahlawat and M. Peet, “A convex Sum-of-Squares approach to analysis, state feedback and output feedback control of parabolic PDEs,” IEEE Transactions on Automatic Control , vol. 62, no. 4, pp. 1636–1651, 2017

    work page 2017

  8. [8]

    Estimator-based output-feedback stabilization of linear multi-delay systems using SOS,

    S. Wu, C.-C. Hua, and M. Peet, “Estimator-based output-feedback stabilization of linear multi-delay systems using SOS,” in Proceedings of the IEEE Conference on Decision and Control , 2019

    work page 2019

  9. [9]

    Full-state feedback of delayed systems using SOS: A new theory of duality,

    M. Peet, “Full-state feedback of delayed systems using SOS: A new theory of duality,” inProceedings of the 11th IFAC Workshop on Time- Delay Systems, 2013

    work page 2013

  10. [10]

    Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations,

    A. Smyshlyaev and M. Krstic, “Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations,” IEEE Trans- actions on Automatic Control , vol. 49, no. 12, pp. 2185–2202, 2004

    work page 2004

  11. [11]

    Feedback control of hyperbolic PDE systems,

    P. Christofides and P. Daoutidis, “Feedback control of hyperbolic PDE systems,” AIChE Journal, vol. 42, no. 11, pp. 3063–3086, 1996

    work page 1996

  12. [12]

    Optimal observer-based fault detection and estimation approaches for T–S fuzzy systems,

    L. Li, S. X. Ding, and X. Peng, “Optimal observer-based fault detection and estimation approaches for T–S fuzzy systems,” IEEE Transactions on Fuzzy Systems , vol. 30, no. 2, pp. 579–590, 2020

    work page 2020

  13. [13]

    Fault detection and diagnosis based on modeling and estimation methods,

    S. Huang and K. K. Tan, “Fault detection and diagnosis based on modeling and estimation methods,” IEEE transactions on neural networks, vol. 20, no. 5, pp. 872–881, 2009

    work page 2009

  14. [14]

    Optimal fault detection design via iterative estimation methods for industrial control systems,

    L. Li, S. X. Ding, Y . Zhang, and Y . Yang, “Optimal fault detection design via iterative estimation methods for industrial control systems,” Journal of the franklin institute , vol. 353, no. 2, pp. 359–377, 2016

    work page 2016

  15. [15]

    Backstepping observers for a class of parabolic pdes,

    A. Smyshlyaev and M. Krstic, “Backstepping observers for a class of parabolic pdes,” Systems & Control Letters , vol. 54, no. 7, pp. 613– 625, 2005

    work page 2005

  16. [16]

    Estimation of heat source term and thermal diffusion in tokamak plasmas using a Kalman filtering method in the early lumping approach,

    S. Mechhoud, E. Witrant, L. Dugard, and D. Moreau, “Estimation of heat source term and thermal diffusion in tokamak plasmas using a Kalman filtering method in the early lumping approach,” IEEE Transactions on Control Systems Technology, vol. 23, no. 2, pp. 449– 463, 2014

    work page 2014

  17. [17]

    Active control of flexible systems,

    M. Balas, “Active control of flexible systems,” Journal of Optimization theory and Applications , vol. 25, no. 3, pp. 415–436, 1978

    work page 1978

  18. [18]

    A dynamic state observer for real-time reconstruction of the tokamak plasma profile state and disturbances,

    F. Felici, M. De Baar, and M. Steinbuch, “A dynamic state observer for real-time reconstruction of the tokamak plasma profile state and disturbances,” in 2014 American Control Conference . IEEE, 2014, pp. 4816–4823

    work page 2014

  19. [19]

    Reduced basis method for H2 optimal feedback control problems,

    A. Schmidt and B. Haasdonk, “Reduced basis method for H2 optimal feedback control problems,” IFAC-PapersOnLine, vol. 49, no. 8, pp. 327–332, 2016

    work page 2016

  20. [20]

    Numerical solution of the infinite- dimensional LQR problem and the associated Riccati differential equations,

    P. Benner and H. Mena, “Numerical solution of the infinite- dimensional LQR problem and the associated Riccati differential equations,” Journal of Numerical Mathematics , vol. 26, no. 1, pp. 1–20, 2018

    work page 2018

  21. [21]

    Bound- ary optimal (LQ) control of coupled hyperbolic PDEs and ODEs,

    A. A. Moghadam, I. Aksikas, S. Dubljevic, and J. F. Forbes, “Bound- ary optimal (LQ) control of coupled hyperbolic PDEs and ODEs,” Automatica, vol. 49, no. 2, pp. 526–533, 2013

    work page 2013

  22. [22]

    A partial integral equation (PIE) representation of coupled linear PDEs and scalable stability analysis using LMIs,

    M. Peet, “A partial integral equation (PIE) representation of coupled linear PDEs and scalable stability analysis using LMIs,” Automatica, vol. 125, p. 109473, 2021

    work page 2021

  23. [23]

    Shivakumar, A

    S. Shivakumar, A. Das, S. Weiland, and M. Peet, “Extension of the Partial Integral Equation representation to GPDE input-output systems,” arXiv, no. 2205.03735, 2022

    work page arXiv 2022

  24. [24]

    l2-gain analysis of coupled linear 2D PDEs using Linear PI Inequalities,

    D. Jagt and M. Peet, “ l2-gain analysis of coupled linear 2D PDEs using Linear PI Inequalities,” in Proceedings of the IEEE Conference on Decision and Control , 2022, pp. 6097–6104

    work page 2022

  25. [25]

    Representation of networks and systems with delay: DDEs, DDFs, ODE–PDEs and PIEs,

    M. Peet, “Representation of networks and systems with delay: DDEs, DDFs, ODE–PDEs and PIEs,” Automatica, vol. 127, p. 109508, 2021

    work page 2021

  26. [26]

    H∞optimal estimation for linear coupled PDE systems,

    A. Das, S. Shivakumar, S. Weiland, and M. Peet, “ H∞optimal estimation for linear coupled PDE systems,” in Proceedings of the IEEE Conference on Decision and Control , 2019

    work page 2019

  27. [27]

    H∞-optimal estimator synthesis for coupled linear 2D PDEs using convex optimization,

    D. S. Jagt and M. M. Peet, “ H∞-optimal estimator synthesis for coupled linear 2D PDEs using convex optimization,” arXiv, no. 2402.05061, 2024

    work page arXiv 2024

  28. [28]

    H∞-optimal observer design for systems with multiple delays in states, inputs and outputs: A pie approach,

    S. Wu, M. M. Peet, F. Sun, and C. Hua, “ H∞-optimal observer design for systems with multiple delays in states, inputs and outputs: A pie approach,” International Journal of Robust and Nonlinear Control , vol. 33, no. 8, pp. 4523–4540, 2023

    work page 2023

  29. [29]

    PIETOOLS 2022: User manual,

    S. Shivakumar, D. Jagt, D. Braghini, A. Das, and M. Peet, “PIETOOLS 2022: User manual,” arXiv, no. 2101.02050, 2021

    work page arXiv 2022

  30. [30]

    Computing optimal upper bounds on the H2-norm of ODE-PDE systems using linear Partial Inequalities,

    D. Braghini and M. Peet, “Computing optimal upper bounds on the H2-norm of ODE-PDE systems using linear Partial Inequalities,” IFAC-PapersOnLine, vol. 56, no. 2, pp. 6994–6999, 2023

    work page 2023

  31. [31]

    G. E. Dullerud and F. Paganini, A course in robust control theory: a convex approach. Springer Science & Business Media, 2013, vol. 36

    work page 2013

  32. [32]

    Dual Representations and $H_{\infty}$-Optimal Control of Partial Differential Equations

    S. Shivakumar, A. Das, S. Weiland, and M. Peet, “H∞-optimal control of coupled ODE-PDE systems using PIE framework and LPIs,” arXiv, no. 2208.13104, 2022

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  33. [33]

    A new treatment of boundary conditions in PDE solution with Galerkin Methods via Partial Integral Equation Framework,

    Y . Peet and M. Peet, “A new treatment of boundary conditions in PDE solution with Galerkin Methods via Partial Integral Equation Framework,”Journal of Computational and Applied Mathematics, vol. 442, p. 115673, 2024

    work page 2024

  34. [34]

    P. L. Gould and Y . Feng, Introduction to linear elasticity . Springer, 1994, vol. 2

    work page 1994