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arxiv: 2508.08844 · v2 · submitted 2025-08-12 · 🧮 math.OC · cs.SY· eess.SY

Fundamental limitations of monotonic tracking systems

Hamed Taghavian This is my paper

Pith reviewed 2026-05-18 23:06 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords monotonic trackingfundamental limitationsSISO linear systemsoutput feedbackplant zeroscontroller orderdecay rategeometric constraints
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The pith

Necessary and sufficient conditions for monotonic tracking reveal exact limits on plant zero locations, minimum controller order, and fastest closed-loop decay rate.

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

The paper studies the monotonic tracking control problem for continuous-time single-input single-output linear systems under output-feedback linear controllers. It supplies necessary and sufficient conditions for solvability and spells out the resulting fundamental limitations on where plant zeros may sit, how low the controller order can go, and how fast the closed-loop poles can decay. When the plant has a pair of complex-conjugate zeros, the three limits are linked by one simple geometric shape in the plane. A reader cares because these bounds show what tracking performance is impossible no matter how the controller is tuned.

Core claim

The paper establishes necessary and sufficient conditions for solvability of the monotonic tracking problem in continuous-time SISO linear systems with linear output-feedback controllers. It determines the precise feasible locations of the plant zeros, the lowest attainable controller order, and the highest attainable decay rate of the closed-loop system, and shows that these quantities are related through a simple geometric shape whenever the plant possesses a pair of complex-conjugate zeros.

What carries the argument

The geometric shape in the complex plane that encodes the joint constraints on feasible zero locations, minimum controller order, and maximum decay rate for plants with complex-conjugate zeros.

If this is right

  • Plant zeros must occupy only certain regions in the complex plane for monotonic tracking to be possible at all.
  • Any stabilizing linear controller must have order at least as high as the minimum value fixed by the zero locations.
  • The closed-loop decay rate is bounded above by a value determined jointly by the zeros and the controller order.
  • For plants with one complex-conjugate zero pair the three bounds collapse into a single geometric relation that can be read off a diagram.

Where Pith is reading between the lines

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

  • The geometric picture may supply a quick visual test that designers can apply before launching numerical synthesis routines.
  • Similar geometric constraints could appear in discrete-time or MIMO versions of the same tracking problem.
  • The decay-rate bound offers a concrete benchmark against which non-monotonic designs can be compared for speed.
  • Knowing the minimal order in advance can prevent fruitless searches over lower-order controllers.

Load-bearing premise

The setting is restricted to continuous-time single-input single-output linear systems under linear output-feedback controllers.

What would settle it

A concrete counterexample consisting of a plant whose zeros lie outside the stated feasible region yet still admits a linear controller that produces exact monotonic tracking at a decay rate faster than the derived bound would falsify the necessary conditions.

Figures

Figures reproduced from arXiv: 2508.08844 by Hamed Taghavian.

Figure 4
Figure 4. Figure 4: The regions on the complex plane where z1,2 satisfy the inequality Arg(z1,2 +α) > θ(n) in (36). Problem 2 is feasible using a controller of order nc = n − no and a closed-loop decay rate σ(H) < −α, if and only if the plant zeros are located in these regions, where no is the order of the plant (fixed). For more information, see Example 5. Corollary 2 and following a similar process as in Example 4 indicates… view at source ↗
read the original abstract

We consider the monotonic tracking control problem for continuous-time single-input single-output linear systems using output-feedback linear controllers in this paper. We provide the necessary and sufficient conditions for this problem to be solvable and expose its fundamental limitations: the exact feasible locations of the plant zeros, the minimum controller order possible, and the fastest decay rate achievable for the closed-loop system. The relationship between these bounds is explained by a simple geometric shape for plants with a pair of complex-conjugate zeros.

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

0 major / 3 minor

Summary. The paper considers the monotonic tracking control problem for continuous-time single-input single-output linear systems using output-feedback linear controllers. It provides necessary and sufficient conditions for solvability and exposes fundamental limitations including the exact feasible locations of the plant zeros, the minimum controller order, and the fastest decay rate achievable for the closed-loop system. For plants with a pair of complex-conjugate zeros, the relationship between these bounds is explained via a simple geometric shape derived from the resulting quadratic constraints.

Significance. If the derivations hold, the work establishes explicit necessary and sufficient conditions for monotonic tracking, along with concrete bounds on zero placement, controller order, and decay rates. This contributes to the understanding of performance limitations in linear control design for non-overshooting responses. The paper receives credit for deriving the results via pole-zero placement and closed-loop characteristic equations, yielding a geometric interpretation without internal contradictions within the stated class of systems.

minor comments (3)
  1. The abstract and introduction could more explicitly reference the key equations (e.g., the characteristic polynomial or quadratic constraints) that underpin the necessity and sufficiency arguments, to improve accessibility for readers.
  2. Section on the geometric shape for complex-conjugate zeros would benefit from an accompanying figure illustrating the feasible region, as the textual description alone may not convey the shape as clearly as intended.
  3. Notation for the decay rate bound and controller order minimum should be consistently defined across the main results and any examples to avoid minor ambiguity in cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the report and the recommendation of minor revision. The provided summary accurately reflects the scope of the work. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives necessary and sufficient conditions for solvability of the monotonic tracking problem, along with bounds on zero locations, controller order, and decay rate, directly from pole-zero placement and closed-loop characteristic equations for continuous-time SISO linear plants under linear output feedback. These steps rely on standard algebraic constraints and geometric interpretations of quadratic forms for complex zeros, without any reduction to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The derivation remains self-contained and independent within the explicitly scoped class of systems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated in the provided text.

axioms (2)
  • domain assumption The plant is a continuous-time single-input single-output linear system.
    Explicitly stated as the class of systems under consideration in the first sentence of the abstract.
  • domain assumption The controller is a linear output-feedback controller.
    Stated as the controller class in the problem formulation within the abstract.

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Reference graph

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