Flatness-Preserving Operations (FPOs)

• Flatness-Preserving Operations (FPOs) are structured transformations that enforce a predefined flatness invariant, ensuring that key properties like differential or spectral flatness remain intact after system modifications.
• In control theory and quantum information, FPOs are implemented via lower-triangular residuals and CPTP maps, respectively, preserving the original flat outputs or uniform spectral distributions for efficient planning and state conversion.
• FPO methodologies extend to coupling strategies, discretization schemes, and even algebraic and geometric operations, providing a unified framework to maintain system invariants across diverse applications.

Searching arXiv for the cited papers and nearby work on Flatness-Preserving Operations. Flatness-Preserving Operations (FPOs) denote structured transformations constrained so that a designated notion of flatness survives the transformation. Recent usage suggests that the term is not standardized across disciplines. In nonlinear control, FPOs are additive residual or coupling modifications that preserve differential flatness and, in the pure-feedback setting, preserve the original flat outputs. In quantum information, FPOs are completely positive trace-preserving maps that do not increase Rényi-entropy spread of a reduced spectrum. Across these settings, the common theme is that admissible operations are defined by compatibility with a pre-existing flat structure rather than by unconstrained model correction or state conversion (Yang et al., 6 Apr 2025, Jasser et al., 20 May 2026).

1. Terminological scope and core preservation principle

The control-theoretic use of FPOs is tied to differential flatness. For a nominal nonlinear system

$\dot x = f(x,u)=\bar f(x,u)+\Delta(x,u),$

the objective is to learn or impose a correction term without destroying the existence of a flatness diffeomorphism or the availability of flatness-based planning and control. The central obstruction is that generic residual parameterizations can destroy flatness even when $\bar f$ is differentially flat (Yang et al., 6 Apr 2025).

The quantum-information use of FPOs is tied to spectral flatness of a reduced density operator. There, flat states are those whose reduced spectrum is uniform on its support, and FPOs are the largest class of CPTP maps that do not increase Rényi entropy spread. The preserved object is not differential flatness of a control system but the absence of additional spectral fluctuations in the entanglement spectrum (Jasser et al., 20 May 2026).

A plausible implication is that the label “flatness-preserving” currently functions as a structural umbrella rather than a single standardized formalism. The preserved invariant changes with the domain, but the admissible operation is always defined by a monotonicity or invariance requirement.

2. Residual FPOs for pure-feedback systems

The most explicit control-theoretic formulation appears for nominal systems admitting a multi-input, multi-output pure-feedback form

$$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$

with each block $x_i\in\mathbb R^m$ and $u\in\mathbb R^m$. Under the regularity conditions

$$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$

and

$$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$

the nominal system is locally differentially flat with flat output

$y=x_1.$

The recursive proof uses the implicit function theorem to solve successively for $x_{i+1}$ and then $u$ as functions of $\bar f$0 (Yang et al., 6 Apr 2025).

An FPO is obtained by constraining the additive residual to be lower-triangular: $\bar f$1 with each $\bar f$2 continuously differentiable. The augmented dynamics become

$\bar f$3

and

$\bar f$4

The preservation mechanism is exact. Because $\bar f$5 does not depend on $\bar f$6, and $\bar f$7 does not depend on $\bar f$8, the Jacobians used in the nominal implicit-function argument are unchanged: $\bar f$9 Hence the same nonsingularity conditions continue to hold, the recursive flatness argument survives intact, the augmented system remains differentially flat, and the original flat output is preserved: $$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$0 This preservation of the original output is practically important because that output often retains direct physical meaning, such as position for a quadrotor (Yang et al., 6 Apr 2025).

3. Constructive recovery, learning, and empirical validation

The residual-FPO framework is constructive rather than merely existential. Assuming local inverse maps $$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$1 for the nominal system,

$$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$2

and

$$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$3

the augmented flatness diffeomorphism is recovered recursively from the nominal one. The construction starts from $$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$4 and then builds $$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$5 and $$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$6 by subtracting the learned lower-triangular residual at each step before applying the appropriate inverse map (Yang et al., 6 Apr 2025).

The learning algorithm enforces the FPO structure directly: $$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$7 Each $$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$8 is chosen to be a smooth function approximator, for example a neural network with smooth activations such as GeLU, so that the learned model remains differentiable. Given trajectories

$$\begin{aligned} \dot x_1 &= \bar f_1(x_1,x_2),\ \dot x_2 &= \bar f_2(x_1,x_2,x_3),\ &\vdots\ \dot x_r &= \bar f_r(x_1,\ldots,x_r,u), \end{aligned}$$9

the parameters are fit by a squared derivative-matching loss, with the true residual approximated from finite differences. Optimization is performed by backpropagation, and after training the augmented flatness diffeomorphism is computed offline using the recursive formula together with Jacobians obtained via automatic differentiation (Yang et al., 6 Apr 2025).

The validation example is a planar quadrotor with states

$x_i\in\mathbb R^m$0

and inputs thrust $x_i\in\mathbb R^m$1 and torque $x_i\in\mathbb R^m$2. After dynamic extension,

$x_i\in\mathbb R^m$3

so the system fits pure-feedback form and has flat output $x_i\in\mathbb R^m$4, except at the singular case $x_i\in\mathbb R^m$5. The disturbance used in simulation is lower-triangular and models drag: $x_i\in\mathbb R^m$6 The learned residual is trained on 3000 trajectories with a 1-hidden-layer network of 32 neurons and GeLU activations; training takes under 1 minute (Yang et al., 6 Apr 2025).

Scenario	Nominal flat model/controller	Learned flat model/controller
Open-loop circle	$x_i\in\mathbb R^m$7 m	$x_i\in\mathbb R^m$8 m
Open-loop lemniscate	$x_i\in\mathbb R^m$9 m	$u\in\mathbb R^m$0 m
Closed-loop circle	$u\in\mathbb R^m$1 m	$u\in\mathbb R^m$2 m
Closed-loop lemniscate	$u\in\mathbb R^m$3 m	$u\in\mathbb R^m$4 m

In closed-loop tracking, the learned flatness controller is compared against nonlinear model predictive control using the same learned model. The NMPC errors are $u\in\mathbb R^m$5 m on the circle and $u\in\mathbb R^m$6 m on the lemniscate, so the learned flatness-based controller achieves performance comparable to NMPC and more than $u\in\mathbb R^m$7 better tracking than the nominal flat controller. The learned flatness-based controller averages $u\in\mathbb R^m$8 ms per control update, while NMPC takes $u\in\mathbb R^m$9 ms, yielding a bit over a $$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$0 speedup (Yang et al., 6 Apr 2025).

4. Coupling and discretization as flatness-preserving constructions

The residual-FPO idea extends from single-system model augmentation to structured interconnection. For a network of differentially flat subsystems in pure-feedback form, lower-triangular dynamic coupling preserves flatness and guarantees that the flat outputs of the subsystems remain the flat outputs of the coupled system. If the uncoupled dynamics satisfy the same regularity assumptions as in the single-system case and the coupling term $$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$1 is lower-triangular, then the joint dynamics remain differentially flat with flat output

$$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$2

Moreover, the joint flatness diffeomorphism can be constructed recursively from the subsystem diffeomorphisms, and its sparsity structure reflects that of the coupling. Corollaries bound the information required by subsystem $$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$3 in terms of ancestor sets in a state-dependent coupling graph, and under strongly lower-triangular coupling the required information is bounded by a fixed graph radius. The framework is validated on planar quadrotors coupled via aerodynamic downwash, where the distributed controller achieves accurate trajectory tracking (Yang et al., 1 Dec 2025).

A different preservation problem arises under digital implementation. Numerical discretization does not in general preserve flatness, whether exact or approximate. The paper on flatness-preserving discretization therefore constructs numerical schemes from discretization maps so that a continuous-time differentially flat system yields a discrete-time difference-flat system. The mechanism is to use dynamic endogenous feedback and a local diffeomorphism $$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$4 to convert the extended system into a linear controllable form $$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$5, discretize in the linear coordinates by a discretization map, and then pull the discretization back through the diffeomorphism. The resulting discrete extended system is diffeomorphic to a linear discrete-time system and is therefore flat (Jindal et al., 14 Nov 2025).

The paper also gives a concrete counterexample showing why ordinary discretization is insufficient: the continuous-time system

$$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$6

is flat with flat output

$$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$7

but its explicit Euler discretization is not flat for any $$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$8. This sharp failure is the motivation for treating flatness preservation as a design objective of the discretization scheme itself rather than a byproduct of numerical approximation (Jindal et al., 14 Nov 2025).

5. FPOs in antiflatness resource theory

In quantum information, FPOs formalize a resource theory in which flatness of the reduced spectrum is free and antiflatness is the resource. A bipartite state belongs to

$$\left|D_{x_{i+1}}\bar f_i(x_1^*,\ldots,x_{i+1}^*)\right|\neq 0,\quad i=1,\ldots,r-1,$$9

precisely when the reduced state is proportional to a projector on its support, equivalently when all nonzero eigenvalues of $$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$0 are equal. The ordering quantity is the Rényi entropy spread

$$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$1

with the key equivalence

$$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$2

Thus flat states are exactly the states with zero spread between any two Rényi orders (Jasser et al., 20 May 2026).

The associated partial order is antiflat majorization: $$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$3 FPOs are then defined as the largest class of CPTP maps that do not increase antiflatness: $$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$4 For pure bipartite states, deterministic convertibility under FPOs obeys the necessary condition

$$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$5

Sufficiency is open: the antiflat order is a selection rule, not a complete Nielsen-style theorem (Jasser et al., 20 May 2026).

Several structural results sharpen this framework. Standard majorization is insufficient because none of $$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$6, $$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$7, or $$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$8 is strictly Schur-convex or strictly Schur-concave. The order admits a one-parameter reformulation: if

$$\left|D_u \bar f_r(x_1^*,\ldots,x_r^*,u^*)\right|\neq 0,$$9

then

$y=x_1.$0

On the iso-purity manifold

$y=x_1.$1

antiflat convertibility is rigid: if $y=x_1.$2 have the same rank and $y=x_1.$3, then their spectra coincide. The framework is compatible with the Capacity of Entanglement

$y=x_1.$4

with logarithmic antiflatness

$y=x_1.$5

and, conditionally on purity ordering, with the Linear Rényi spread

$y=x_1.$6

The paper also expresses $y=x_1.$7 as a second derivative of the Kullback-Leibler divergence along the escort trajectory and identifies a continuous Pareto frontier of maximally antiflat jump spectra rather than a single universal maximizer (Jasser et al., 20 May 2026).

6. Operator-preserver analogues in algebra and geometry

Several papers study preservation classes that are not always called FPOs formally but follow the same preserver logic. For volume polynomials, the main result is that covolume polynomials are exactly the polynomial differential operators that preserve volume polynomials. If $y=x_1.$8, then $y=x_1.$9 is a realizable covolume polynomial over $x_{i+1}$0 iff for any realizable volume polynomial $x_{i+1}$1 over $x_{i+1}$2,

$x_{i+1}$3

is again a realizable volume polynomial. The limiting version extends this from realizable volume polynomials to volume polynomials in general. The paper also gives a symbol theorem: if the symbol of a homogeneous linear operator $x_{i+1}$4 is a realizable volume polynomial, then $x_{i+1}$5 sends realizable volume polynomials to realizable volume polynomials (Grund et al., 27 Jun 2025). A parallel symbol criterion is stated directly for homogeneous linear operators $x_{i+1}$6: if

$x_{i+1}$7

is a volume polynomial, then $x_{i+1}$8 preserves volume polynomials (Grund et al., 23 Mar 2025).

In commutative algebra, the relevant “flatness” is algebraic flatness rather than differential flatness. Hochster and Jeffries prove that if $x_{i+1}$9 is reduced, every maximal ideal of $u$0 contains only finitely many minimal primes of $u$1, and $u$2 has the stable prime extension property, then $u$3 is flat over $u$4. They also show the necessity of the finite-minimal-primes hypothesis via a reduced quasilocal counterexample with infinitely many minimal primes and develop intersection flatness as a tool for reducing stable prime extension questions to fiberwise checks in graded settings (Hochster et al., 2020). This suggests a related but distinct preservation paradigm: a stronger extension property forces flatness under explicit reducedness and finiteness hypotheses.

7. Preservation-type operations in polytope and map theory

The literature on simple polytopes and maps supplies further preservation-type constructions that are structurally analogous to FPOs but preserve different invariants. Bosio introduces two operations on simple polytopes, biflip and puzzle-move, that produce polytopes with diffeomorphic moment-angle manifolds. A biflip is a sequence of two compatible flips on simplicial faces with the same bounding facets, and a puzzle-move is a cut-and-reglue operation along a hyperplane using a harmless automorphism of the cut face. Any two puzzle-equivalent polytopes are Gr-equivalent, and any two polytopes related by one or more biflips are Gr-equivalent. These operations are best viewed as moment-angle-manifold-preserving rather than as literal flatness-preserving operations (Frédéric, 2017).

For plane graphs and maps, the relevant framework is local orientation-preserving symmetry-preserving operations. A lopsp operation is defined by a connected tiling $u$5 of the Euclidean plane together with centers $u$6 and $u$7 of a $u$8 and a $u$9 clockwise rotation symmetry, respectively. The invariant classifying such operations is the double chamber decoration, and the resulting operation is sound in the sense that the output does not depend on the chosen cut path. If $\bar f$00 is $\bar f$01-connected for $\bar f$02 and $\bar f$03 is a $\bar f$04-connected lopsp operation, then $\bar f$05 is also $\bar f$06-connected (Goetschalckx et al., 2020).

A complementary algorithmic treatment reduces generation of lopsp operations to generation of plane quadrangulations. In that framework, classical operations such as dual, ambo, truncation, leapfrog, and join are realized as lopsp operations, every orientation-preserving automorphism of a map $\bar f$07 induces one of $\bar f$08, and a lopsp operation can be written as an lsp operation exactly when it has an orientation-reversing automorphism fixing the marked vertices $\bar f$09 (Camp et al., 2024). These constructions reinforce a broad preservation motif: local replacements are admissible only when they respect a designated global invariant, whether that invariant is differential flatness, spectral flatness, graded diffeomorphism type, or orientation-preserving symmetry.