FPE-MPC: Diverse Predictive Control Approaches

• FPE-MPC is a collection of predictive control architectures that incorporate auxiliary mechanisms like Fokker–Planck equations, fractional dynamics approximations, and funnel-based tracking.
• Each variant targets objectives such as state distribution regulation, robust handling of fractional-memory effects, and adaptive persistent excitation to manage computational challenges.
• The approaches balance approximation accuracy, computational cost, and stability, emphasizing the need for clear local definitions and mechanism selection.

Searching arXiv for the cited papers to ground the article and verify bibliographic details. FPE-based Model Predictive Control (FPE-MPC) is not a single canonical MPC formulation. In recent arXiv usage, the label denotes several distinct but structurally related families of receding-horizon control methods in which the predictive controller is organized around an auxiliary mechanism beyond standard sequence optimization. In the narrowest and most literal sense, it refers to Fokker–Planck-equation-based MPC, where the controlled object is a state probability density function rather than a nominal trajectory (Honji et al., 1 Sep 2025). In other contexts, the label has been used for finite-dimensional approximation of fractional-order dynamics (Sopasakis et al., 2016), for flexible-step persistent-excitation schemes that interleave online learning and stabilization (Pietschner et al., 1 Oct 2025), and more loosely for filter-aware, predictive-filter, funnel, or prescribed-performance MPC architectures (Kayalibay et al., 2023, Yan et al., 29 Mar 2026, Göbel et al., 26 May 2025, Dennstädt, 25 Feb 2026). A closely related but distinct line is the physics-informed MPC surrogate PI-MPCS, which approximates an MPC policy with dynamics- and Lyapunov-regularized learning and is explicitly not an FPE-based MPC method in the usual sense (Rivera et al., 2024).

1. Terminological scope and principal lineages

The term “FPE-MPC” is context-dependent. The literature represented here associates it with multiple constructions rather than a single standardized acronym expansion.

Usage of “FPE-MPC”	Core mechanism	Representative paper
Fokker–Planck equation	MPC shapes the predicted PDF through an FPE model	(Honji et al., 1 Sep 2025)
Finite-dimensional approximation	Fractional-order infinite-memory dynamics are truncated and controlled via tube MPC	(Sopasakis et al., 2016)
Flexible-step persistent-excitation	A flexible number of planned inputs is applied; PE is invoked when feasibility is lost	(Pietschner et al., 1 Oct 2025)
Future prediction of estimation or filter performance	MPC constrains future estimator degradation through trackability or predictive filtering	(Kayalibay et al., 2023, Yan et al., 29 Mar 2026)
Funnel or prescribed-performance control	MPC enforces time-varying error envelopes through funnel boundaries or funnel penalty costs	(Göbel et al., 26 May 2025, Dennstädt, 25 Feb 2026)

This multiplicity matters because several papers are conceptually adjacent while being mathematically non-equivalent. In particular, the PI-MPCS surrogate is “closely related” only in the broad sense of being a physics-informed MPC surrogate; it contains no Fokker–Planck equation, no explicit PDE-based predictive model, and no FPE-constrained optimal control problem (Rivera et al., 2024). Conversely, the soft-robotics paper uses the term literally: the FPE is the predictive model, and the MPC acts on the evolution of the distribution itself (Honji et al., 1 Sep 2025).

A plausible implication is that any use of “FPE-MPC” requires local definition before comparison across papers. Without that disambiguation, claims about stability proofs, computational complexity, or robustness may refer to fundamentally different objects: trajectories, tubes, funnels, filters, or full state densities.

2. Finite-dimensional approximation for fractional-order systems

One established lineage uses “FPE-based MPC” to denote MPC built from a finite-dimensional approximation of discrete-time fractional-order dynamics (Sopasakis et al., 2016). The underlying issue is that Grünwald–Letnikov fractional differences depend on the entire signal history,

$$\Delta_h^\alpha x(t) = \sum_{j=0}^{\infty} (-1)^j {\alpha \choose j} x(t-jh),$$

so the exact discretized model is infinite-dimensional.

The approximation step truncates the fractional memory after $\nu$ terms,

$$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$

which yields, after algebraic rearrangement and state augmentation,

$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$

The neglected tail is not ignored; it is represented as a bounded additive disturbance,

$$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$

with $D_\nu \to \{0\}$ as $\nu\to\infty$. This bounded-disturbance interpretation is central: the approximation order $\nu$ becomes both an accuracy parameter and a robustness-design parameter (Sopasakis et al., 2016).

The controller is a standard tube-based robust MPC on the augmented LTI model. The actual input is

$u_k=v_k+Ke_k, \qquad e_k=\tilde{x}_k-\tilde{z}_k,$

with nominal dynamics

$\tilde{z}_{k+1}=A\tilde{z}_k+Bv_k,$

and tightened nominal constraints

$\nu$0

The error tube is generated by

$\nu$1

and the minimal robust positively invariant set is

$\nu$2

The resulting optimization problem is a conventional finite-horizon quadratic MPC on the nominal model,

$\nu$3

but the stability analysis must account for the infinite-dimensional origin of the plant. The paper proves constraint satisfaction, recursive feasibility, exponential convergence to a neighborhood of the origin, and, under additional small-disturbance and contraction conditions, asymptotic stability of the origin (Sopasakis et al., 2016).

The principal trade-off is explicit. Increasing $\nu$4 reduces approximation error and shrinks the invariant tube, but increases computation substantially. This suggests that, within this lineage, FPE-MPC is less a new optimal-control principle than a tractable embedding of fractional-memory dynamics into robust MPC.

3. Fokker–Planck-equation-based distributional MPC

In the most literal usage, FPE-MPC is a distribution-shaping controller in which the Fokker–Planck equation propagates the state probability density function and the MPC chooses inputs to make that density match a reference density (Honji et al., 1 Sep 2025). The controlled object is therefore not a nominal state trajectory but the predicted PDF itself.

The stochastic plant is written as an Itô system, $\nu\to\infty$8 and the associated FPE is $\nu\to\infty$9 with $\nu$0 The paper emphasizes that this is an advection-diffusion equation for the PDF, so the stochastic term does not need to be handled directly once the FPE is used (Honji et al., 1 Sep 2025).

The concrete application is a 2-link 3-tendon soft finger with viscoelastic uncertainty. The state is $\nu$1 and parameter uncertainty in $\nu$5, $\nu$6, and $\nu$7 is modeled by log-normal distributions. The FPE is discretized with the Chang–Cooper scheme, which the paper states has second-order accuracy in time and space and guarantees positivity. For the numerical implementation, the state of interest is the joint-angle vector $\nu$8, the grid is $\nu$9 points over $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$0, the simulation time is $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$1 s, and the time resolution is $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$2 s (Honji et al., 1 Sep 2025).

The MPC objective is the squared PDF mismatch, $\nu$2 with prediction horizon = 1 and control horizon = 1. The reference PDF is typically Gaussian, allowing the controller to regulate both mean and variance in principle. The reported simulations show that the controller can bring the joint angles near target means within the $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$3 confidence bounds in feasible cases, whereas variance shaping is more difficult, especially for narrower target PDFs (Honji et al., 1 Sep 2025).

This formulation differs sharply from deterministic MPC and from many stochastic MPC schemes. The predictive model is a PDE over densities, the decision criterion is distributional, and the main computational burden lies in repeated FPE solves. The paper explicitly notes the curse of dimensionality, the need for faster MPC methods such as C/GMRES, possible use of Physics-Informed Neural Networks, and future integration with feedback sensing (Honji et al., 1 Sep 2025). This suggests that literal FPE-MPC is especially natural when uncertainty is intrinsic to the plant and distribution-level objectives are primary.

4. Flexible-step and persistently exciting formulations

Another usage associates FPE-MPC with flexible-step execution and persistent excitation, especially for unknown or hard-to-stabilize discrete-time systems (Fürnsinn et al., 2022, Pietschner et al., 1 Oct 2025). In this lineage, the distinctive idea is not density propagation or fractional-memory truncation, but the replacement of per-step Lyapunov decrease by a finite-window descent mechanism.

The generalized-Lyapunov formulation introduces a generalized discrete-time control Lyapunov function of order $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$4 and requires only average decrease over a window,

$$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$5

rather than decrease at every sampling instant. This implies the existence of an index $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$6 at which the descent condition holds, after which the controller applies the first $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$7 elements of the optimized sequence and only then re-optimizes (Fürnsinn et al., 2022). The method is explicitly distinguished from variable-horizon MPC: the prediction horizon stays fixed, while the number of implemented optimal inputs varies from iteration to iteration.

The unknown-system extension considers

$$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$8

with unknown $$\Delta_{h,\nu}^{\alpha} x_k = \sum_{j=0}^{\nu} c_j^\alpha x_{k-j},$$9, and solves a finite-horizon OCP using the current estimate $\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$0 together with a generalized control Lyapunov function constraint,

$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$1

If the OCP is feasible, a flexible number $\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$2 of planned inputs is applied such that

$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$3

If the OCP becomes infeasible, the controller switches to a persistently exciting input block of order $\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$4, updates the data set

$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$5

and recomputes an estimator for $\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$6, for example through a least-squares pseudoinverse formula (Pietschner et al., 1 Oct 2025).

The paper’s main theorem states that if $\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$7 for every $\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$8, then for every initial condition $\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k.$9, the state under the unknown-system algorithm converges to the origin (Pietschner et al., 1 Oct 2025). The exploration logic is therefore opportunistic rather than front-loaded: persistent excitation is used only when infeasibility forces exploration. The paper explicitly describes this as a modern, online, feasibility-driven version of FPE-MPC rather than a classical pre-exploration-based one (Pietschner et al., 1 Oct 2025).

A plausible implication is that this lineage treats “FPE” less as a model class and more as an architectural principle for coupling stabilization, identification, and implementation frequency. Its hallmark properties are recursive feasibility, convergence under stabilizability, and reduced need to re-optimize at every time step.

5. Filter, funnel, and prescribed-performance variants

A broader set of papers maps FPE-MPC to predictive filtering, command projection, and prescribed-performance envelopes rather than to a literal FPE or flexible-step PE mechanism (Kayalibay et al., 2023, Yan et al., 29 Mar 2026, Göbel et al., 26 May 2025, Dennstädt, 25 Feb 2026). These formulations share a common structure: the predictive optimizer does not merely minimize a stage cost, but also restricts the admissible evolution to preserve estimation quality, safety, or a time-varying performance bound.

In filter-aware MPC, the additional object is trackability, defined as discounted expected future estimator error,

$$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$0

The practical approximation replaces $$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$1 by $$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$2 and learns a neural approximation $$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$3 using TD($$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$4) rollouts. The MPC then constrains planned states to remain trackable with high probability, or, in practice, below a deterministic threshold on $$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$5 (Kayalibay et al., 2023). The paper positions this as a middle ground between state-space MPC and full belief-space planning.

In the Predictive Safety–Stability Filter (PS$$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$6F), the architecture is explicitly two-layered. A nominal MPC “copilot” produces a certified safe-stable reference trajectory and an implicit Lyapunov function, while a secondary filtering OCP projects an external command onto the admissible first-input set

$$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$7

so that

$$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$8

The paper proves recursive feasibility and asymptotic stability and states that the filter introduces no additional conservatism beyond the nominal MPC region of attraction (Yan et al., 29 Mar 2026). This is described as closely aligned with predictive enforcement or command-projection interpretations of FPE-MPC.

The funnel and prescribed-performance lineages make the auxiliary mechanism an explicit time-varying error envelope. In model predictive funnel control (MPFC), the lower layer is a continuous exact funnel controller with collapsing boundary

$$\tilde{x}_{k+1}=A\tilde{x}_k+Bu_k+Gd_k, \qquad d_k\in D_\nu,$$9

while the upper layer optimizes the funnel parameters $D_\nu \to \{0\}$0 rather than a full control sequence. The resulting optimization dimension is constant and does not depend on the prediction horizon, and the paper proves initial feasibility, recursive feasibility, bounded closed-loop cost, and asymptotic stability (Göbel et al., 26 May 2025).

The dissertation on MPC for output tracking with prescribed performance goes further by embedding the funnel directly into the stage cost. The basic funnel stage cost is

$D_\nu \to \{0\}$1

and finite cost implies that the trajectory remains inside the funnel. The thesis proves initial and recursive feasibility without terminal constraints or long horizons, then extends the architecture to robust funnel MPC, learning-based robust funnel MPC, and sampled-data funnel MPC (Dennstädt, 25 Feb 2026).

These variants are not identical, but they are unified by a common design move: predictive optimization is subordinated to an additional envelope, filter, or performance certificate that restricts what trajectories are admissible. This suggests why they are sometimes grouped under a broad FPE-MPC label despite very different mathematical formulations.

6. Physics-informed MPC surrogates as an adjacent but distinct direction

The paper “Fast Physics-Informed Model Predictive Control Approximation for Lyapunov Stability” introduces PI-MPCS, a deterministic neural surrogate trained to imitate an MPC controller while enforcing approximate dynamics consistency and Lyapunov-regularized stability behavior (Rivera et al., 2024). It is closely related in motivation to several FPE-MPC families—especially those that seek reduced online computation without discarding structure—but it is explicitly not an FPE-based MPC method in the senses of Fokker–Planck, finite-dimensional projection of a PDE, or any canonical FPE-based controller formulation.

The surrogate maps the current state directly to a control correction,

$D_\nu \to \{0\}$2

using a fully connected neural network,

$D_\nu \to \{0\}$3

with learnable parameters $D_\nu \to \{0\}$4. The training data are tuples $D_\nu \to \{0\}$5 generated by an MPC, and the total loss is

$D_\nu \to \{0\}$6

with experimental weights

$D_\nu \to \{0\}$7

Here $D_\nu \to \{0\}$8 is control imitation, $D_\nu \to \{0\}$9 is a discrete dynamics residual based on forward Euler,

$\nu\to\infty$0

$\nu\to\infty$1 is a Lyapunov decrease penalty based on a learned quadratic profile $\nu\to\infty$2, and $\nu\to\infty$3 is an optional feasibility penalty (Rivera et al., 2024).

The runtime controller preserves the quadcopter’s fixed feedforward and stabilizing terms,

$\nu\to\infty$4

The surrogate itself is a 4-layer deterministic multilayer perceptron with sigmoid activations, 312 learnable parameters, Kaiming uniform initialization, Adam at learning rate $\nu\to\infty$5, and 200 epochs of training. On the 2D quadcopter landing task, the paper reports mean runtimes of 3.2037 s for MPC and 1.7344 s for PI-MPCS, corresponding to about a 2× speedup in the abstract and a 45% reduction in runtime in the results section (Rivera et al., 2024).

The stability interpretation is deliberately limited. The paper estimates a quadratic Lyapunov profile by a convex optimization over $\nu\to\infty$6 and uses the Lyapunov loss as a training regularizer, but it does not provide a full closed-loop proof that the learned surrogate guarantees asymptotic stability (Rivera et al., 2024). This distinguishes PI-MPCS from FPE-MPC lineages that place the stability argument directly inside the receding-horizon control law.

7. Recurrent guarantees, computational motives, and open distinctions

Across these disparate formulations, several technical objectives recur: constraint satisfaction, recursive feasibility, asymptotic stability or convergence to the origin, reduced online optimization, and explicit treatment of uncertainty, model mismatch, or limited computational resources (Sopasakis et al., 2016, Göbel et al., 26 May 2025, Yan et al., 29 Mar 2026, Dennstädt, 25 Feb 2026). What changes is the mechanism by which those properties are pursued.

In finite-dimensional approximation for fractional-order systems, tractability is obtained by truncating infinite memory and treating the truncation as bounded disturbance; accuracy improves with $\nu\to\infty$7, but online complexity grows (Sopasakis et al., 2016). In literal Fokker–Planck MPC, the state-density evolution is modeled explicitly, which permits mean and variance shaping but incurs PDE-solving costs and dimensionality limits (Honji et al., 1 Sep 2025). In flexible-step persistent-excitation schemes, stabilization is decoupled from full identification and re-optimization frequency can fall below the sampling frequency, but feasibility management and excitation design become central (Pietschner et al., 1 Oct 2025). In filter-aware and predictive-filter variants, the planner preserves estimator quality or command safety through projected admissible sets, which is less costly than full belief-space planning yet more structured than blind state-space MPC (Kayalibay et al., 2023, Yan et al., 29 Mar 2026). In funnel and prescribed-performance approaches, feasibility is encoded through time-varying boundaries or barrier-like penalties, allowing rigorous tracking guarantees without the terminal ingredients of classical MPC (Göbel et al., 26 May 2025, Dennstädt, 25 Feb 2026).

A further computational theme appears in reduced-order PDE-constrained MPC. The reduced-basis framework for parametrized parabolic PDEs replaces repeated full-order finite-horizon solves by a certified surrogate, computes a posteriori bounds for the control and cost, and adaptively selects a stabilizing prediction horizon online (Dietze et al., 2021). Although this is not presented as a canonical FPE-MPC formulation, the paper is explicitly relevant when FPE is interpreted as a full-order PDE-based predictive model and the key question is how to preserve Lyapunov-type guarantees under model reduction (Dietze et al., 2021).

The main misconception addressed by this literature is that “FPE-MPC” names a unique method. The evidence instead points to a cluster of architectures that share an MPC backbone but differ in what is being predicted and enforced: a nominal trajectory, an invariant tube, a funnel boundary, a filter-performance surrogate, a projected safe set, or a probability density function. The most precise usage therefore attaches the acronym to its operative mechanism—Fokker–Planck, finite-dimensional approximation, flexible-step persistent excitation, or funnel/prescribed-performance—before making theoretical or experimental comparisons.