---
title: 'FPE-MPC: Diverse Predictive Control Approaches'
url: https://www.emergentmind.com/topics/fpe-based-model-predictive-control-fpe-mpc
type: topic
---

# FPE-MPC: Diverse Predictive Control Approaches

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 [2509.01065]. In other contexts, the label has been used for **finite-dimensional approximation** of fractional-order dynamics [1605.08719], for **flexible-step persistent-excitation** schemes that interleave online learning and stabilization [2510.00961], and more loosely for **filter-aware**, **predictive-filter**, **funnel**, or **prescribed-performance** MPC architectures [2304.10246], [2603.27893], [2505.20090], [2602.22458]. 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 [2410.16173].

## 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 | [2509.01065] |
| Finite-dimensional approximation | Fractional-order infinite-memory dynamics are truncated and controlled via tube MPC | [1605.08719] |
| Flexible-step persistent-excitation | A flexible number of planned inputs is applied; PE is invoked when feasibility is lost | [2510.00961] |
| Future prediction of estimation or filter performance | MPC constrains future estimator degradation through trackability or predictive filtering | [2304.10246], [2603.27893] |
| Funnel or prescribed-performance control | MPC enforces time-varying error envelopes through funnel boundaries or funnel penalty costs | [2505.20090], [2602.22458] |

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** [2410.16173]. 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 [2509.01065].

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 [1605.08719]. 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 [1605.08719].

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
$$
\tilde{z}_k \in \tilde{X}\ominus S, \qquad v_k \in U\ominus KS.
$$
The error tube is generated by
$$
e_{k+1}=A_K e_k+Gd_k, \qquad A_K=A+BK,
$$
and the minimal robust positively invariant set is
$$
S_\infty^\nu = \lim_{k\to\infty} \bigoplus_{i=0}^{k} A_K^i G D_\nu.
$$

The resulting optimization problem is a conventional finite-horizon quadratic MPC on the nominal model,
$$
V_N^\star(\tilde{z}_k)= \min_{\mathbf{v}_k\in\mathcal{V}_N(\tilde{z}_k)}
\left( V_f(\tilde{z}_{k+N|k}) + \sum_{i=0}^{N-1}\ell(\tilde{z}_{k+i|k},v_{k+i|k}) \right),
$$
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 [1605.08719].

The principal trade-off is explicit. Increasing $\nu$ 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 [2509.01065]. The controlled object is therefore not a nominal state trajectory but the predicted PDF itself.

The stochastic plant is written as an Itô system,
```latex
d\boldsymbol{x} = \boldsymbol{f}(\boldsymbol{x}, \boldsymbol{u}, t)\,dt + \boldsymbol{\sigma}(\boldsymbol{x}, t)\,d\boldsymbol{W}(t),
```
and the associated FPE is
```latex
\frac{\partial p(\boldsymbol{x})}{\partial t} = \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2}{\partial x_i \partial x_j} \left\{a_{i,j}(\boldsymbol{x},t)p(\boldsymbol{x},t)\right\} - \sum_{i=1}^n \frac{\partial}{\partial x_i} \left\{f_i(\boldsymbol{x},\boldsymbol{u},t)p(\boldsymbol{x},t)\right\},
```
with
```latex
\boldsymbol{a}(\boldsymbol{x},t)=\boldsymbol{\sigma}\boldsymbol{\sigma}^\top.
```
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 [2509.01065].

The concrete application is a **2-link 3-tendon soft finger** with viscoelastic uncertainty. The state is
```latex
\boldsymbol{x} = \left[\boldsymbol{q}^\top\ \dot{\boldsymbol{q}}^\top\ \boldsymbol{\tau}_\mathrm{joint}^\top\right]^\top \in \mathbb{R}^{3n_j},
```
and parameter uncertainty in $k_\mathrm{v}$, $c_\mathrm{v}$, and $c_\mathrm{p}$ 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 $\boldsymbol{q}=[q_1\ q_2]^\top$, the grid is $21\times 21=441$ points over $-0.15 \le q_i \le 0.15$, the simulation time is $10$ s, and the time resolution is $0.1$ s [2509.01065].

The MPC objective is the squared PDF mismatch,
```latex
J[k] = \sum_{i,j=1}^n \left(p_{i,j}[k] - {}^\mathrm{ref}p_{i,j}[k]\right)^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 $95\%$ confidence bounds in feasible cases, whereas variance shaping is more difficult, especially for narrower target PDFs [2509.01065].

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 [2509.01065]. 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 [2211.02780], [2510.00961]. 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 $m$ and requires only average decrease over a window,
$$
\frac{1}{m}\Big(\sum_{j=1}^mV(x^j)\Big) - V(x^0) \leq - \alpha(x^0),
$$
rather than decrease at every sampling instant. This implies the existence of an index $l\in\{1,\dots,m\}$ at which the descent condition holds, after which the controller applies the first $l$ elements of the optimized sequence and only then re-optimizes [2211.02780]. 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
$$
x(t+1)=Ax(t)+Bu(t),
$$
with unknown $(A,B)$, and solves a finite-horizon OCP using the current estimate $(\hat A,\hat B)$ together with a generalized control Lyapunov function constraint,
$$
\sum_{k=1}^N \sigma_k V(x_k)\le (1-\alpha)V(x_0).
$$
If the OCP is feasible, a flexible number $\ell\in[1:N]$ of planned inputs is applied such that
$$
V(x_\ell)-V(x_0)\le -\alpha V(x_0).
$$
If the OCP becomes infeasible, the controller switches to a **persistently exciting** input block of order $d=\dim\mathbb{X}+1$, updates the data set
$$
\mathcal{D}_\tau=\big(x(0),(u(0),x(1)),\ldots,(u(\tau-1),x(\tau))\big),
$$
and recomputes an estimator for $(A,B)$, for example through a least-squares pseudoinverse formula [2510.00961].

The paper’s main theorem states that if $\Omega(A,B,\bar{x})\neq\emptyset$ for every $\bar{x}$, then for every initial condition $x(0)$, the state under the unknown-system algorithm converges to the origin [2510.00961]. 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 [2510.00961].

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 [2304.10246], [2603.27893], [2505.20090], [2602.22458]. 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,
$$
\track(\state, \carry) = \mathbb{E}\Big[\sum_{t=1}^\infty \discount^{t-1}\errorf(\state_t,\carry_t) \mid \state_1 = \state,\carry_1=\carry\Big].
$$
The practical approximation replaces $\track(\state_t,\carry_t)$ by $\track(\state_t,\carryperf_{\state_t})$ and learns a neural approximation $\phi(\state)$ using TD($\lambda$) rollouts. The MPC then constrains planned states to remain trackable with high probability, or, in practice, below a deterministic threshold on $\phi(\state)$ [2304.10246]. The paper positions this as a middle ground between state-space MPC and full belief-space planning.

In the **Predictive Safety–Stability Filter (PS\(^2\)F)**, 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
$$
\mathbb S(x):=\{u(0;x)\mid \mathbf u\in\mathcal U_M^a(x)\},
$$
so that
$$
u = \arg\min_{\zeta\in\mathbb S(x)} |u_{\mathrm{ext}}-\zeta|^2.
$$
The paper proves recursive feasibility and asymptotic stability and states that the filter introduces no additional conservatism beyond the nominal MPC region of attraction [2603.27893]. 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
$$
\varphi(t;c,T)=c(T-t),
$$
while the upper layer optimizes the funnel parameters $(c,T)$ 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 [2505.20090].

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
$$
\ell_{\varphi}(t,y,u)=
\begin{cases}
\dfrac{\|y-y_{\mathrm{ref}}(t)\|^2}{\varphi(t)^2-\|y-y_{\mathrm{ref}}(t)\|^2}+\lambda_u\|u\|^2, & \|y-y_{\mathrm{ref}}(t)\|\neq \varphi(t),\\[1.2ex]
+\infty, & \text{else},
\end{cases}
$$
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** [2602.22458].

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 [2410.16173]. 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,
$$
\hat{u}_c = h(s; \mu),
$$
using a fully connected neural network,
$$
\hat{u}_c = h(s; \mu) = (h_k \circ \sigma \circ ... \circ h_2 \circ \sigma \circ h_1)(s), \qquad h_j(z) = W_j z + b_j,
$$
with learnable parameters $\mu = \{W_1, b_1, ..., W_k, b_k\}$. The training data are tuples $(s,u_c,s_+)_i$ generated by an MPC, and the total loss is
$$
L = w_1 L_1 + w_2 L_2 + w_3 \tilde{L}_3 + w_4 \tilde{L}_4,
$$
with experimental weights
$$
[w_1, w_2, w_3, w_4] = [1, 1000, 1, 100].
$$
Here $L_1$ is control imitation, $L_2$ is a discrete dynamics residual based on forward Euler,
$$
\hat{s}_+(s, \hat{u}) = s + \Delta T_c \cdot f(s, \hat{u}),
$$
$L_3$ is a Lyapunov decrease penalty based on a learned quadratic profile $V(s)=s^T P s$, and $L_4$ is an optional feasibility penalty [2410.16173].

The runtime controller preserves the quadcopter’s fixed feedforward and stabilizing terms,
$$
\hat{u} = u_e + u_s + \hat{u}_c, \qquad
u_e = [mg/2, mg/2]^T,\qquad u_s = \kappa s.
$$
The surrogate itself is a **4-layer** deterministic multilayer perceptron with **sigmoid activations**, **312 learnable parameters**, **Kaiming uniform** initialization, **Adam** at learning rate $10^{-3}$, 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 [2410.16173].

The stability interpretation is deliberately limited. The paper estimates a quadratic Lyapunov profile by a convex optimization over $P$ 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 [2410.16173]. 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 [1605.08719], [2505.20090], [2603.27893], [2602.22458]. 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 $\nu$, but online complexity grows [1605.08719]. 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 [2509.01065]. 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 [2510.00961]. 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 [2304.10246], [2603.27893]. 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 [2505.20090], [2602.22458].

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 [2111.00597]. 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 [2111.00597].

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.

Source: https://www.emergentmind.com/topics/fpe-based-model-predictive-control-fpe-mpc