---
title: Feasibility Value Function (FVF)
url: https://www.emergentmind.com/topics/feasibility-value-function-fvf
type: topic
---

# Feasibility Value Function (FVF)

The Feasibility Value Function (FVF), denoted \(V_f\), is a predictive metric introduced to rigorously quantify finite-horizon feasibility in hierarchical model predictive control (MPC) architectures. Its construction leverages slack variables to relax state and reference-dependent constraints, providing an analytically tractable and contractible interface between hierarchical control layers. The zero-level set of \(V_f\) precisely delineates the admissible region for the lower-level MPC, enabling provably safe modular design and execution, notably when model and cost structures are sequestered across control layers [2504.12036].

## 1. Formal Definition and Mathematical Structure

Let \(x(k)\in\R^{n_x}\) represent the current lower-layer state, and \(r^{\rm H}_{\cdot|k_{\rm H}}\in\R^{n_r\times N_{\rm H}}\) a high-level reference trajectory, constant over fast-time blocks of size \(N_{\rm L}\). Slack vectors \(\xi_{l|k}=[{\xi^x_{l|k}}^{\top},{\xi^{\Delta x}_{l|k}}^{\top}]^{\top}\geq0\) relax state and reference-dependent constraints. The predictive feasibility value function is defined as
\[
V_{f}(x(k),\,r^{\rm H}_{\cdot|k_{\rm H}}):=\min_{\substack{x_{\cdot|k},\,u_{\cdot|k},\\\xi_{\cdot|k}\geq0}} \sum_{l=0}^{N(k)-1}\|\xi_{l|k}\|_{1}
\]
subject to the finite-horizon system dynamics and relaxed constraints:
\[
\begin{cases}
x_{0|k}=x(k),\\
x_{l+1|k}=f(x_{l|k},u_{l|k}),\quad l=0,\dots,N(k)-1,\\
c_x(x_{l|k})\le \xi^x_{l|k},\\
c_{\Delta x}(x_{l|k},r^{\rm H}_{\cdot|k_{\rm H}})\le \xi^{\Delta x}_{l|k},\\
c_u(u_{l|k})\le 0,
\end{cases}
\]
where \(N(k):=N_{\rm H}N_{\rm L}-k\). By definition, \(V_{f}(x,r)=0\) if and only if the corresponding hard-constrained MPC is feasible; otherwise, \(V_{f}(x,r)>0\) quantifies the minimal slack violation required [2504.12036].

## 2. Theoretical Properties and Relationship to Viability Kernels

The FVF admits a direct interpretation in viability kernel theory. The set \(\mathcal{K}_0=\{(x,r)\mid V_{f}(x,r)=0\}\) forms the finite-horizon viability set in state-reference space. For state/reference pairs outside \(\mathcal{K}_0\), the value \(V_{f}(x,r)>0\) quantifies, in an \(ℓ_1\)-slack sense, the minimal aggregate violation necessary for admissibility under system constraints. This function acts as a cost-to-go: it is zero inside the viability kernel, strictly positive outside, and monotonically non-increasing under optimal (soft-constrained) control trajectories. The objective \(\sum_{l=0}^{N(k)-1}\|\xi_{l|k}\|_{1}\) uniquely serves as the tightest violation margin to accommodate model and reference inconsistencies over the given horizon [2504.12036].

## 3. Contract-Based Hierarchical Control Architecture

In the contract-based hierarchical control setting, a high-level planner (operating at slow timescale \(T_{\rm H}=N_{\rm L}T_{\rm L}\)) issues reference trajectories \(r^{\rm H}_{\cdot|k_{\rm H}}\) to a low-level, fast-sampled MPC. The low-level MPC minimizes
\[
J_{\rm MPC}(x_{\cdot|k},u_{\cdot|k},r^{\rm H}_{\cdot|k_{\rm H}})
+
w_{\xi}\sum_{l=0}^{N(k)-1}\|\xi_{l|k}\|_{1}
\]
subject to the same relaxed system and reference constraints.

A contract function \(h_{\rm C}(x,r)\) is exchanged offline: it is an explicit (often neural network-based) approximation of \(V_f\). Online, the high-level planner incorporates this contract in its optimization:
\[
\min_{x^{\rm H},\,r^{\rm H}}J^{\rm H}(x^{\rm H},r^{\rm H})+w_{h}\,h_{\rm C}(x(k),r^{\rm H}_{\cdot|k_{\rm H}})
\]
with the option to enforce \(h_{\rm C}(x,r)\leq0\) for strict lower-level feasibility. This enables the upper layer to proactively avoid references that violate the constrained capabilities of lower controllers, despite not possessing detailed model or constraint information from that layer [2504.12036].

## 4. Explicit Function Approximation and Implementation

The FVF can be efficiently approximated via either look-up tables (LUT) or neural networks (NN), to facilitate online evaluation within the planner. For NNs, input features comprise a concatenated vector \(\left[x(k);\,r^{\rm H}_{0|k_{\rm H}};\ldots;r^{\rm H}_{N_{\rm H}-1|k_{\rm H}}\right] \in\R^{n_x+n_rN_{\rm H}}\), with 2–4 hidden layers of 50–200 ReLU or tanh neurons, and a scalar output \(\hat h_{\rm C}(x,r)\). Training proceeds by sampling states \(x^{(i)}\) and reference sequences \(r^{(i)}\), then solving the slack value problem offline to generate targets \(V_f(x^{(i)},r^{(i)})\). The standard loss is
\[
\mathcal{L}(\theta)=\frac{1}{M}\sum_{i=1}^{M}\left(\hat h_{\rm C}(x^{(i)},r^{(i)};\theta)-V_f(x^{(i)},r^{(i)})\right)^2+\lambda\|\theta\|_2^2,
\]
where \(\lambda\) is a regularization parameter. By the universal-approximation theorem and the Lipschitz continuity of \(V_f\), a sufficiently wide/deep NN can achieve \(\sup_{x,r}|\hat h_{\rm C}(x,r)-V_f(x,r)|\leq\varepsilon\) for arbitrary small \(\varepsilon>0\). In practice, a positive safety margin \(\delta>\varepsilon\) may be set, and feasibility is conservatively enforced via \(\hat h_{\rm C}(x,r)\leq\delta\implies V_f(x,r)=0\) [2504.12036].

## 5. Case Study: Autonomous Driving Application

In the autonomous driving example, the lower-level controller employs a single-track dynamic vehicle model discretized at \(T_{\rm L}=50\) ms, with states \(x=(p_x,p_y,\psi,v,\dot\psi,\beta)\) and controls \(u=(\delta,a)\). Hard box constraints on velocity, steering, and acceleration are augmented by soft “tube” constraints:
\[
|p_{x,l|k}-p^{\rm ref}_{x,l|k_{\rm H}}|\leq d_{\max},\quad|p_{y,l|k}-p^{\rm ref}_{y,l|k_{\rm H}}|\leq d_{\max}
\]
surrounding the planner’s path. The high-level planner operates on a simplified "constant-velocity + heading" model at a slower timescale and incorporates both quadratic target-tracking and nonconvex obstacle-avoidance costs.

Offline, for sampled \((x,r^{\rm H})\), the FVF is solved and stored as a LUT or approximator \(\hat h_{\rm C}(x,r^{\rm H})\). Online, the planner evaluates candidate reference pairs (\(\psi,v\)), discards those with \(\hat h_{\rm C}>0\), and propagates only admissible trajectories. The paper illustrates two sample runs: one with \(\hat h_{\rm C}>0\) leading to a collision (constraint violation), and one with \(\hat h_{\rm C}=0\) where the controller enforces the corridor and avoids obstacles [2504.12036].

## 6. Significance for Modular and Confidential Control Design

The introduction of FVF enables modular, decoupled design in hierarchical control systems. By using an explicit, contract-based interface, the high-level planner need not have explicit access to the lower-level model, cost, or constraint definitions. This modularity supports model confidentiality and IP protection—a substantive concern in industrial and safety-critical domains—while maintaining system-wide feasibility guarantees. The FVF’s role as a cost-to-go proxy for constraint satisfaction also links it to viability theory and enables further generalizations for scenarios with time-varying, nonlinear, or nonconvex constraints [2504.12036].

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**Summary Table: Core Properties of the Feasibility Value Function**

| Property                  | Mathematical Description                          | Control-Theoretic Significance            |
|---------------------------|---------------------------------------------------|-------------------------------------------|
| Zero-level set            | \(\{(x,r): V_f(x,r)=0\}\)                         | Finite-horizon viability kernel           |
| Value outside kernel      | \(V_f(x,r)>0\)                                    | Minimal total constraint violation needed |
| Contractability           | Explicit LUT or NN approximation possible         | Modularization; enables offline exchange  |
| Monotonicity              | Non-increasing under optimal relaxed policy       | Cost-to-go behavior, feasibility margin   |

Source: https://www.emergentmind.com/topics/feasibility-value-function-fvf