Hierarchical Longitudinal Control

• Hierarchical longitudinal control is a modular, multi-layered approach that splits control tasks into high-level planning and low-level actuation to optimize vehicle dynamics.
• It integrates model-based, optimization-based, and learning-based techniques to enhance performance, robustness, and real-time safety under uncertainty.
• The architecture enables independent design and safety guarantees through contracts, control barrier functions, and end-to-end learning strategies for complex autonomous systems.

A hierarchical longitudinal control architecture is a modular, multi-layered approach to regulating the longitudinal dynamics (e.g., speed, spacing, energy) of vehicles or robotic systems. By decomposing the overall control task into semantically distinct temporal or functional strata, these architectures realize improved performance, robustness, safety, modularity, and extensibility under uncertainty and real-time constraints. Major instantiations include model-based, optimization-based, and learning-based frameworks, some tailored for specific domains such as traffic, autonomous driving, and robotics.

1. Structural Principles of Hierarchical Longitudinal Control

Hierarchical longitudinal control architectures consistently comprise at least two layers, differentiated by update frequency, temporal horizon, and operational role:

• High-level (outer-loop) controller: Operates at a slower rate, producing abstract reference trajectories, setpoints, or plans for the system's desired longitudinal behavior (e.g., speed profile, inter-vehicle gaps). Its optimization criterion often spans a longer temporal horizon and incorporates strategic objectives such as fuel efficiency (Hammerl et al., 15 Nov 2025), reference feasibility (Berkel et al., 16 Apr 2025), or high-level reward maximization (Zhang et al., 25 Jan 2025).
• Low-level (inner-loop) controller: Runs at a higher rate, directly actuating the system to track the supplied references. It addresses fast-timescale plant dynamics, disturbance rejection, and constraint satisfaction, possibly using model predictive control (MPC), PID regulation, or reinforcement learning (Shi et al., 2020, Li et al., 2023, Liu et al., 2020).

This separation enables independent design, analysis, and real-time deployment, while systematically handling uncertainty, constraints, and multi-objective criteria by allocating them appropriately across levels.

2. Data-driven and Model-based Hierarchical Approaches

A canonical instance is the Data-driven Hierarchical Control (DHC) architecture tailored for systems with uncertain dynamics, as introduced in "A Data-driven Hierarchical Control Structure for Systems with Uncertainty" (Shi et al., 2020). The architecture is encapsulated by the following loop:

• Data-driven model identification (DMDc): At each control step, a Dynamic Mode Decomposition with control (DMDc) algorithm fits a linear map between sequences of past refined reference inputs and resulting state outputs, solving

$$\begin{align*} \text{minimize } &\|\Xi_P - [A\, B]\Omega\|_F\ \text{with closed-form:}\quad &[A\, B] = \Xi_P\Omega^\dagger \end{align*}$$

where $\Xi$, $X$, $\Omega$ collect windowed reference and state data.

• High-level reference-shaping controller: Uses the model estimates $(\bar{A}_k, \bar{B}_k)$ to filter the nominal reference $r_{k+1}$:

$$\hat{r}_{k+1} = \bar{A}_k\,\hat{r}_k + \bar{B}_k\,r_{k+1}.$$

The low-level controller and true plant then track $\hat{r}_{k+1}$.

• Stability and robustness: The closed-loop retains Lyapunov stability provided the inner loop is stable. Explicit sensitivity bounds (both classical worst-case and tight bounds) on $\| \delta Z \|/\|Z\|$ with respect to noisy data are derived, quantifying robustness.
• Online adaptation: Minimal data is required for online implementation, supporting real-time deployment in safety-critical domains (e.g., altitude control for aerial robots).

Simulation (planar quadrotor with mass/inertia uncertainty) and hardware validation (Crazyflie quadrotor under strong ground-effect) show that augmenting standard low-level controllers with online DHC yields up to 50% reduction in altitude error without destabilization or excessive effort (Shi et al., 2020).

3. Optimization and Contract-based Hierarchies

Optimization-centric hierarchies, exemplified by contract-based hierarchical MPC, further modularize the reference-generation and constraint-verification interface (Berkel et al., 16 Apr 2025). The structure is:

• High-level planner: Solves a nonconvex trajectory generation problem, outputting desired speed or position references on a coarse timescale.
• Low-level soft-constrained MPC: Tracks held-constant references, enforcing state, input, and deviation constraints with slack variables:

$$\min_{x,u,\xi}\sum_{l=0}^{N-1} \ell(x_l,u_l,r^H) + w_\xi \sum_{l=0}^{N-1} \|\xi_l\|_1$$

with $\xi_l^*=0$ only when hard constraints are feasible.

• Contract via predictive feasibility value function: An offline-computed contract $h_C(x, r^H)\approx h^*(x,r^H)$, where $h^*(x,r^H)$ is the minimal slack cost, is approximated by a neural network. The high-level planner queries $h_C$ at each candidate $r^H$; only references with $h_C=0$ (hard feasible) are deployed, guaranteeing constraint satisfaction of the MPC.
• Computational layering: Offline contract computation is intensive (sampling, LPs, NN training), but online evaluation is efficient (sub-millisecond NN inference), making the approach viable for real-time planning (Berkel et al., 16 Apr 2025).

Specialization to longitudinal vehicle control demonstrates that contract-based filtering prevents infeasible velocity profiles, as verified in autonomous driving scenarios.

4. Hierarchical Architectures in Cooperative and Distributed Longitudinal Control

Further dimensions are introduced by cooperative, multi-agent contexts, as in sequencing-enabled on-ramp merging for connected automated vehicles (CAVs) (Li et al., 2023). The architecture integrates:

• Upper-level sequencing controller: Solves a mixed-integer program to allocate CAVs to a merging sequence optimizing both macro-scale (mainline/ramp density) and micro-scale (spacing deviation, sequencing precision) objectives under assignment/non-overtaking constraints.
• Lower-level distributed MPC: Each vehicle solves an MPC for longitudinal tracking of assigned virtual car-following slots, including soft safety penalties and terminal constraints for speed/acceleration matching. Predicted trajectories of predecessors are communicated via V2V, embedding a distributed implementation.
• Stability properties: Asymptotic local stability and ℓ₂-norm string stability are explicitly proved for the resulting virtual platoon, with quantifiable enlargement of the terminal-set controllable region over classical MPC, directly addressing the large spacing deviations common in merging scenarios.

This architecture achieves real-time viability and outperforms baseline FIFO or distance-based methods in both transient convergence and steady-state feasibility, according to extensive simulation (Li et al., 2023).

5. Learning-based Hierarchical Longitudinal Control

Hierarchical control is exploited in deep reinforcement learning (DRL) to address temporal abstraction and credit assignment for long-horizon traffic scenarios (Zhang et al., 25 Jan 2025). The key structure is:

• High-level decision maker: At a low frequency (~1 Hz), outputs abstract longitudinal goals (e.g., target speed adjustment) and lateral plans (e.g., lane change intent) optimized for long-term cumulative reward.
• Low-level controller: Operates at higher frequency (~2 Hz), converting high-level goals plus raw observations into discrete actuator commands (e.g., $\pm1$ m/s² acceleration), addressing short-term safety and smoothness.
• Two-step training: High-level policy is first trained using a rule-based low-level controller to ensure goal attainment; low-level controller is subsequently trained on frozen high-level goals. Both employ double DQN, with appropriately decomposed state and action spaces.
• Performance: This approach yields significantly higher escape rates from "trap" traffic configurations and better average reward and velocity than flat DRL or single-level approaches (e.g., +297% reward, +30% speed in test) (Zhang et al., 25 Jan 2025).

Explicitly separated timescales enable effective exploration and exploitation of long-term strategies not accessible in monolithic DRL.

6. Safety, Stability, and Economic Layers in Hierarchical Longitudinal Control

Recent architectures for autonomous truck platoons introduce a tri-layered hierarchy with explicit guarantees on safety, string stability, and operational efficiency (Hammerl et al., 15 Nov 2025):

• Safety projection filter: Runs at maximal control bandwidth, enforcing forward invariance of a velocity-aware headway set by a high-order control barrier function (CBF), accounting for bounded actuator lag. Each control command is projected into the set of safe accelerations as determined by the CBF constraint.
• Spacing-regulation (lag-aware PID) controller: Shapes spacing error dynamics into a tunable second-order system (parameters: damping $\zeta$, natural frequency $\omega_n$), with explicit compensation for actuator lag. Gain selection ensures closed-loop $L_\infty$ string stability provided $\omega_n\tau_a \le 0.3$–$0.35$.
• Economic optimizer: At a slow, event-triggered rate, selects the leader's cruise speed by minimizing a long-term cost $J(v)$ combining aggregate platoon fuel use (with drag coefficients modulated by inter-vehicle gaps) and schedule alignment, subject to comfort and actuation constraints.

Propositions show convergence to the Optimal Velocity Model with Relative Velocity (OVRV) and provide explicit stabilization time bounds in terms of system and tuning parameters. Simulation shows suppression of string oscillations, strict collision avoidance, and 0.32–0.39 L/100 km fuel economy in eight-truck platoons, exceeding classic baselines (Hammerl et al., 15 Nov 2025).

7. End-to-End Learning Architectures and Differentiable Hierarchies

Advanced learning-based hierarchies implement fully differentiable control stacks mapping raw sensory input to longitudinal actuation (Li et al., 2023):

• Visual predicate extractor: CNN encodes a BEV scene image into a vector of learned semantic predicates.
• Visual Automaton Generative Network (vAGN): Interprets predicates as weighting matrices for a learned discrete state machine over latent behavior modes (e.g., cruise, brake). State transitions are learnable and differentiable.
• Dynamic Movement Primitives (DMP): The vAGN output modulates DMP parameters (damping, stiffness) for a point-attractor second-order ODE on position, producing continuous acceleration commands. Stability is enforced via regularization ($\alpha_y, \beta_y > 0$, critical damping conditions).

End-to-end training by behavior cloning demonstrates low displacement error (ADE = $5.9\pm1.4$ m) and goal-tracking distance (8.2 m), with near-human comfort and safety. Sample efficiency is enhanced by the intrinsic stability of DMPs; architectural hyperparameters govern a trade-off between explainability and representational power (Li et al., 2023).

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In summary, hierarchical longitudinal control architectures integrate disparate temporal, functional, or informational modules while guaranteeing performance, safety, and scalability across a range of real-world domains. They enable robust feedback adaptation under uncertainty, modular interfaces for planner-controller decoupling, provable safety (CBFs, contracts), and generalization to multi-agent, data-driven, and learning-augmented applications (Shi et al., 2020, Berkel et al., 16 Apr 2025, Li et al., 2023, Liu et al., 2020, Hammerl et al., 15 Nov 2025, Zhang et al., 25 Jan 2025, Li et al., 2023).