Receding-Horizon Control Framework

• Receding-Horizon Control is a feedback method that solves a finite-horizon optimal control problem at each step to compute real-time control actions.
• The framework integrates formal methods like temporal logic and automata to enforce complex state/input constraints and mission-specific goals.
• Recent advances extend RHC to hybrid, stochastic, and data-driven settings, enhancing its applicability in robotics, autonomous systems, and beyond.

Receding-Horizon Control (RHC) Frameworks are a class of feedback control methodologies that compute control actions through an online sequence of optimal planning problems, each solved over a finite future time window that moves forward in time as the system evolves. This approach, often referred to interchangeably with Model Predictive Control (MPC), offers a principled way to incorporate complex performance objectives, state/input constraints, uncertainty, and logical or combinatorial specifications into real-time, dynamically adaptive control laws. Recent developments—across finite as well as infinite-dimensional systems, deterministic and stochastic settings, and data-driven scenarios—demonstrate the versatility and theoretical depth of receding-horizon paradigms.

1. Problem Formulation and Core Principles

At its core, the receding-horizon control protocol is implemented as follows:

• At each decision instant, the controller solves an optimal control problem forecasting the future evolution of the system over a finite prediction horizon $N$.
• The problem is formulated to minimize a (possibly time-varying) cost (e.g., stage costs in LQR, cumulative rewards, sparsity-inducing penalties, or regret relative to a clairvoyant policy) subject to the plant dynamics and operational constraints.
• Only the first element (or a prescribed portion) of the optimal input sequence is applied; the process is repeated at the next time step with updated state information and (potentially) new environment data.

The RHC method enables

• explicit treatment of state/input and other constraints,
• direct handling of non-standard objectives (e.g., energy budgets, logical/temporal goals, risk/variance, or sparsity), and
• integration with estimation layers (observers, parameter estimators, data-driven identification).

Mathematically, the RHC law often takes the form: $$u^*_k = \text{first}(\arg\min_{u_{k:k+N-1}} J_k(y_{k:k+N}, u_{k:k+N-1}))$$ subject to dynamics, constraints, and possibly other side information (e.g., predicted disturbances, constructed automata, or learned models).

2. Advanced Specifications: Temporal Logic and Formal Constraints

Classical cost-based objectives in RHC have been extended to enforce logical and temporal task satisfaction. For systems modeled as finite deterministic transition systems, high-level tasks are often encoded as linear temporal logic (LTL) formulas. These logical formulas (e.g., $$G F\,\mathsf{survey} \,\wedge\, G\,\neg\,\mathsf{unsafe}$$) can require, for example, persistent satisfaction of surveillance and safety constraints (Ding et al., 2012).

Key features of these approaches include:

• Construction of a product automaton, $P = T \times B$, incorporating system dynamics $T$ and a Büchi automaton $B$ that accepts precisely the set of runs satisfying the LTL formula.
• Definition of an “energy” function $V(p)$ on $P$ (analogous to a Lyapunov function) that quantifies distance to an accepting condition, and imposition of terminal constraints that ensure monotonic progress toward satisfaction of LTL goals.
• Receding-horizon optimization that simultaneously maximizes local (possibly time-varying) rewards and enforces constraints guaranteeing infinite-horizon satisfaction of the logical specification.

This framework (see table below) enables the synthesis of controllers blending real-time reward optimization with formal methods for correctness.

Component	Description	Functionality
Product Automaton	$P = T \times B$ for dynamics $T$ and LTL automaton $B$	Encodes combined dynamics/specs
Energy Function $V$	Shortest-path to accepting set $F^*$ on $P$	Enforces progress towards LTL goal
Constraints	Terminal constraints on $V$	Guarantees satisfaction

3. Extensions for Hybrid, Probabilistic, and Nonlinear Systems

The RHC framework has been generalized along multiple axes beyond deterministic linear systems:

• Energy-Constrained Systems: The Interval-wise RHC strategy (Arvelo et al., 2012) adapts the prediction horizon to always encompass whole energy-constrained intervals, preventing partial coverage, and enforces contractive final state constraints to guarantee analytic asymptotic stability.
• Stochastic and Probabilistic Models: Hybrid frameworks have been developed for
  • continuous-time stochastic nonlinear systems with probabilistic state constraints—featuring a two-layer design (deterministic planning + stochastic optimal control with exit constraints) (Shah et al., 2012);
  • linear systems with probabilistic parametric uncertainty via generalized polynomial chaos expansions, converting the original stochastic system into a deterministic system with higher state dimension (Bhattacharya et al., 2014);
  • state-dependent jump linear systems subject to chance constraints managed via cumulative distribution function or ellipsoidal approximations—ensuring mean square boundedness using Lyapunov-based analysis and linear matrix inequalities (Chitraganti et al., 2014).
• Data-Driven and Model-Free Variants: Algorithms in this category leverage operational data to update consistency sets of feasible models in a receding-horizon fashion. Robust stability and constraint satisfaction are established using tools from semidefinite programming, set membership, and duality theory (Zheng et al., 7 Oct 2025, Zheng et al., 7 Oct 2025).

4. Optimization Structures, Constraints, and Guarantees

The RHC optimization problem typically merges multi-stage objectives (tracking error, energy, variance, risk, sparsity) with physical and operational constraints (input/state bounds, safety, logic, or energy) and terminal ingredients (e.g., value function, contractive set, terminal penalty):

• Terminal Constraints: Utilized for ensuring recursive feasibility and performance guarantees. Examples include contractive constraints $\|x(k+h_c)\|^2 \leq \beta^{i+1} \|x(0)\|^2$ (Arvelo et al., 2012), Lyapunov-based inequalities, and enforcement of logical acceptance conditions in automata.
• Recursive Feasibility and Stability: Proven typically either by explicit construction of invariant (possibly contractive) sets, verifying that feasible solutions exist for all future steps, or Lyapunov-based decrease arguments.
• Probabilistic Safety and Risk Constraints: Formulated via chance constraints translated into deterministic conditions using quantile/ellipsoidal approximations; used in settings with unbounded noise or stochastic switching (Chitraganti et al., 2014).
• Learning-Based and Regret-Mininimizing Objectives: Dynamic regret relative to a clairvoyant (perfect hindsight) policy is minimized in recent RHC schemes, with stability and bounded regret established through new monotonicity properties of the clairvoyant cost sequence and convex-concave optimization (Martin et al., 2023).

5. Data-Driven, Sparse, and Output-Based RHC Architectures

Recent advances address settings where system identification is partial or infeasible:

• Data-Driven Robust RHC: Instead of relying on a fixed (possibly inaccurate) plant model, the controller refines its knowledge recursively—using the accumulated execution data to tighten the polytopic consistency set for the plant. Robust optimization is posed over this set, with the dual constraints solved via semidefinite or linear programming (Zheng et al., 7 Oct 2025, Zheng et al., 7 Oct 2025). These algorithms guarantee robust constraint satisfaction and improved contractivity/ultimate boundedness compared to non-updating data-driven methods.
• Sparse and Minimum Attention MPC: To reduce actuator activity, RHC objectives have been augmented with explicit $\ell_0$ (zero-norm) sparsity constraints on input changes. This is formulated as nonconvex optimization (penalizing the number of nonzero input changes), which is solved by alternating minimization: quadratic programming for continuous variables and closed-form sparse projection for the combinatorial variables (Teja et al., 28 Jul 2025).
• Output-Based RHC for Infinite-Dimensional Systems: For systems where only output (not full state) measurements are available—common in PDE control—receding-horizon architectures are coupled with Luenberger (or observer-based) state estimation. By designing sets of localized actuators and sensors as indicator functions, and updating control via the receding horizon of the estimated state, exponential stabilization can be rigorously proven even on infinite-dimensional parabolic systems (Azmi et al., 16 Jul 2024).

6. Applications and Empirical Performance

RHC frameworks have been applied to a broad range of domains and validated empirically:

• Autonomous Systems and Robotics: Receding horizon estimation and control, enhanced by structured parameter blocking, yield robust slip compensation for mobile robots navigating uncertain terrain (Wallace et al., 2018).
• Cyber-Physical Systems under Interval-Wise Power Constraints: Variable-horizon, contractive RHC ensures asymptotic stability despite energy budget constraints (Arvelo et al., 2012).
• Mixed Traffic and Platooning: By embedding real-time, data-driven prediction of human-driven vehicles’ trajectories, safety-prioritized RHC enables safe and efficient platoon formation in CAV/HDV mixed highway environments, leveraging recursive least-squares identification (Mahbub et al., 2022).
• Multi-Agent and Game-Theoretic Systems: Distributed RHC frameworks apply to aggregative games, ensuring convergence to equilibrium under periodic constraints via Lyapunov-based monotonicity (Fele et al., 2022).
• Data Routing, Energy and Process Systems, and Beyond: Performance improvements, versatility (e.g., in hybrid or infinite-dimensional PDE-controlled systems (Azmi et al., 2019, Azmi et al., 16 Jul 2024)), and resiliency against model uncertainty are consistently observed in simulations.

Empirical results are typically presented in terms of tracking error, control effort, sparsity measures, state contraction/decay rates, constraint satisfaction frequency, or regret versus optimal/clairvoyant baselines—demonstrating the capacity of RHC/MPC to outperform both purely open-loop and less reactive static or greedy strategies across diverse settings.

7. Theoretical and Computational Considerations

RHC framework design and analysis rely on:

• Finite-horizon approximations to infinite-horizon (stationary) optimal control, with error bounds derived from dynamic programming and turnpike property theory (Breiten et al., 2018, Kunisch et al., 2018).
• Convex, nonconvex, and minimax optimization, with tractable relaxations using duality, semidefinite programming, sum-of-squares (SOS) certificates, and structural insights from system level synthesis.
• Recursive feasibility and Lyapunov-based arguments underpinning formal guarantees of closed-loop stability, constraint satisfaction, and (in stochastic settings) probabilistic safety or almost-sure convergence.
• Algorithmic strategies—ranging from alternating minimization, block-structured estimation (for adaptive parameters), structured prediction, and policy gradient methods—that afford computational tractability and scalability.
• Handling of nonlinearity and infinite-dimensionality is managed either via local linearization, lifting techniques (e.g., polynomial chaos, Koopman), or carefully crafted observer-controller separations.

The comprehensive integration of optimization, learning, control-theoretic, and formal verification tools marks receding-horizon control frameworks as a cornerstone of both contemporary theoretical research and practical application in constrained, uncertain, and complex systems.