Receding-Horizon Control (RHC) Frameworks

• Receding-Horizon Control is a framework that iteratively solves finite-horizon optimal control problems to effectively manage state and input constraints.
• It leverages mathematical optimization with robust terminal costs to ensure stability, recursive feasibility, and convergence in both centralized and distributed settings.
• RHC is widely applied in multi-agent coordination, robotics, energy systems, and safety-critical domains, offering scalable and real-time solutions.

Receding-Horizon Control (RHC) Frameworks

Receding-Horizon Control (RHC), also commonly referred to as Model Predictive Control (MPC) in the optimal and robust control literature, is a family of feedback control methodologies built on the repeated solution of finite-horizon optimal control problems. At each time step, a (typically constrained, multi-step) optimization is solved using the latest available state information; only the first control input of the optimal sequence is implemented before the process repeats at the next time interval. RHC frameworks have become central in the design of constrained control, coordination, estimation, and learning strategies for a variety of systems including multi-agent networks, robotics, energy systems, and large-scale infrastructure.

1. Mathematical Foundations and Problem Statement

The RHC paradigm formalizes control synthesis as a rolling sequence of finite horizon optimal control subproblems. At time step $k$, for a discrete-time system $x_{k+1}=f(x_k, u_k)$ with $x_k \in \mathbb{R}^n$, $u_k \in \mathbb{R}^m$, and possibly constraints $x_k \in \mathcal{X}$, $u_k \in \mathcal{U}$, the canonical receding-horizon problem is

$$\begin{aligned} \min_{u_{0:N-1}} \quad & J_k(u_{0:N-1}) = \sum_{h=0}^{N-1} \ell(x_{h|k}, u_{h|k}) + V_f(x_{N|k}) \ \text{subject to} \quad & x_{h+1|k} = f(x_{h|k}, u_{h|k}), \; x_{0|k} = x_k \ & u_{h|k} \in \mathcal{U}, \; x_{h|k} \in \mathcal{X}, \; x_{N|k} \in \mathcal{X}_f \end{aligned}$$

where $N$ is the horizon, $\ell$ is typically a stage (tracking or regulation) cost, and $V_f$ is a terminal penalty. The resulting $u_{0:N-1}^*$ is recomputed at each timestep, and only $u_0^*$ is applied. Iteration over this basic routine constitutes the core of any RHC framework.

Several essential features distinguish the RHC approach:

• It enables direct handling of state and control constraints.
• The performance relies on proper selection of horizon $N$, cost weights, terminal cost, and, if present, terminal constraint sets.
• With sufficiently long horizons and suitable terminal costs, it is possible to guarantee closed-loop properties such as recursive feasibility, stability, and even exponential convergence to optimal control in linear-quadratic or convex setups (Breiten et al., 2018, Kunisch et al., 2018).

2. Structural Design: Constraints, Optimality, and Distributed Schemes

RHC schemes support a variety of structural extensions:

• Constraints: RHC is inherently suited for problems with polyhedral, nonlinear, or nonconvex state/input constraints, such as actuator saturations, safety regions, or output constraints. For example, consensus of linear multi-agent systems with input limits is enforced by constraining $u_k \in \mathcal{U}$ and using invariant terminal sets (Li et al., 2016).
• Optimality and Inverse Optimality: For linear systems, the use of optimal or "inverse optimal" state-feedback protocols (e.g., Riccati-based, ARE/LQR) ensures that the closed-loop law minimizes a quadratic cost and achieves consensus or stabilization under prescribed performance indices coupled to the system and network topology (Li et al., 2016).
• Distributed RHC: In multi-agent networks, RHC can be split into local agent-level subproblems with distributed assignment of stage and terminal costs. This decomposition leverages graph symmetry, separable Lyapunov factors, and ensures local optimizations require only neighbor information, sharply reducing communication/computation overhead (Li et al., 2016, Li et al., 2014).
• Feasibility and Recursive Feasibility: Terminal set and cost design ensures that, if a feasible sequence exists at one time, shifted sequences plus the terminal control remain feasible at subsequent steps, allowing indefinite repeatability and real-time rollout (Li et al., 2016, Kunisch et al., 2018).

3. Consensus, Coordination, and Learning in Multi-Agent and Networked Systems

RHC is particularly effective for networked systems, including formations, swarms, and large-scale resource allocation:

• Consensus via Distributed RHC: For consensus of discrete-time agent networks $x_i(k+1)=A x_i(k)+B u_i(k)$, each agent solves a horizon-$N$ RHC problem that penalizes deviations from neighbor states both along the horizon and at the terminal point. The explicit form of the optimal protocol is computable via network-coupled Difference Riccati Equations (DREs), and consensus is characterized via Schur criteria on lifted system matrices dependent on graph Laplacian eigenvalues (Li et al., 2014).
• Explicit Feedback and Scalability: By embedding coupled neighbor penalties directly into the stage/terminal cost, the protocol enables pre-computation and explicit algebraic feedback (no online optimization required), yielding favorable complexity and scalability for large networks (Li et al., 2014).
• Learning and Adaptive RHC: Adaptive frameworks such as 'receding horizon learning' integrate online system identification with RHC for unknown or partially known dynamics, yielding provable global asymptotic convergence under minimal structural assumptions and solely utilizing measured input/output data (Ebenbauer et al., 2020). Robust data-driven RHC further combines set-membership identification and robust contractivity optimization for unknown plants with bounded disturbances (Zheng et al., 7 Oct 2025).

4. RHC for Estimation, Safety, and Hybrid Systems

Modern RHC frameworks are not limited to direct control, but have been extended to estimation and safety-critical contexts:

• Estimation and Output Feedback: For state and parameter estimation in the presence of modeling uncertainty (e.g., robot slip), RHC can be coupled with receding horizon estimation (RHE), yielding online moving-horizon optimization-based observers. Structured noise blocking and overlapping-block parameterization can be introduced to improve identifiability and estimation-tracking performance in time-varying environments (Wallace et al., 2018).
• Safety via Barrier Functions: High-order control barrier functions (CBFs), possibly coupled with Lyapunov functions (CLFs), can be enforced as constraints in the RHC subproblem to guarantee forward invariance of safety sets, thereby certifying safety, even under learned or adaptive parameterizations via reinforcement learning (RL) (Sabouni et al., 2024).
• Hybrid and Infinite-Dimensional Systems: RHC extends naturally to PDEs and infinite dimensional problems by discretizing actuators and coupling with output-based observers, enabling stabilization via micro-localized actuation and measurement, and providing theoretical guarantees under spectral controllability/observability conditions (Azmi et al., 2019, Azmi et al., 2024).

5. Computational Methods and Real-Time Implementation

Efficient real-time feasibility of RHC frameworks is achieved through:

• Sparse and Structured Nonlinear Programming: Direct multiple-shooting, sequential quadratic programming (SQP), and barrier-augmented cost formulations enable high-frequency (10–100 Hz) solution of large-scale safety-critical or high-dimensional planning problems, e.g., autonomous vehicle navigation in dense traffic (Zheng et al., 2023).
• Model Order Reduction (MOR): Data-driven reduced-order models, such as Proper Orthogonal Decomposition (POD) or certified Galerkin approaches, drastically reduce the computational burden of solving repeated large-scale optimal control subproblems, preserving (certified) closed-loop stability and suboptimality (Azmi et al., 22 Aug 2025, Azmi et al., 2024).
• Decomposition and Parallelization: Tractable planning in infrastructure systems is obtained by decomposing large-scale network optimizations (e.g., battery placement) via Benders decomposition, enabling practical solution of otherwise intractable horizon-spanning problems (Fortenbacher et al., 2016), and via event-driven, localized updates for discrete-event and persistent monitoring problems (Chen et al., 2019, Welikala et al., 2021).

6. Theoretical Properties: Stability, Suboptimality, and Robustness

RHC frameworks are underpinned by rigorous theoretical results:

• Stability and Turnpike Theory: For linear-quadratic or parabolic systems, explicit error estimates demonstrate that the RHC feedback control converges exponentially to the infinite-horizon optimal control as the prediction horizon increases, provided the system is stabilizable and detectable (Breiten et al., 2018, Kunisch et al., 2018). This is often formalized via Lyapunov function arguments and Bellman’s principle.
• Robustness and Uncertainty: Moment constraints, chance constraints, and comprehensive handling of model/data uncertainty are realized via polynomial chaos expansions, robust one-step contractivity optimization, and set-membership identification; these approaches yield provable ultimate boundedness or safety, even when only partial structural knowledge is available (Bhattacharya et al., 2014, Zheng et al., 7 Oct 2025, Shah et al., 2012).
• Estimation and Observer Coupling: Output-based stabilization is attained through proofs that combine the exponential convergence of block observers (e.g., Luenberger observers) with receding-horizon squeezing properties, ensuring that the closed-loop state converges to equilibrium exponentially, even under finite-dimensional output sampling (Azmi et al., 2024).

7. Applications and Impact

RHC frameworks have broad impact across domains:

• Multi-agent Coordination: Optimal consensus and distributed stabilization in dynamically heterogeneous, input-constrained networks (Li et al., 2016, Li et al., 2014).
• Robotics and Autonomous Vehicles: Path tracking and adaptive slip estimation in robotics, real-time dense traffic planning using spatiotemporal barrier-augmented costs, and manipulator-based aerial vehicle interception (Wallace et al., 2018, Zheng et al., 2023, Wu et al., 14 May 2025).
• Cyber-Physical Infrastructure: Real-time distributed persistent monitoring, ride-sharing systems, and optimal energy storage management under uncertainty and economic constraints (Chen et al., 2019, Welikala et al., 2021, Fortenbacher et al., 2016).
• Stochastic and Infinite-Dimensional Systems: RHC schemes for nonlinear stochastic systems with probabilistic state constraints, and parabolic PDEs with sparse actuator structures and model-order reduction (Shah et al., 2012, Azmi et al., 22 Aug 2025).

The flexibility, rigorous performance guarantees, and computational viability of RHC frameworks ensure their continuing adoption as the default paradigm for constrained, real-time, and safety-critical control in networked and large-scale systems.