FAIR Data Principles: A Data Stewardship Guide

• FAIR Data Principles are guidelines ensuring that research data is Findable, Accessible, Interoperable, and Reusable.
• They enhance data management by encouraging standardized metadata practices and robust access protocols.
• Adoption of FAIR principles boosts collaboration, reproducibility, and transparency in academic and scientific research.

Finite-Control-Set Model Predictive Control (FCS-MPC) is a discrete-time optimal control paradigm in which the input at each sample is constrained to a finite (typically small and enumerative) set. This finite input structure arises fundamentally from the switched or quantized nature of actuators in key technological domains such as power electronics and energy conversion, including multilevel inverters, DC-DC converters, and microgrid interfaces. FCS-MPC frameworks have been systematically developed to provide direct, non-modulated gate-level control of switching devices, enabling both rapid dynamic response and explicit treatment of hard nonlinearities and constraints—properties not available in classical continuous-input MPC.

1. Mathematical Formulation and Problem Class

FCS-MPC problems are formulated on discrete-time dynamical systems of the form

$$x_{k+1} = A_{u_k}\,x_k + B_{u_k}\,u_k + f_{u_k}, \quad y_k = C_{u_k}\,x_k + d_{u_k},$$

with state $x_k \in \mathbb{R}^{n_x}$, measured output $y_k \in \mathbb{R}^{n_y}$, and control $u_k$ selected from a finite set $$\mathcal{U} = \{u^1, \dots, u^{N_s}\} \subset \mathbb{R}^{n_u}$$. The system matrices and affine terms are indexed by the control mode $u$, characterizing the switched-affine regime (Xu et al., 2024).

At each sampling instant, FCS-MPC solves an open-loop optimal control problem over horizon $N$:

$$\min_{\{u_{i|k}\}_{i=0}^{N-1} \subset \mathcal{U}} \sum_{i=0}^{N-1} \ell\big(x_{i|k}, u_{i|k}\big) + V_f\big(x_{N|k}\big)$$

subject to system dynamics and constraints. The first element of the optimal input sequence is applied; the process repeats at the next time step (receding horizon).

The cost function $\ell(\cdot, \cdot)$ encodes tracking performance and/or switching penalties, and $V_f$ is a terminal cost, often designed to ensure recursive feasibility and stability in the absence of continuous input regularity.

2. Limit-Cycle and Practical Stability

A fundamental distinction in FCS-MPC arises due to the quantized nature of $x_k \in \mathbb{R}^{n_x}$0. Classical MPC with a continuous control set can enforce asymptotic convergence to an exact equilibrium, but FCS-MPC under typical cost functions only yields practical stability: trajectories converge to a (potentially non-singleton) invariant set surrounding the desired steady state. Within this set, the switching law may produce arbitrary chattering, resulting in unpredictable switching patterns and output ripple (Xu et al., 2024, Xu et al., 2022).

Advances in limit-cycle stabilization for FCS-MPC leverage the design of cost functions and terminal ingredients to guarantee asymptotic convergence to a specific precomputed limit cycle $x_k \in \mathbb{R}^{n_x}$1 of period $x_k \in \mathbb{R}^{n_x}$2. The FCS-MPC problem then tracks deviations $x_k \in \mathbb{R}^{n_x}$3 and employs $x_k \in \mathbb{R}^{n_x}$4-periodic terminal sets and costs that enforce recursive feasibility and a Lyapunov-type decrease relative to the cycle (Xu et al., 2024, Xu et al., 2022).

The theoretical framework includes:

• Construction of $x_k \in \mathbb{R}^{n_x}$5-periodic invariant tubes $x_k \in \mathbb{R}^{n_x}$6 containing the cycle states.
• Periodic terminal costs $x_k \in \mathbb{R}^{n_x}$7 and terminal control law $x_k \in \mathbb{R}^{n_x}$8.
• Lyapunov-type decrease conditions of the form

$x_k \in \mathbb{R}^{n_x}$9

for all $y_k \in \mathbb{R}^{n_y}$0.

This construction enables both recursive feasibility and asymptotic convergence (tracking error $y_k \in \mathbb{R}^{n_y}$1) to the chosen orbit rather than to arbitrary points within a set, yielding predictable steady-state behavior and quantized switching frequency (Xu et al., 2024, Xu et al., 2022).

3. Terminal Set, Cost Construction, and Computation

The terminal ingredients play a decisive role in achieving convergence or practical stability:

• Quadratic terminal costs via LMIs: For $y_k \in \mathbb{R}^{n_y}$2, solve for positive definite $y_k \in \mathbb{R}^{n_y}$3 such that

$y_k \in \mathbb{R}^{n_y}$4

for each $y_k \in \mathbb{R}^{n_y}$5. Terminal costs $y_k \in \mathbb{R}^{n_y}$6 and sets constructed accordingly (Xu et al., 2024).

• Ellipsoidal invariant tubes: Parametrize each section as $y_k \in \mathbb{R}^{n_y}$7, optimize $y_k \in \mathbb{R}^{n_y}$8 via SDP, and ensure inter-stage forward-invariance under $y_k \in \mathbb{R}^{n_y}$9 (Xu et al., 2024).
• Polytopic invariant tubes: Employ set-iteration from $u_k$0, iterating

$u_k$1

for $u_k$2 mod $u_k$3, terminating at fixed points (Xu et al., 2024).

Such procedures enable systematic, convex computation of the terminal sets required for rigorous Lyapunov arguments in the finite-control setting, in contrast to the continuous-set case.

4. Algorithmic Structure, Complexity, and Real-Time Implementation

The cardinality of the input space $u_k$4 and the prediction horizon $u_k$5 induces a combinatorial complexity $u_k$6 for brute-force enumeration. Mitigation techniques include:

• Problem-specific problem decomposition (converter-by-converter or subsystem-by-subsystem).
• Parallelization and warm-starting in online search.
• Tail-cost approximation via offline estimated infinite-horizon value functions (e.g., using approximate dynamic programming and iterated Bellman inequalities), which allows short-horizon (e.g., $u_k$7 or $u_k$8) FCS-MPC to match long-horizon performance with lower complexity (Stellato et al., 2015).
• Mixed-integer programming solvers with explicit enumeration optimized for embedded hardware (e.g., FPGAs) (Stellato et al., 2015).
• Sphere decoding for least-squares–like cost reformulations (Klädtke et al., 2024), binary relaxation methods (Makarow et al., 2024), or semidefinite relaxations to obtain near-optimal feasible input sequences efficiently (Hartmann et al., 2024).

For applications with extremely stringent real-time constraints (e.g., $u_k$9s), tailored hardware implementations achieve deterministic and sub-millisecond solve latencies even for finite-control sets of moderate size (Stellato et al., 2015).

5. Applications, Case Studies, and Design Trade-Offs

FCS-MPC is widely used in power electronics, electrical drives, and grid-forming/converter-dominated microgrids:

• Power converter and inverter control: Direct current and voltage regulation, switching frequency limitation or targeting, and THD minimization (Stellato et al., 2015, Hartmann et al., 2023).
• Microgrids and DER interface: Simultaneous voltage/frequency regulation and decentralized droop-based power sharing using FCS-MPC as the inner loop; this approach circumvents classical PI-tuning and yields faster transients and superior steady-state regulation (Olajube et al., 2024).
• Precision motion/actuation: Limit-cycle tracking for quantized power amplifiers, producing minimized output ripple and predictable switching (Xu et al., 2022).
• Hybrid microgrids: Unified FCS-MPC architectures for converters, battery interfaces, and PV MPPT, using tailored cost functions and explicit candidate enumeration (Yi et al., 2018).

Design trade-offs center on the balance between horizon length, achievable performance (e.g., ripple, tracking error), computational burden, and the complexity of terminal set/cost construction. FCS-MPC with tailored limit-cycle tracking and terminal design guarantees lowest attainable ripple at the cost of offline mixed-integer search and online combinatorial optimization.

6. Extensions: Relaxations, Data-Driven Models, and Learning

Several approaches extend beyond model-based FCS-MPC:

• SDP and convex relaxations: For long-horizon problems, semidefinite relaxation may produce good candidate sequences in predictable time, complementing branch-and-bound or other combinatorial solvers (Hartmann et al., 2024).
• Partial outer convexification and sum-up rounding: Binary/SOS1 relaxations allow efficient continuous optimization followed by rounding with bounded tracking error, with theoretical guarantees of practical asymptotic stability under fast switching (Makarow et al., 2024).
• Data-driven or learning-based approaches: Real-time parameter adaptation via recursive least squares, use of direct data-driven predictive control with sphere decoding, or black-box surrogate models (e.g., via Koopman operators) maintain performance in the presence of significant modeling uncertainty (Brosch et al., 2019, Klädtke et al., 2024, Hanke et al., 2018).
• Imitation learning: Policy distillation of FCS-MPC experts using neural network surrogates for fast sequence selection at runtime has demonstrated comparable closed-loop performance at significantly reduced computational cost (Sheng et al., 13 Apr 2026).

7. Advantages, Limitations, and Research Directions

Key advantages of FCS-MPC with rigorous terminal design and cycle tracking include:

• Predictable, precisely quantized steady-state behavior (fixed switching frequency/waveform).
• Recursive feasibility and Lyapunov-based convergence or practical stability guarantees (even for switched/quantized systems).
• Unified controller structure for diverse converter architectures, supporting constraint handling and modular system expansion (Olajube et al., 2024, Yi et al., 2018).

Limitations and open research challenges include:

• Scalability to high-dimensional systems or long horizons due to the exponential growth of feasible input sequences.
• Additional offline complexity for terminal set and cost computation, especially for precomputed optimal cycles.
• Robustness to dynamic parameter uncertainties and time-varying operating conditions; further development is required for uncertainty-aware terminal set design (Xu et al., 2024).
• Extension to hybrid and stochastic systems, incorporating explicit uncertainty models, or integrating learning-based policies with formal guarantees remains an active area of research.

FCS-MPC thus constitutes a mature, theoretically grounded paradigm for quantized switched-system control with direct impact on the performance envelope of modern power electronics, drive systems, and converter-rich microgrids (Xu et al., 2024, Stellato et al., 2015, Hartmann et al., 2023, Yi et al., 2018, Olajube et al., 2024, Xu et al., 2022).