Finite-Control-Set Model Predictive Control (FCS-MPC)

• FCS-MPC utilizes a finite set of pre-defined inputs for optimization rather than a continuous range, simplifying control actions in discrete systems.
• Primarily used in power electronic converters, FCS-MPC efficiently manages rapid actuation and physical constraints specific to these systems.
• It combines plant modeling and real-time optimization, eschewing explicit modulation for direct constraint enforcement in pivotal applications.

Finite-Control-Set Model Predictive Control (FCS-MPC) is a class of model predictive control techniques in which the control inputs are selected from a finite, enumerated set, rather than optimized continuously. This paradigm is dominant in the control of power electronic converters, electrical drives, and switched systems where actuator signals are inherently discrete (e.g., inverter switch positions). FCS-MPC directly ties plant modeling, real-time optimization, and constraint satisfaction into a feedback loop suited to the rapid actuation requirements and strict physical limitations of modern power devices.

1. Mathematical Foundations and Basic Problem Formulation

FCS-MPC computes a control policy by solving, at each sampling interval $k$, a mixed-integer optimization problem predicting system evolution over a finite horizon $N$:

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

$$\text{s.t.}\quad x_{k+i+1} = f(x_{k+i}, u_{k+i}),\;\; x_{k} \text{ given},\;\; x_{k+i} \in \mathbb{X}$$

• $\mathcal{U}$: finite set of admissible switch positions (the "finite control set")
• $\ell(\cdot,\cdot)$: stage cost, typically including tracking and switching frequency terms
• $V_f(\cdot)$: terminal cost, aiding stability
• $f(\cdot,\cdot)$: discrete or discretized plant dynamics

For linear, time-invariant dynamics $x_{k+1} = Ax_k + Bu_k$, classic FCS-MPC is a mixed-integer quadratic problem. The cardinal feature distinguishing FCS-MPC from continuous-set MPC is that $u_{k+i}$ is chosen from a small (often exponentially large in the control bits) list of physical switching actions, not from a convex set.

FCS-MPC avoids explicit modulation and enables direct enforcement of constraints such as maximum current, voltage, or switching frequency and is widely amenable to parallelizable, low-latency numerical implementation (Hartmann et al., 2023, Klädtke et al., 2024).

2. Modeling Approaches and System Classes

2.1 Physical and Data-Driven Models

• Physics-Based Models: Euler or exact discretization of converter models, electrical drives, and microgrid components with parameters derived from first principles (e.g., $N$0—$N$1, $N$2, $N$3 from physical circuit laws) (Hartmann et al., 2023, Stellato et al., 2015, Yi et al., 2018).
• Koopman and Black-Box Models: System representation leveraging the Koopman operator enables use of low-order, data-driven surrogate models. A set of finite Koopman matrices $N$4 for each input $N$5 encodes input-dependent linear evolution in a lifted state, permitting efficient simulation of nonlinear or high-order plants (Hanke et al., 2018).
• On-line Identification: Recursive Least-Squares (RLS) and similar online methods update plant models in real time to capture parameter drift, saturation, dead time, and unmodeled effects (Brosch et al., 2019, Hanke et al., 2018).

2.2 System Types

FCS-MPC is established for two-level and multi-level inverters, DC/DC and AC/DC converters, switched-mode amplifiers, microgrids, and flying capacitor or neutral-point-clamped systems. It applies to both LTI and switched nonlinear affine systems, with state and input constraints formalized in polyhedral sets or explicit lists (Xu et al., 2024, Hartmann et al., 2024, Yi et al., 2018).

3. Solution Methodologies and Computational Algorithms

3.1 Enumeration, Exact Algorithms, and Decomposition

• Full Enumeration: Directly simulates all $N$6 sequences, feasible only for $N$7 unless problem dimensionality is extremely low (Stellato et al., 2015).
• Mixed-Integer Programming (MIP): Poses the problem as a MIQP or binary program, solvable by commercial or custom branch-and-bound solvers. Practical for horizons up to $N$8 for moderate $N$9 (Klädtke et al., 2024, Hartmann et al., 2024).
• Sphere Decoding: Exploits quadratic cost structure and Cholesky-based lattice search, recursively pruning suboptimal branches using current best cost bounds. Significantly accelerates solution for problems structured as integer least squares (Hartmann et al., 2023, Klädtke et al., 2024).
• Approximate Dynamic Programming (ADP): Computes offline a quadratic underestimator (value function) for the infinite-horizon cost, used as a tail cost to reduce online horizon length with minimal performance loss (Stellato et al., 2015).

3.2 Relaxations and Fast Switching

• SDP Relaxation: Lifts polynomial objective (e.g., for direct torque FCS-MPC) into moment-matrix SDP optimizing over a convex relaxation, followed by rounding to retrieve a feasible sequence. Deployed in tandem with node-limited branch-and-bound to guarantee feasible and robust receding-horizon policies (Hartmann et al., 2024).
• Binary and Convex Relaxation: Binary variable relaxations enable continuous, near-MPC behavior, with post-processing via sum-up rounding and oversampled "fast" switching—proven to attain arbitrarily tight proximity to the relaxed state-space trajectory as switching frequency increases (Makarow et al., 2024).

3.3 Neural Approximators

• Policy Distillation: High-order (long-horizon) FCS-MPC controllers can be effectively emulated by neural networks trained on expert data (via DAgger, domain randomization), yielding $$\min_{\{u_{k+i}\in \mathcal{U}\}}\ J = \sum_{i=0}^{N-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N})$$0 speedup with negligible performance loss across wide parameter regimes, rapid load steps, and topological variants (Sheng et al., 13 Apr 2026).

4. Constraint Handling and Advanced Objectives

4.1 Frequency and Loss Constraints

• Indirect Penalties: Standard practice penalizes $$\min_{\{u_{k+i}\in \mathcal{U}\}}\ J = \sum_{i=0}^{N-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N})$$1 to discourage high switching frequency, but with operating point sensitivity and no explicit upper bound (Hartmann et al., 2023).
• Explicit Hard Bounds: Augments the prediction model with an IIR filter state for switching frequency, enforces $$\min_{\{u_{k+i}\in \mathcal{U}\}}\ J = \sum_{i=0}^{N-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N})$$2 via slack variables penalized in the cost, and solves the resulting MIQP or integer least squares with specialized pruning for slack-penalty bounds (Hartmann et al., 2023).
• Extensibility: The slack-plus-sphere-decoder technique generalizes to additional constraints, such as neutral-point balancing and total harmonic distortion (Hartmann et al., 2023).

4.2 Limit Cycle and Steady-State Optimality

Classical FCS-MPC may exhibit arbitrary switching in steady state, yielding unpredictable ripple and limit cycles. By embedding a precomputed periodic reference (limit cycle) into both stage and terminal costs, convergence to minimal-ripple, pre-specified periodic patterns is established via Lyapunov/MPC arguments; generalizations include switched affine systems, with periodic terminal tubes ensuring recursive feasibility (Xu et al., 2024, Xu et al., 2022).

Constraint Approach	Mechanism	Reference
Switching freq. bound	IIR filter + slack var. + MIQP/ILS	(Hartmann et al., 2023)
Current/voltage limit	State/output polytope, hard constraint	(Olajube et al., 2024, Yi et al., 2018)
Steady-state periodic	Limit-cycle reference in cost	(Xu et al., 2024, Xu et al., 2022)

5. Applications and Practical Implementation

5.1 Power Electronic Drives and Microgrids

FCS-MPC is established for current/torque control in PMSM, induction motors, flying capacitor or neutral-point-clamped converters, multi-level inverters, and for AC/DC hybrid microgrids (Stellato et al., 2015, Hanke et al., 2018, Yi et al., 2018). Fast, one-step enumeration is feasible in embedded hardware (FPGA/DSP) for sampling times down to $$\min_{\{u_{k+i}\in \mathcal{U}\}}\ J = \sum_{i=0}^{N-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N})$$3s (Stellato et al., 2015, Olajube et al., 2024).

5.2 High-Precision Amplifiers and Motion Systems

Embedding limit-cycle tracking eliminates unpredictable ripple, a necessity for high-precision current-actuated amplifiers in lithography and motion applications (Xu et al., 2022).

5.3 Advanced Converter Topologies

Domain-randomized neural surrogates allow robust low-latency switching for converters with multiple modes, e.g., FC-3L boost and NPC-3L buck, while guaranteeing key regulation and balancing objectives under parameter drift (Sheng et al., 13 Apr 2026).

5.4 Control of Islanded Microgrids

A unified FCS-MPC architecture enables voltage, frequency, and power-sharing regulation in DC, AC, and hybrid microgrids, removing the need for nested PI, PWM, or droop loops, and providing superior transient and sharing accuracy (Yi et al., 2018, Olajube et al., 2024).

Application	Notable Implementation Details	Reference
Drives, inverters	ADP tail cost, FPGA hard real time, $$\min_{\{u_{k+i}\in \mathcal{U}\}}\ J = \sum_{i=0}^{N-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N})$$4	(Stellato et al., 2015)
Microgrids	N=1, one-step prediction, no PI/PWM/droop, multi-bus ops	(Yi et al., 2018)
Three-level conv.	Constraints on neutral point, limits via slack variables	(Hartmann et al., 2023)
Neural approximation	N=5, fast policy inference, DAgger data augmentation	(Sheng et al., 13 Apr 2026)

6. Stability, Feasibility, and Theoretical Guarantees

6.1 Asymptotic and Practical Stability

• Terminal Ingredients: Time-varying or periodic terminal sets and costs (ellipsoidal or polytopic invariant tubes) guarantee recursive feasibility and convergence to limit cycles or steady-state references (Xu et al., 2024, Xu et al., 2022).
• Fast-Switching Theorem: With sufficiently high oversampling and sum-up rounding post-relaxation, practical stability arbitrarily close to the ideal behavior of the convexified MPC can be achieved; state tracking errors can be bounded as a function of the switching step size (Makarow et al., 2024).
• Stability Under Constraints: Bounded stability is formally proved for decentralized FCS-MPC in grid-connected inverter networks, with explicit Lyapunov and bounding conditions (Olajube et al., 2024).

6.2 Limitations and Open Issues

• Computational complexity grows exponentially with prediction horizon and finite set cardinality; mitigated by sphere decoding, ADP tails, and neural surrogates, but often constrains horizon $$\min_{\{u_{k+i}\in \mathcal{U}\}}\ J = \sum_{i=0}^{N-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N})$$5 to $$\min_{\{u_{k+i}\in \mathcal{U}\}}\ J = \sum_{i=0}^{N-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N})$$6 in real time (Klädtke et al., 2024, Stellato et al., 2015, Sheng et al., 13 Apr 2026).
• For hard constraints on nonlinear metrics (e.g., total harmonic distortion), custom slack-augmented formulations and branch-and-bound or SDP relaxations become necessary (Hartmann et al., 2023, Hartmann et al., 2024).
• Offline limit-cycle enumeration or terminal set computation can become intractable as the size of the switching alphabet or cycle period grows (Xu et al., 2022).

7. Extensions and Future Directions

• Data-Driven FCS-MPC: Direct data-driven predictive control, through either implicit predictors (modified sphere decoding) or hybrid black-box model identification, is an emerging trend, reducing model-dependence and enabling adaptation to high-order/convoluted plant dynamics (Klädtke et al., 2024, Hanke et al., 2018).
• Real-Time Optimality and Scalability: Advances in lattice search, SDP relaxations, and parallel neural inference progressively enable longer online horizons or tighter constraint satisfaction within hard timing budgets (Hartmann et al., 2024, Sheng et al., 13 Apr 2026).
• Integration with Hierarchical and Decentralized Schemes: Modular embedding of FCS-MPC inside larger droop or hierarchical microgrid systems allows scalable, robust control architectures for heterogeneous renewable resources (Olajube et al., 2024, Yi et al., 2018).

FCS-MPC thus embodies a highly structured, versatile, and rapidly advancing methodology that unites the discrete-acting nature of power electronics and switched systems with modern predictive optimization and data-driven control strategies. Its ongoing evolution is tightly coupled to computational advances, the emergence of data-centric identification, and the theoretical synthesis of constrained, finite-alphabet predictive tracking.