Model-Free Predictive Control

• Model-Free Predictive Control is a methodology that uses data-driven techniques, bypassing detailed system models to predict and optimize control actions in real time.
• It employs frameworks like ultra-local models, dynamic linearization, and explicit data predictions to design control laws without traditional system identification.
• Practical implementations demonstrate MFPC's robust stability, computational efficiency, and adaptability in managing nonlinear, multi-variable, and time-delayed processes.

Model-Free Predictive Control (MFPC) refers to a set of control methodologies that perform prediction and optimization of control actions without requiring a detailed first-principles model of the plant dynamics. Instead, MFPC strategies rely on data-driven representations, ultra-local models, algebraic estimation, or adaptive identification techniques to forecast the effects of control actions and compute optimal feedback in real time. This paradigm circumvents the traditional dependence on system identification, state-space modeling, or symbolic derivation of gradients, providing an attractive approach for nonlinear, time-delayed, or otherwise complex systems that are difficult or expensive to model.

1. Mathematical Frameworks and Core Principles

MFPC approaches vary in their abstraction and methodology, yet share the principle of bypassing explicit model identification. Common frameworks include:

• Ultra-local models: The system is approximated by a simple input-output differential equation, e.g., $\dot{y} = \mathcal{F} + \alpha u$ where $\mathcal{F}$ represents both unmodeled dynamics and disturbances and $\alpha$ is selected for scale compatibility (Join et al., 1 Feb 2025, Fliess et al., 2020).
• Dynamic linearization (MFAC/MFAPC): Nonlinear input-output relationships are locally linearized with pseudo-gradient (PG) vectors estimated from data, yielding predictive models of the form $y(k+1) = y(k) + \Theta_1(k) H(k)$ with no requirement for a physics-based model (Zhang, 2019, Zhang, 2019, Zhang, 2020).
• Parametrized control profiles: The typical piecewise constant control parameterization is replaced by flexible mappings $u = U(p)$, with $p$ a vector of decision variables capturing switching instants, basis coefficients, etc. (Alamir, 2017).
• Data-driven explicit prediction: Control laws are derived directly from input-output data via, e.g., the Willems fundamental lemma or least-squares regression, resulting in explicit affine or nonlinear control actions without system identification (Sassella et al., 2021, Zhou et al., 2020).

Optimization problems in MFPC typically take the form: $$\min_{p} \, J(p) \quad \text{subject to} \quad g(p) \leq 0, \; p \in [p_{\min}, p_{\max}]$$ with $J(p)$ and $g(p)$ evaluating predicted trajectories under data-driven, algebraically estimated, or parametrically defined dynamics.

2. Derivative-Free and Adaptive Predictive Control Strategies

A distinguishing feature of MFPC is the use of derivative-free optimization, trust-region methods, multi-step prediction horizons, and contraction mapping for stability:

• Trust-region optimization: Quadratic approximations are built around cost and constraint functions, with adaptive region updates $\alpha_{\text{new}} = \beta^+\alpha$ or $\beta^-\alpha$ depending on improvement (Alamir, 2017).
• Adaptive prediction: Multi-step forecasting (e.g., $N$-step PG vector prediction) allows MFAPC controllers to anticipate future changes and provide more robust tracking, particularly under time delays (Zhang, 2019, Zhang, 2019, Zhang, 2020, Zhang, 2020).
• Recursive feasibility and stability: Bounded-input, bounded-output (BIBO) and monotonic tracking error convergence are guaranteed via matrix norm, spectral radius conditions, Lyapunov functions, and tube-based robust MPC (Xie et al., 2022, Zhou et al., 2020).

In the absence of analytic gradients, the cost and constraint evaluations are performed directly by simulation or integral algebraic estimation, which is particularly vital for systems defined functionally (ODEs, simulation codes) or with non-differentiable transitions.

3. Implementation Techniques and Computational Aspects

MFPC methods emphasize real-time applicability, computational economy, and practical deployment:

• User-defined modular architecture: Dynamic models, control profiles, and objective/constraint functions are implemented as separate subroutines (e.g., in Matlab Coder), enabling rapid prototyping and scaling to complex applications (Alamir, 2017).
• Algebraic estimation of unmodeled dynamics: Unknown terms such as $\mathcal{F}$ in ultra-local models are estimated via integrals of input/output data, e.g., $$F_{\text{est}}(t) = -\frac{6}{\tau^3}\int_{t-\tau}^t [\ldots]d\sigma$$, providing robust estimation against noise and uncertainty (Fliess et al., 2020, Join et al., 1 Feb 2025).
• Data-driven explicit control law computation: By leveraging persistently exciting input sequences and data Hankel matrices, explicit piecewise affine controllers for LTI systems are synthesized directly from measured data, avoiding the need for system identification or online QP solving (Sassella et al., 2021).
• Fast execution pathways: MFPC packages are designed to generate highly efficient compiled routines (e.g., mex-function in Matlab), supporting interruptibility and real-time feedback (Alamir, 2017).

4. Case Studies, Examples, and Comparative Analysis

Applications of MFPC span a wide range of nonlinear, multi-variable, and time-delayed scenarios:

Application	Key MFPC features	Result/Comparison
Nonlinear crane control	Reduced parametrization, trust-region	Identical performance with major compute-time reduction
Cancer therapy scheduling	Arbitrary control profile (phase-based)	Flexible policy design for schedule optimization
Two-tank system & reactor	Algebraic ultra-local model MFPC	Effective without explicit system model (Join et al., 1 Feb 2025)
MIMO time delay systems	EDLM-based MFAPC, relaxed assumptions	Handles varied delays, eliminates static error (Zhang, 2020)
Quadrotor/robot motion	Ultra-local model for multi-output control	Robust tracking under disturbances (Fliess et al., 2020)

Compared to model-based and deep learning approaches:

• MFPC achieves competitive performance without extensive model identification or machine learning infrastructure. For example, MFPC is only slightly outperformed by HEOL (a hybrid approach needing process expertise) and does not require complex ANN identification for satisfactory control (Join et al., 1 Feb 2025).
• Real-time learning and adaptive correction capabilities of MFPC are validated in both simulation and hardware experiments (Alamir, 2017, Fliess et al., 2020).

5. Stability, Robustness, and Tuning

MFPC frameworks embed rigorous stability analysis, often relying on spectral conditions, Lyapunov functions, or regularization constraints:

• BIBO stability: MFAPC and tube-based MPC schemes provide monotonic convergence of the error and guarantee all signals remain bounded under bounded disturbance (Zhang, 2019, Zhang, 2019, Xie et al., 2022).
• Offset-free tracking: Explicit integral actions or learned disturbance compensation are employed to ensure zero steady-state error despite model-plant mismatches (Son et al., 2020, Xie et al., 2022).
• Parameter tuning: Step factor matrices and weighting matrices (e.g., $\lambda$ in cost functions) can be tuned for convergence speed versus robustness, with explicit design guidelines provided via closed-loop pole analysis (Zhang, 2020, Zhang, 2020, Zhang, 2019).
• Robustness to noise and uncertainty: Simulation and averaging approaches, as in explicit data-driven law derivation, mitigate the effect of measurement noise, with RMSE converging to the model-based oracle as sample size increases (Sassella et al., 2021).

6. Limitations, Extensions, and Future Directions

Limitations of current MFPC methods include:

• Parameter and structure selection: MFAPC approaches may require careful configuration of pseudo-orders and step factors for optimal performance (Zhang, 2019, Zhang, 2020).
• Computational demand for multi-step/multi-variable prediction: Prolonged prediction horizons and matrix inversions may constrain practical deployment on fast or resource-limited platforms (Zhang, 2019, Zhang, 2020).
• Extension to constrained and hybrid systems: Certain MFPC variants do not yet include formal constraint or stability analyses (noted as an area for further research in (Join et al., 1 Feb 2025)).

Directions for further investigation include:

• Integration of MFPC with reinforcement learning and neural certificates for safety, stability, and asymptotic performance improvement (Hashimoto et al., 18 Jul 2025).
• Extension to MIMO systems, high-dimensional action spaces, hybrid systems, and broader classes of nonlinearity, leveraging flexible parametrizations and data-driven estimators.
• Development of hybrid strategies combining algebraic MFPC for baseline control with ANN-driven identification or adaptive tuning where warranted by system complexity (Join et al., 1 Feb 2025).

In summary, Model-Free Predictive Control constitutes a rigorously developed, computationally efficient, and flexibly parametrized alternative to traditional MPC, especially suited for real-time, nonlinear, and uncertain dynamic systems. Its robustness, stability guarantees, and practical effectiveness have been documented across diverse control scenarios, illustrating its significance for both applied engineering and control theory research.