[Paper Review] Vers une commande multivariable sans modèle

Michel Fliesś, Cédric Join+2 more|arXiv (Cornell University)|Mar 7, 2006

Control Systems and IdentificationEngineering15 references18 citations

TL;DR

This paper proposes a model-free multivariable control framework for finite-dimensional linear and nonlinear systems using differential algebra and real-time derivative estimation of noisy signals. By formulating systems via phenomenological differential equations updated in real time and employing generalized proportional-integral (GPI) controllers, the method achieves accurate tracking without requiring a priori system models, validated through linear and nonlinear simulation examples with high robustness to noise.

ABSTRACT

A control strategy without any precise mathematical model is derived for linear or nonlinear systems which are assumed to be finite-dimensional. Two convincing numerical simulations are provided.

Motivation & Objective

• To develop a control methodology for multivariable systems without requiring an explicit mathematical model.
• To overcome the challenges of traditional black-box identification by using real-time derivative estimation of noisy signals.
• To enable robust, real-time control using generalized proportional-integral (GPI) controllers based on phenomenological differential equations.
• To extend model-free control from single-input single-output to multi-input multi-output systems.
• To validate the approach through numerical simulations of both linear and nonlinear multivariable systems.

Proposed method

• The system is represented by a set of phenomenological differential equations of the form $ y^{(n_j)}_j = F_j + \sum_{i=1}^m \alpha_{j,i}u_i + \beta_j $, where $ F_j $ is derived from measured signals and parameters.
• Differential algebra is used to derive input-output relations for nonlinear systems, enabling the construction of control-relevant models without full system identification.
• Real-time derivative estimation of noisy signals is achieved through a recently developed numerical method, allowing accurate computation of derivatives without filtering or smoothing.
• The control law is implemented using generalized proportional-integral (GPI) controllers, which are robust and effective for trajectory tracking in the absence of a system model.
• For non-square systems, only $ m $ outputs are selected to form a square, invertible system for control design.
• The method relies on a paradigm shift: replacing global mathematical models with local, time-varying differential equations updated continuously based on real-time data.

Experimental results

Research questions

• RQ1 Can multivariable control be achieved without prior knowledge of the system's mathematical model?
• RQ2 How can accurate derivative estimation of noisy signals be achieved in real time for control applications?
• RQ3 Can phenomenological differential equations be used effectively to represent complex multivariable systems without full system identification?
• RQ4 How does the proposed method compare to traditional black-box identification in terms of robustness and control performance?
• RQ5 What is the performance of the GPI controller in tracking reference trajectories when no system model is available?

Key findings

• The proposed model-free control approach successfully achieves accurate trajectory tracking in both linear and nonlinear multivariable systems without requiring a priori system models.
• Real-time derivative estimation of noisy signals enables reliable computation of system derivatives, which is critical for control design in the absence of analytical models.
• The use of generalized proportional-integral (GPI) controllers ensures robustness and high performance even with limited system information.
• The method demonstrates strong resilience to measurement noise, as validated by simulation results in both linear and nonlinear cases.
• The phenomenological differential equation formulation allows for dynamic, time-varying modeling that adapts to changing system behavior in real time.
• The approach is computationally efficient and suitable for real-time implementation, as demonstrated by numerical simulations.