Source-linked AI summary
Model-free control
Michel Fliess, Cédric Join
TL;DR
The paper addresses control of industrial plants, including tracking a prescribed temperature and relating model-free control to classic PID practice. It presents practitioner-tuned intelligent controllers, reports straightforward tuning and favorable comparisons, and identifies unresolved general understanding and simulation-model requirements.
Problem
The paper considers obtaining a prescribed time-dependent temperature and examines control settings where general understanding remains missing for some delay or non-minimum-phase examples.
Method
The approach uses low-order output derivatives and a practitioner-selected non-physical parameter, while relating intelligent controllers to classic PIDs.
Results
The unknown plant parts and disturbances vanish from the resulting third-order linear differential equation, making controller tuning straightforward for tracking y∗; comparisons were reported as favorable.
Takeaways & Limitations
The model-free standpoint could reduce the importance of questions about mathematical modeling, system structure, and parameter identification if reinforced by further applications.
Takeaways & Limitations
Digital simulations require a reasonably accurate mathematical model even though the model-free strategy yields straightforward industrial regulation.
Abstract
from arXiv · showhide
"Model-free control" and the corresponding "intelligent" PID controllers (iPIDs), which already had many successful concrete applications, are presented here for the first time in an unified manner, where the new advances are taken into account. The basics of model-free control is now employing some old functional analysis and some elementary differential algebra. The estimation techniques become quite straightforward via a recent online parameter identification approach. The importance of iPIs and especially of iPs is deduced from the presence of friction. The strange industrial ubiquity of classic PID's and the great difficulty for tuning them in complex situations is deduced, via an elementary sampling, from their connections with iPIDs. Several numerical simulations are presented which include some infinite-dimensional systems. They demonstrate not only the power of our intelligent controllers but also the great simplicity for tuning them.
1 Introduction
The paper presents model-free control and intelligent PID controllers through an ultra-local model that avoids requiring a good global plant model. It develops estimation and tuning principles and reports applications spanning nonlinear, delayed, infinite-dimensional, and faulty systems.
- Model-free control: Model-free control replaces an unknown complex model with an ultra-local relation between an output derivative, an updated term F, and the control input.F subsumes poorly known plant components and disturbances, while α is selected so αu and y^(ν) have comparable magnitudes.
- Model-free control: The practitioner usually selects a low derivative order, typically ν = 1 and only seldom ν = 2.The derivative order is chosen for the application rather than fixed universally.
- Estimation: Online estimation approximates F as piecewise constant and applies algebraic identification techniques expressed through low-pass filters such as iterated time integrals.The estimation requires only a quite short time lapse and can be implemented with discrete linear filters.
- Applications: The paper reports robustness to strong noise corruption and preserved performance under aging and actuator faults without new calibration.These results are presented alongside simulations involving nonlinear, varying-delay, heat-equation, and non-minimum-phase systems.
- Intelligent controllers: The iPID closes the loop so that unknown plant parts and disturbances disappear from the resulting third-order linear differential equation, simplifying gain tuning.The gains KP, KI, and KD are tuned to obtain suitable tracking of the reference trajectory.
- Relation to classic PIDs: The paper relates classic PID industrial capabilities to intelligent controllers and reports favorable comparisons, while using functional analysis and elementary differential algebra in its foundations.The comparison is presented as an explanation of classic PID ubiquity and tuning behavior.
2 Model-free control: general principles
Model-free control uses a causal intelligent controller around an ultra-local model, with reference tracking reduced to selecting a suitable controller functional. The paper develops iPID, iPI, and iP variants while noting implementation and non-minimum-phase difficulties.
- Generalities: The intelligent controller is built around an ultra-local model and uses a causal functional of the tracking error.C(e) depends on the past and present error, not the future.
- Reference trajectories: Reference trajectories can cause severe difficulties for non-minimum-phase systems, and suitable trajectory planning may require imposing an ordinary differential equation.Flatness can help choose a reference trajectory when part of the system is known, but flatness is not generally available or easy to verify.
- Generalities: Combining the ultra-local model with the controller yields a functional equation whose design goal is asymptotically perfect tracking.The controller functional C is selected to ensure the tracking condition.
- Implementation: A general functional setting may not yield easily implementable tools, motivating the more specific controller forms developed afterward.The paper presents this as a shortcoming corrected by the subsequent construction.
- Intelligent PID controllers: For ν = 2, the iPID eliminates F from the tracking equation, reducing tuning to stability of a third-order linear differential equation with constant coefficients.Appropriate KP, KI, and KD values are used to fulfill the tracking condition.
- Intelligent PI and P controllers: The iPI and iP are obtained by restricting the intelligent controller, with KI often set to 0 for the iP.The paper connects the frequent practical use of iPs to friction and notes that their lack of error integration removes the need for anti-windup algorithms.
3 Online estimation of F
Online estimation replaces the unknown term F with a locally constant approximation and identifies its parameter using algebraic techniques and operational-calculus transformations. The resulting formulas can be implemented as discrete linear filters and support noise-robust control.
- Online approximation: F is approximated by a piecewise constant function, reducing its estimation to identifying a constant parameter over a sufficiently small time interval.The same approach supports estimation within the intelligent controllers.
- Algebraic identification: The algebraic identification procedure estimates the constant parameter φ after eliminating initial conditions through differentiation in the operational variable.The identification is described as linear in φ.
- Operational calculus: Multiplication by s^-M removes positive powers of s, corresponding to time derivatives, while remaining negative powers represent iterated time integrals.The resulting time-domain expressions can be implemented as discrete linear filters.
- Operational calculus: The estimation procedure may replace s^-M with a suitable rational function of s when useful in practice.This is presented as an implementation option within the operational-calculus formulation.
- Noise attenuation: Noise is modeled as a quick fluctuation whose integral over any finite interval is infinitesimal, providing the basis for the stated robustness to corrupting noise.The paper attributes this robustness to the estimation and denoising procedure.
4 When is the order ν = 1 enough?
The first-order ultra-local model is generally effective when friction contributes a ẏ term, but choosing ν = 1 can degrade control when friction is absent. Numerical simulations contrast satisfactory iPI behavior with friction against strong degradation for a harmonic oscillator.
- Friction and ν = 1: With friction, numerical simulations using an iPI corrector produce satisfactory results for the selected parameters c = 3, α = 1, KP = 16, and KI = 25.The friction term is represented by c ẏ in the elementary constant linear system.
- Friction and ν = 1: For a harmonic oscillator with c = 0, an iPI produces a strong degradation of performance.The paper links this case to the absence of ẏ in the unknown equation.
- Friction and ν = 1: When friction is absent, taking ν = 1 can create an algebraic loop that adds numerical instabilities and deteriorates control behavior.The simulation comparison motivates choosing the model order according to the system’s friction-related structure.
5 Numerical experiments
Numerical experiments compare classic PID controllers with intelligent variants across frictional, changing, delayed, nonlinear, and infinite-dimensional systems. The intelligent controllers generally show strong tracking and robustness, while performance depends on system features such as friction and non-minimum-phase structure.
- 5.1 Partially known systems: The iPID estimates and compensates nonlinearities and perturbations such as friction, yielding excellent results where the PID performs poorly.In the partially known system comparison, the iPID is excellent and the iP is correct; the practitioner may prefer the simpler iP synthesis.
- 5.2 Robustness with respect to system’s changes: Without new calibration, changing systems preserve intelligent-controller performance, including under aging and an actuator fault.A pole change worsens PID performance while iP performance remains excellent, and a power-loss fault is accommodated faster by the iP than by the PID.
- 5.3 A non-linear system: The iP shows excellent trajectory tracking across the operating domain for an unstable nonlinear system, while the classic PID tracks poorly for small reference trajectories.The comparison uses the ultra-local model ẏ = F + u for the iP.
- 5.4 Delay systems: For a delay system with unknown physical delay, an iP based on ẏ = F + u produces quite satisfactory results.The simulations use an iP with K_P = 1 and Gaussian noise with standard deviation 1.
- 5.5 A one-dimensional semi-linear heat equation: The model-free synthesis controls a one-dimensional semi-linear heat equation using an elementary ultra-local model and an iP, with four simulations described as convincing.The objective is to obtain a specified time-dependent temperature at x = x_c.
6 Connections between classic and intelligent controllers
The paper connects sampled classic PI/PID controllers with intelligent controllers, explaining classic regulators’ industrial use through gain correspondences and sampling. The equivalence is tied to computer implementation rather than continuous-time control.
- Industrial interpretation: Classic PIDs’ industrial ubiquity is related to intelligent controllers through gain tuning, although tuning may be difficult in complex situations.The authors present this connection as an explanation for classic PIDs’ use in rather arbitrary industrial settings.
- Sampled controller connections: Sampling establishes correspondences between classic PI/PID controllers and intelligent controllers through suitable gain settings.The paper extends the calculations from PI to PID and PI2D controllers and summarizes the resulting gain relationships in Table 1.
- Sampling condition: The PI equivalence does not hold for continuous-time PIs and iPs; it depends strictly on time sampling and computer implementation.The paper notes that taking the sampling interval h toward zero demonstrates this dependence.
- Controller extensions: The section extends the sampled-controller comparison from PI controllers to PID controllers and then to PI2D controllers.The PI2D connection is stated as a correspondence between iPIs and PI2s.
7 Conclusion
The conclusion identifies open theoretical and comparative questions while reporting favorable comparisons and outlining possible consequences of model-free control for control theory. It also marks boundaries involving multivariable systems, delays, non-minimum-phase systems, and simulation models.
- Open problems: The paper did not study multivariable systems because concrete case studies were lacking, and it calls for closer examination of them.This is presented as an open question rather than a resolved capability.
- Open problems: A general understanding of delay and non-minimum-phase examples remains missing despite successful treatment of some individual cases.The conclusion states that this gap also exists in other recent settings.
- Evidence and comparisons: Comparisons reported in the paper were favorable to the model-free-control setting.The conclusion states that comparisons with existing approaches should be explored further, while noting that those conducted so far were favorable.
- Possible consequences: The conclusion suggests that model-free control could reduce the importance of mathematical modeling and robustness efforts as its applications develop.These possible consequences are linked to diminishing reliance on good models and continuously updated values of F.
- Scope and limitations: Straightforward industrial regulation can coexist with a need for reasonably accurate mathematical models in corresponding digital simulations.The paper describes this as a dichotomy and notes that advanced parameter identification and numerical analysis may be necessary for simulation.
A.1 Functionals
This appendix treats systems as causal functionals of past and present inputs and initial conditions, and introduces Volterra series as a representation for broad classes of nonlinear systems.
- Causal functionals: The appendix restricts attention to causal systems whose outputs depend on past and present values, not future values, together with initial conditions.The system is represented as a causal, or non-anticipative, functional.
- Volterra representation: Volterra series provide a popular engineering representation of rather arbitrary nonlinear systems.The appendix notes that solutions of quite arbitrary differential equations may be expressed as Volterra series.
- Volterra representation: The Volterra representation uses integrals of products of input values weighted by kernel functions.The displayed series includes terms involving kernels hν and input factors u(τν) through u(τ1).
A.2 The Stone-Weierstraß theorem
The appendix formulates continuous causal functionals on a compact input-function domain as a Banach algebra and invokes the Stone-Weierstraß theorem to establish density under standard algebraic conditions.
- Function space: The input-function domain C is a compact subset of continuous functions equipped with the topology of uniform convergence.C consists of functions from I to R.
- Density result: A subalgebra of continuous causal functionals is dense when it contains a nonzero constant and separates points in I × C.This conclusion follows from the Stone-Weierstraß theorem.
A.3 Algebraic differential equations
The section characterizes functionals satisfying polynomial differential equations and establishes that this class is dense in the space S. It then uses a local implicit-function representation to connect such equations to an input-output form.
- Algebraic differential equations: A consists of functionals satisfying a polynomial differential equation involving finitely many derivatives of y and u.The equation is expressed as E(y, ẏ, . . . , y(a), u, u̇, . . . , u(b)) = 0, with E polynomial and real coefficients.
- Algebraic differential equations: These functionals are differential algebraic over the differential field R⟨u⟩, whose typical elements are rational functions of u and its derivatives.The field is equipped with a derivation, and its elements may involve derivatives through arbitrary finite orders.
- Algebraic differential equations: Sums and products of differentially algebraic elements remain differentially algebraic, while constant elements belong to A.Constants satisfy ẏ = 0.
- Density in S: A is dense in S: distinct points can be separated using time functions or derivatives of functions related to the input.For differing times, y = t separates the points; for equal times, Lerch’s theorem and the Cauchy formula provide a derivative order that separates them.
- Local representation: For a SISO system well approximated by the differential equation, the implicit function theorem locally solves for y^(ν) as a function of lower and higher output derivatives and input derivatives.This local expression can then be rewritten in the form of Equation (1).