Source-linked AI summary
Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?
Michel Fliess, Cédric Join
TL;DR
Writing simple, reliable differential equations for complex nonlinear systems is difficult; this paper introduces model-free and restricted-model control using numerical differentiation, validated by numerical experiments, and reports improved applicability and performance for classical PIDs on finite-dimensional systems.
Problem
The paper addresses the difficulty of writing simple and reliable differential equations for concrete plants and controlling finite-dimensional complex systems.
Method
The approach combines model-free and restricted-model control with newly developed numerical differentiation, framed by differential algebra.
Results
The authors report improved applicability and performance for classical PIDs, at least for finite-dimensional systems known to be nonminimum phase.
Takeaways & Limitations
Model-free and restricted-model control may simplify control design for finite-dimensional systems and question conventional assumptions about PID control.
Takeaways & Limitations
The analysis assumes that the system is left invertible and considers finite-dimensional systems known to be nonminimum phase.
Abstract
from arXiv · showhide
We are introducing a model-free control and a control with a restricted model for finite-dimensional complex systems. This control design may be viewed as a contribution to "intelligent" PID controllers, the tuning of which becomes quite straightforward, even with highly nonlinear and/or time-varying systems. Our main tool is a newly developed numerical differentiation. Differential algebra provides the theoretical framework. Our approach is validated by several numerical experiments.
Mi hel FLIESS ∗,∗∗
The paper introduces model-free control and control with a restricted model for finite-dimensional complex systems, framing the design as intelligent PID control with straightforward tuning. It uses newly developed numerical differentiation within a differential-algebra framework and validates the approach through numerical experiments.
- The paper introduces model-free control and control with a restricted model for finite-dimensional complex systems.
- The control design contributes to intelligent PID controllers whose tuning becomes straightforward for highly nonlinear and/or time-varying systems.
- The main tool is a newly developed numerical differentiation, with differential algebra providing the theoretical framework.
- The approach is validated by several numerical experiments.
1. INTR ODUCTION
The introduction motivates model-free control by the difficulty of constructing precise plant equations and presents model-free and restricted-model approaches based on continuously updated input-output information. It previews intelligent controllers, numerical differentiation, and simulations reporting better behavior than classical controllers across several system types.
- Motivation: Constructing simple, reliable differential equations for concrete plants is difficult because friction, heat effects, ageing, and production-related characteristic dispersions must be accounted for.These difficulties help explain industrial reluctance to use control techniques based on precise mathematical models.
- Approach: The paper develops two cases: model-free control and control using a restricted or partial plant model.The restricted-model formulation represents the plant as a known differential equation plus an unknown term G.
- Model-free control: Model-free control uses continuously updated local modeling derived solely from the system’s input-output behavior, rather than seeking a black-box model valid over a broad operating range.The authors distinguish this terminology from black-box identification because the local model is updated continuously.
- Restricted-model control: When the known part of a restricted model is flat, an intelligent controller can be derived that removes the effects represented by the unknown term G.This controller is called an intelligent controller, or i-controller.
- Validation: The paper reports simulations in linear and nonlinear, monovariable and multivariable settings, including a non-minimum-phase example, with the controllers behaving much better than classical controllers.The introduction also notes that numerical differentiation of noisy signals and partially known modeling are addressed in later sections.
3. NUMERICAL DIFFERENTIA TION · 3.2 Noises
The section treats noise as rapid fluctuations around zero that can be attenuated by low-pass filters, including iterated integrals. It then outlines local model-based control assumptions and notes implementation and applicability limitations.
- 3. NUMERICAL DIFFERENTIA TION: Numerical differentiation develops principles for polynomial and analytic signals, including algebraic differentiation and truncated Taylor expansions.Polynomial signals yield linearly identifiable derivative quantities through triangular linear systems; analytic signals are handled by truncating their convergent Taylor expansions.
- 3.2 Noises: Noises are viewed as quick fluctuations around zero and are attenuated by low-pass filters such as iterated integrals.This filtering treatment is presented as part of the numerical differentiation approach.
- 3.2 Noises: The local modeling procedure assumes that the system is left invertible, using a square subsystem when there are more outputs than inputs.The model includes output derivatives, estimated numerical values, and nonphysical constant parameters αj,i that are of the same magnitude.
- 3.2 Noises: Time sampling supplies numerical values to avoid algebraic loops, while reference trajectories are determined as in flatness-based control.The numerical value is an estimate at time instant κ.
- 3.2 Noises: The control design can produce divergent numerical control values for non-minimum phase systems, limiting the applicability of the techniques.The cited remarks identify this as a difficulty of the control design rather than a numerical performance result.
4.2 Contr ol lers
The section specializes the intelligent PID controller to an intelligent PI controller and discusses extensions involving generalized proportional-integral structures and iterated integrals. Compared with classical PID controllers, the approach avoids identification procedures and supports more flexible reference trajectories that avoid overshoots and undershoots.
- Controllers: For ν = 1 in Eq. (2), Eq. (3) is replaced by the intelligent PI controller, or i-PI.
- Controllers: The controller structure can be extended to generalized proportional-integral controllers, or GPIs.
- Controllers: Replacing the unique integral term K_I with a finite sum of iterated integrals is mathematically possible, but omitting integral action is not recommended practically.The iterated integrals have gains λ = 1, . . . , Λ.
- Controllers: Flatness-based reference trajectories are more flexible than those usually used in industry, thereby avoiding overshoots and undershoots.
5.1 A stable monovariable line ar system
On a stable monovariable linear system, the i-PI controller performs only slightly better than a classical PID in the nominal case but is markedly more resilient to ageing, faults, and power loss. The example also argues that conventional robust-control criteria and general diagnosis theory may be less necessary for the proposed control approach.
- 5.1.3. Numerical simulations: The i-PI controller behaves only slightly better than the classical PID controller in the nominal simulation.
- 5.1.3. Numerical simulations: The same conclusion holds when the control suffers a 50% power loss.
- 5.1.3. Numerical simulations: The example suggests that introducing delay systems for tuning classical PID controllers may be useless despite their involved identification procedure.
5.3 A multivariable line ar system · 5.4 A n unstable monovariable nonline ar system
The paper demonstrates excellent stabilization for a multivariable linear system using a multivariable i-PID, and for an unstable monovariable nonlinear system using i-PID with anti-windup under input constraints.
- 5.3 A multivariable line ar system: The multivariable controller uses decoupled control laws with KP1 = 1, KI1 = KD1 = 0, KP2 = KI2 = 50, and KD2 = 10.
- 5.3 A multivariable line ar system: Setting F1 = F2 = 0 provides a comparison case for the multivariable-system results.The comparison is specifically associated with Fig. 6-(b).
- 5.4.1. i-PID: The simulations for the unstable monovariable nonlinear system are described as excellent.This result is reported in Fig. 7.
- 5.4.2. Anti-windup: With input constraints −2 ≤ u ≤ 0.4, the constrained performances are mediocre without an anti-windup mechanism.
- 5.4.2. Anti-windup: The proposed anti-windup solution keeps the integral term constant as soon as the control variable becomes saturated.This modifies the classical part of the i-PID controller.
5.5 Bal l and b e am
The ball-and-beam example uses an i-PID controller on a nonlinear, difficult-to-handle system with bounded control and control-rate inputs. Bézier and sine reference trajectories both achieve excellent tracking, while the discussion notes possible improvements through flatness-based reference modification and a theoretical limitation for the sine case.
- System and setup: The nonlinear ball-and-beam system is difficult to handle because it is not linearizable by static state feedback.Its dynamics are given by 19 ÿ = Bẏu^2 − BG sin u, with u = θ as the control variable.
- System and setup: The controller uses ÿ = F + 100u, with saturation −π/3 < u < π/3 and −π < u̇ < π.These bounds were chosen to satisfy the experimental conditions as well as possible.
- Tracking results: Bézier-polynomial and sine trajectories both receive excellent tracking from the i-PID controller.The figures also display the control variable and estimates of F.
- Possible improvement: Better performances could be readily achieved using flatness-based control with a modified reference trajectory.This is presented as a possible improvement beyond the reported tracking results.
- Theoretical limitation: The sine term places the example outside the theory outlined in Section 2, although this difficulty may be circumvented using tg u.The limitation concerns the sine function appearing in the system equation.
5.6 The thr e e tank example
The three-tank example applies a zero-order hold with a decoupled model-based control equation. The experiments show trajectory tracking, excellent derivative estimation despite additive corrupting noise, and nominal controls close to those from a flatness-based design.
- Setup: The popular three-tank example uses a zero-order hold and the decoupled equation ẏ_i = F_i + 200u_i for i = 1, 2.The tanks and connecting pipes are assigned numerical physical parameters, including S = 0.0154 m², S_p = 5.10−5 m², and g = 9.81 m.s−2.
- Results: The resulting trajectories display tracking, while derivative estimation remains excellent despite additive corrupting noise.The trajectory-tracking results are shown in Fig. 14-(a), and derivative estimation under noise is shown in Fig. 14-(b).
- Results: The nominal controls are not very far from those computed with a flatness-based viewpoint.This comparison concerns the nominal controls displayed in Fig. 14-(c).
6.3 Non-minimum phase systems
The section adapts the control design to non-minimum phase systems because the flat output differs from the measured output. The modified approach achieves excellent tracking in exact-model and unmodeled-effects experiments, including additive corrupting noise.
- 6.3 Non-minimum phase systems: For a non-minimum phase system, the flat output is not the measured output, so the control design must be modified.The modification addresses the mismatch between the flat output and the measured output in the state-variable representation.
- 6.3.1. Control of the exact model: The GPI controller uses coefficients chosen to stabilize the tracking-error dynamics.The controller is introduced for the exact model, with coefficients γ, K_P, K_I, and K_II selected for error stabilization.
- 6.3.1. Control of the exact model: The exact-model controller shows excellent performance for a = 1, b = −1, and c = −0.5, even with additive corrupting noise.The results are displayed in Figures 16-(a) and (b).
- 6.3.2. Unmodeled effects: Estimating unmodeled effects is essential: leaving them unestimated substantially affects tracking, whereas estimation yields excellent results despite additive corrupting noise.The unmodeled effects may represent friction or an actuator fault, and the comparison demonstrates the superiority of the modified approach.
7. CONCLUSION
The results with intelligent PID controllers suggest improved practical applicability and performance over classic PIDs for finite-dimensional non-minimum-phase systems. Model-free control and control with a restricted model also challenge conventional modeling principles and motivate extensions to uncontrolled systems.
- 7. CONCLUSION: Intelligent PID controllers may greatly improve the practical applicability and performance of classic PIDs for finite-dimensional systems known to be non-minimum phase.This conclusion is restricted to systems within their operating range that are known to be non-minimum phase.
- 7. CONCLUSION: The gains of intelligent PIDs are straightforward to tune because the unknown part is eliminated and design reduces to a pure integrator of order 1 or 2.The conclusion contrasts this with identification techniques for classic PID regulators, which are often imprecise and difficult to handle.
- 7. CONCLUSION: Model-free control and control with a restricted model seem to question the principles of modeling in applied sciences when controlling a concrete plant.The authors suggest this could represent a fundamental epistemological change, while emphasizing that it requires further discussion and analysis.
- 7. CONCLUSION: A natural extension to uncontrolled systems is being developed through various questions in financial engineering.Preliminary studies are cited as part of this extension.
Time (
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