International audienceThe use of model-free control (MFC) spreads now more and more in industry. Nevertheless, control unknown delayed systems with this method remains an open problem. In this contribution, we present the use of model-free control in this context and we propose a solution to improve the effectiveness of this approach using a parameter estimation

Doublet, maxime

Hamelin, Frédéric

Join, Cédric

Doublet, Maxime

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HAL Id: hal-01369776https://hal.archives-ouvertes.fr/hal-01369776Submitted on 22 Sep 2016HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Model-free control for unknown delayed systemsMaxime Doublet, Cédric Join, Frédéric HamelinTo cite this version:Maxime Doublet, Cédric Join, Frédéric Hamelin. Model-free control for unknown delayed systems.3rd Conference on Control and Fault-Tolerant Systems, SysTol 2016, Sep 2016, Barcelone, Spain.￿hal-01369776￿Model-free control for unknown delayed systemsDoublet M.1, Join C.1,2,3, Hamelin F.11 Université de Lorraine, CRAN, UMR 7039 Vandœuvre-lès-Nancy 54500, France2 ALIEN (ALgèbre pour Identification et Estimation Numériques), 24-30 rue Lionnois, BP 60120, 54003 Nancy, France3 Projet NON-A, INRIA Lille-Nord-Europe, FranceAbstract— The use of model-free control (MFC) spreads nowmore and more in industry. Nevertheless, control unknown delayedsystems with this method remains an open problem. In thiscontribution, we present the use of model-free control in this contextand we propose a solution to improve the effectiveness of thisapproach using a parameter estimation.Key words : Model-free control, parameter estimation,delayed systems, unknown delayI. INTRODUCTIONA lot of dynamic systems present a delay more or lessimportant. We can find this kind of systems in manydomains as physics, chemistry, aeronautics,...[7], [5]. Inpresence of delays, different modifications arise (differentschemes of models, problems of stablity in control,...) [4].Indeed, sometimes systems become unstable or have a lossof performance as over/under shoots or output swings andthe consequences could be disastrous.There exist many ways to tackle and to solve the problem ofdelayed systems in the literature, only two are quoted here.The first one is the Smith predictor [6] [13], which is basedon a classic control adjusted with respect to the delay. Thismethod works only if the delay is relatively well-known.On the other hand, we have the model predictive control[11] and [10], which is based on a cost function that wehave to minimize to find the optimal control. This costfunction depends on different parameters as the referencevariable, the controlled variable,...This method requires agood knowledge of the model. In this paper, we will useanother method.Around ten years ago, a new approach, usually known asmodel-free control, has been proposed by M. Fliess and C.Join [2] and [8]. This method is based on an ultra-localmodel estimated at each time. We will show how thismethod can be applied on delayed systems for which thedelay is unknown.Usually, model-free control is based on the scheme presentedin fig 1. We have to estimate F to make the input/outputbehaviour of the system be equivalent to an integratorcascade of order v. With a delayed system, this equivalentmodel does not remain valid. Intuitively, in this context, anFig. 1. MFC principleequivalent model should be :Y (s)U(s)=e−τssv(1)If the delay τ is unknown, we cannot use this model (1).So, in this contribution we will present a new solution tosolve this problem considering only stable delayed systems.This paper is organized as follows, section 2 presentsthe model-free control. Section 3 explains our contributionto improve the effectiveness of classic MFC. The last sectiongives a concrete application and a conclusion is made in thelast section.II. MODEL-FREE CONTROL : USUAL APPROACHWith model-free control method, we want the input/outputbehaviour of the considered system be ruled by equation (2)without knowing the exact model of the system :y(v)(t) = F (t) + αucsm(t) (2)where :• y(t) and ucsm(t) are respectively the controlled outputand input.• v is the time derivative order, usually no more than 2because it is sufficient to describe the behaviour of thesystem.• F (t) groups all the unknown signals (disturbance,noise,...) and the imperfections of the model. It isestimated at each sample time.• α was until now considered as a constant [8]. It can beviewed as a scaling between y(v)(t) and ucsm(t).Assuming the estimation of F (t), the control law is designedas :ucsm(t) =1α(−F (t)− C(e(t)) + y(v)ref (t))(3)with e(t) = y(t) − yref (t) and C(e(t)) is a functionnal ofthe error. It is chosen such as y(v) + C(e) is stable.In the following, we approach F (t) by a constantfunction (noted F̂ ) in a short interval [t − T, t] where T isthe estimation horizon. By sliding this interval, it becomes[0, T ] with σ ∈ [0, T ] the time variable. It exists severalways to estimate F (t). In this paper, we propose thecontinuous estimation of F (t) using algebraic technics [3].To do this, we take equation (2) with v = 1, because it issufficent in the Laplace domain :sY (s)− y(0) = F̂s+ αUcsm(s) (4)Then, we derive this equation with respect to s to eliminatethe initial condition y(0) which is a constant term :F̂s2= −Y (s)− sdY (s)ds+ αdUcsm(s)ds(5)Now, by applying the inverse Laplace transform on theinterval [0, T ] we get :σF̂ = −y(σ) + ( ddσσy(σ))− ασucsm(σ) (6)And finally, by evaluating (6) in T , we have :T F̂ = −y(T ) + ( ddσσy(σ))(T )− αTucsm(T ) (7)Remark : We use equation (7) to estimate F because thefirst derivative of y(t) is generally not reachable (and in amore general context the vth derivative).III. α ONLINE ADAPTATION AND SIMULATIONSWith the classic approach, the control is designed as givenin equation (3). But now, with α variable, we will use thiscontrol law :ucsm(t) =−F̂ (t)α̂(t)−Kpe(t) +ẏref (t)α̂(t)(8)We consider in the following (v) = 1. Furthermore, in thefollowing, C(e) will just be a proportionnal corrector Kp asin equation (8).We do not divide Kp by α to make the correction applied tothe error uniform during all the simulation. Furthermore, wecan see that if α is constant, with a simple variable change,we come back in the classic approach with :K̃p =Kpα(9)In the following, we will present how the adaptation of α ismade.A. α online adaptationIn the most proposed works, α is constant. Since a fewyears, the problem of its online estimation is discussed [1].So the most important contribution of this paper is topropose an estimation algorithm of α.As said before, α is a scaling between y(v)(t), with v = 1,and ucsm(t). If the magnitude of the vth derivative of theoutput is going too high, the initial chosen value for α canbecome irrelevant. In this paper, we show that for importantdelays we have a loss of effectiveness with α constant. Itis why we propose this new approach for α online estimation.When α is variable, it leads that equations (5) and(7) become respectively :F̂s2= −Y (s)− sdY (s)ds+ α(s) ∗ dUcsm(s)ds(10)And :T F̂ = −y(T ) + ( ddσσy(σ))(T )− α(T )Tucsm(T ) (11)Where ∗ is the convolution product.Assuming that α(t) is constant for a short time lapse, theestimation of F (t) remains valid.The idea here, is to consider the error between y(t) andyref (t). From (3) it comes :e(t) =1Kp(−F̂ (t) + ẏref (t)α(t)− ucsm(t))(12)Control objective is to track yref (t) (in ideal way e(t) = 0)and then :ucsm(t) =−F̂ (t) + ẏref (t)α(t)(13)So it comes the estimation of α :α̂(t) =−F̂ (t) + ẏref (t)ucsm(t)(14)And, the new control law becomes (8).Now, we show some simulations explaining from wherecomes the difference between classic model-free control andour proposal.Delayed systems are considered :Y (s)U(s)= G(s)e−τs (15)with G(s) a stable transfer function. U(s)e−τs is equivalentto u(t−τ) with respect to the inverse Laplace transform. Asexplained in the introduction, we do not know the delay τ .We will explain the influence of the variable parameter. Weset :∆u(t) = u(t)− u(t− τ) (16)and with respect to the Laplace transform :∆U(s) = U(s)− U(s)e−τs (17)So, it comes :Y (s) = G(s)U(s)e−τs = G(s)(U(s)−∆U(s)) (18)We have to focus on the value of the ∆u(t) term because ifthis term is neglictable we come back in a classical situationwithout delay. The simulations presented here will show thisvalue for both cases with α constant and α variable using aparticular system which is a stable second order :Y (s)U(s)=e−τss2 + 1.4s+ 1(19)Fig. 2. In clear line |u(t)− u(t− τ)| and in black ||y(t)− yref (t)|| infunction of the delay τ in [s] for α constant.Fig. 3. In clear line |u(t)− u(t− τ)| and in black ||y(t)− yref (t)|| infunction of the delay τ in [s] for α variable.On figures 2 and 3, we represent the value of ∆u(t) termand the quadratic error with respect to delay τ . On figure 2we can see that with α constant, the system becomes quicklyunstable (black line), that is why there is nothing more after adelay of around twenty samples (0.2s). But, before this limitthe value of ∆u is small enough so it is why the systemremains stable (see [9] for more explanations).On figure 3, we see that ∆u(t), the blue line, stays smalla very long time thanks to the α adaptation. The systemremains stable as long as ∆u is unimportant. It means thatwe have more or less the required model given in equation(21). After a limit, around 200 samples (2s), the systembecomes unstable.So, when the estimation algorithm is used, we have shownthat term ∆u(t) can be neglected and we come back in awell-known situation :Y (s) ≈ G(s)U(s) (20)B. Implementation issueIn this short subsection, we explain how theimplementation for our simulations was made. Wewant to implement numerically the control. To do that, wehave to discretize the equations given previously.In a first time, the reference trajectory is obtained byfiltering a step signal with a second order filter. Indeed, itis done to make the signal smoother and mainly, to have noproblem with the derivative of the reference which appearsin (3) or (12).Next, the more important thing is to discretize the Festimation, given in equation (6), so it gives :F̂ (k) =−y(k) +A(k)− kTeα(k − 1)ucsm(k − 1)kTe(21)with Te the sampling period and where :A(k) = ky(k)− (k − 1)y(k − 1) (22)F̂ (k) is also filtered with a second order to be robust inpresence of noise and then comes the adaptation of α :α̂(k) =−F̂f (k) + ẏref (k)ucsm(k − 1)(23)where F̂f (k) is the filtered estimation of F̂ (k).And finally we get the control law :ucsm(k) =−F̂f (k)α̂(k)−Kpe(k) +ẏref (k)α̂(k)(24)with :ẏref (k) =yref (k)− yref (k − 1)Te(25)In the next section, we show some simulations using thisimplementation.IV. SIMULATIONS AND RESULTSHere, we apply the algorithm explained in the previoussection on an unknown delayed system and the efficiencyof this approach is shown.As said before, we consider here the following particularsystem, corresponding to (13):Y (s)U(s)=e−τss2 + 1.4s+ 1(26)With :• Y (s) and U(s) the output and the input.• τ an unknown delay.This is a stable second order delayed system. We controlthe system with the model-free control and we will presenta comparison between classic model-free control andmodel-free control with α adaptation.We do not need to consider a more complex system becausecontrol an unknown delayed system with good performancesremains an open problem.Each simulation has been done with the same parametersand the x-axis is graduated in time. Every time, "optimal"parameters were chosen, in particular α for the classicMFC. On figure 4, we can see the outputs obtained forFig. 4. Comparison between classic MFC and our proposal for a delay of1 samplea delay of one sample (0.01s) in both cases. We see thatresults are different because there is an over threshold withour proposed method but both results remain correct. Fora delay of ten samples (0.1s) (figure 5), result remainsequivalent for the proposed algorithm ; whereas resultobtained with usual model-free control is degraded (someswings appear). This phenomenon increases with the delayand the system finally diverges as we can see in figure 6.Fig. 5. Comparison between classic MFC and our proposal for a delay of10 samplesFig. 6. Comparison between classic MFC and our proposal for a delay of20 samplesWith this three figures, we can see that when α isvariable we clearly extend the range of accepted delay infront of the classic method without changing any parametersin the simulation for all cases.Remark : It is important to note that we do not careabout the initial condition of α for the algorithm. Indeed, αis automatically adapted and it does not affect the output asshown in figure 7. On the contrary, if the chosen value forα in the classic method is not as well as needed (chosingthis value by testing), some problems appear (figure 8). Onfigure 7 we can see that α becomes exactly identical inboth cases with α(0) = 100 and α(0) = 1. It is clearly anadvantage in front of the classic method.Fig. 7. Comparison of output with different initial conditions of α for adelay of 10 samples.Fig. 8. Comparison of output with different α constant for a delay of 10samples.Now, some additive simulations to show the accepted rangeof delay for our proposed method.As seen before, with the classic method of model-freecontrol, a delay of around 10 samples is acceptable but nomore in our particular case. After this limit, the systembecomes unstable and the output diverges (figure 6). Withthe proposed algorithm we see that a limit far more high isacceptable, a bit less than 200 samples. On figure 9, we cansee the different outputs for different delays (50 and 100samples) with our method. In front of the classic MFC, abigger delay is accepted, so we can say that our proposalimprove the effectiveness of the classic MFC.Fig. 9. Comparison of output with different delays.Here, a comparison with a classic PI controller is made asin [12] but using a MSE calculation. The controller hasthis form : C(e) = (Pk + Pis )(E(s)). We consider thatthe delay is in : τ ∈ [0; τmax], with τmax = 10[s] (100samples), but we do not know its exact value. With a MSEcalculation, we find the best tunning for Pk and Pi in theworst case, i.e. when the delay τ = τmax. We find thefollowing parameters : Pk = 0.1 and Pi = 0.08. With thistunning for the controller, we know that the system remainsstable for any delay in the defined interval.Fig. 10. Comparison between our proposed method and the "best" PIcontroller for a delay of 100 samples.On figure 10, we can see the output of our system witha delay of 100 samples with the tunning of the controllerwhich garantees the smallest MSE. With the PI method wefind MSEPI = 0.287. On the opposite, with our proposedMFC and for the same delay, we find MSEMFC = 0.139.With a MSE which is two times higher with the PI controller,we can say that our proposed method presents clearly anadvantage against classic methods like PI controller whenthe delay is unknown.To conclude this section, we show two simulations(figure 11 and 12) to prove the robustness of our method inpresence of an output additive noise.Fig. 11. Results for a delay of 10 samples in the presence of noise.Fig. 12. Results for a delay of 50 samples in the presence of noise.V. CONCLUSIONIn this contribution, unknown delayed systems wereconsidered. We present a solution to control this kind ofsystem using model-free control method. Furthermore,an improvement of this method was explained. Indeed,a method to adapt online the parameter α was presented.An explanation of the difference between usual model-free control and our proposal was done thanks to somesimulations. With different simulations and different delays,the efficiency of this approach has been shown when thedelay is unknown.More, a comparison with a classic PI controller has beendone. Thanks to a MSE calculation, we have shown that ourproposal presents better results than the classic PI controller.For the future, a more mathematical proof will bepresented and this method will be applied on systemsdescribed by partial derivative equations because a linkbetween unknown delayed systems and distributed systemscan be made.REFERENCES[1] Francisco Javier Carrillo and Frédéric Rotella. Some contributions toestimation for model-free control. In 17th IFAC Symposium on SystemIdentification SYSID 2015 Ű Beijing, China, 2015.[2] C.Join, J.Masse, and M.Fliess. A model-free control for an enginethrottle: a preliminary study. Journal Européen des Systèmes Automa-tisés, Hermès, 42 (2-3), 337-354, 2008.[3] H.Thabet, M.Ayadi, and F.Rotella. An algebraic control approachbased on the estimation of an ultra-local broida model. In 4thInternational Conference on Systems and Control, Sousse, Tunisia.,April 28-30, 2015.[4] J.P.Richard. Time-delay systems : an overview of some recentadvances and open problems. Automatica, 39, 1667-1694, 2003.[5] K.Gu and S.I.Niculescu. Survey on recent results in the stabilityand control of time-delay systems. Journal of dynamic systemsmeasurement and control, 125(2), 158-165, 2003.[6] K.J.Astrom, C.C. Hang, and B.C. Lim. A new smith predictor forcontrolling a process with an integrator and long dead-time. IEEEtransactions on automatic control, 39(2), 343-345, 1994.[7] L.Denis-Vidal and C.Jauberthie. Identifiability of a nonlinear delayed-differential aerospace model. IEEE Transactions on automatic control,56(1), 154-158, 2006.[8] M.Fliess and C.Join. Model-free control. International Journal ofControl, 86(12), 2228-2252, 2013.[9] M.Fliess and C.Join. Stability margins and model-free control: A firstlook. In 13th European Control Conference, ECC’14, 454-459, 2014.[10] M.Nikolaou. Model predictive controllers : A critical synthesis oftheory and industrial needs. Advances in Chemical Engineering, 26,131-204, 2001.[11] M.R.Garcia, C.Vilas, L.O.Santos, and A.A.Alonso. A robust multi-model predictive controller for distributed parameter systems. Journalof Process Control, 22, 60-71, 2012.[12] P.A.Gédouin, E.Delaleau, J.M.Bourgeot, C.Join, S.Arab-Chirani, andS.Calloch. Experimental comparison of classical pid and model-freecontrol: Position control of a shape memory alloy active spring.Control Engineering Practice, 19, 433-441., 2011.[13] S.Majhi and D.P.Atherton. Obtaining controller parameters for a newsmith predictor using autotuning. Automatica, 36, 1651-1658, 2000.

Model-free control for unknown delayed systems

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Model-free control for unknown delayed systems

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