In this paper, we presented a self-tuning control algorithm based on a three layers perceptron type neural network. The proposed algorithm is advantageous in the sense that practically a previous training of the net is not required and some changes in the set-point are generally enough to adjust the learning coefficient. Optionally, it is possible to introduce a self-tuning mechanism of the learning coefficient although by the moment it is not possible to give final conclusions about this possibility. The proposed algorithm has the special feature that the regulation error instead of the net output error is retropropagated for the weighting coefficients modifications.

Behar, A. Aguado

Hernández, A. Ordaz

Ponce, A. Noriega

Sitar, V. Rauch

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

1 IntroductionThe use of Artificial Neural Nets for systems identificationand control, has been the subject of a vast amount of publica-tions in recent years. It is possible tomention for instance, thepaper of Narendra and Parthasaraty [5] in which severalpossible structures of the neural controller that suppose ana priori knowledge of the plant dynamic structure are pro-posed. In Bhat and McAvoy [3] and in many other papers,a scheme based in an inverse dynamic neural model is used.A similar approach is adopted by Aguado and del Pozo [2]but here the inverse neural regulator is complemented bya self-tuning PID algorithm which guarantees that the sta-tionary error goes to zero. In the abovementioned paper andin many others, it is required an exhaustive previous trainingof the neural net, which is a serious obstacle for the practicalimplementation of the proposed solution in the industry. Atthe same time, the controller structure in some cases, as in thementioned paper of Narendra and Parthasaraty [5], is verycomplex, including several nets each one with two hiddenlayers of many neurons which must be trained using largedata samples.An important contribution to enhance the possibilities ofneural nets in the solution of practical problems is the paperby Cui and Shin [4]. In that work a direct neural controlscheme is proposed for a wide class of non-linear processesand it is shown that in many cases, the net can be traineddirectly retropropagating the regulation error instead of thenet output error. However, in that paper it is not discussedthe influence of some training parameters, particularly thelearning coefficient, over the closed loop dynamics. It is alsonot remarked that practically with a direct neural controlscheme the training stage can be substituted by a permanentand real time adaptation of the weighting coefficients of theneural net.In the present paper, it is proposed a self-tuning neuralregulator, inspired in the ideas of Cui and Shin [4], but withthe particular feature that the previous training is substitutedby a permanent adjustment of the weighting coefficientsbased on the control error. At the same time, it is shown theinfluence of the learning coefficient over the closed loopdynamics and some criteria are given about how to choosethat parameter. Finally, some examples are given where thepossibilities of the proposed method for difficult non-linearsystems control is shown, specially when there exists a consid-erable pure time delay.2 Self tuning neural controlerstructureIn Fig. 1, it is shown the proposed scheme of the self--tuning neural regulator. The neural net which assume theregulator function, is a 3 neuron layers perceptron (one hid-den layer) which weighting coefficients are adjusted by amod-ified retropropagation algorithm. In this case, instead of thenet output error:     e t u t u tu d  (1)it is used the process output error:     e t y t y ty r  (2)to adjust the weighting coefficients.In Fig. 2, it is represented the neural net controller struc-ture. The output layer has only one neuron because by themoment, we are limiting the analysis to one input-one outputprocesses. In the input layer, the present and some previousvalues of the regulation error are introduced, it means that:        x t e t e t e t ny y y  1  . (3)For the cases that has been simulated until now, the valuen  2 was sufficient to obtain an adequate closed loop per-formance. It means that the number of input neurons canbe 3. A similar number of hidden layer neurons was equallysatisfactory.3 Weighting coefficients adaptationalgorithmIn Fig. 2, the weighting coefficients wji and vj for the hid-den layer and output layer input connections respectively, areshown. In what follows, we will detail the adaptation algo-©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 49Czech Technical University in Prague Acta Polytechnica Vol. 44  No. 1/2004Neural Networks for Self-tuning ControlSystemsA. Noriega Ponce, A. Aguado Behar, A. Ordaz Hernández, V. Rauch SitarIn this paper, we presented a self-tuning control algorithm based on a three layers perceptron type neural network. The proposed algorithmis advantageous in the sense that practically a previous training of the net is not required and some changes in the set-point are generallyenough to adjust the learning coefficient. Optionally, it is possible to introduce a self-tuning mechanism of the learning coefficient althoughby the moment it is not possible to give final conclusions about this possibility. The proposed algorithm has the special feature that theregulation error instead of the net output error is retropropagated for the weighting coefficients modifications.Keywords: neural networks, feedforward, back-propagation, networks, self-tuning control.eyRN Pyr u yFig. 1.: Scheme of the self-tuning neural regulatorrithm for that coefficients, which ensures the minimisation ofa regulation error  e ty function.The output of the j hidden layer neuron may be calcu-lated by means of:hej S j11, j 1 2 3, , , (4)where:S w xj ji ii	13(5)At the same time, the output layer neuron output valuewill be: u te r 11(6)where:r v hj jj~~	13. (7)As a criteria to be minimised, we defined the function:   E t e kykt~~	12 21(8)where it is supposed that the time has been discretized by us-ing an equally-spaced small time interval.The minimisation procedure consists, as it is known, in amovement in the negative gradient direction of the func-tion E(t) with respect to the weighting coefficients vj andwji. The E(t) gradient is a multi-dimensional vector whosecomponents are the partial derivatives  E tvj,  E twji, it meansthat:    E tE tvE twjji(9)Let us first obtain the partial derivatives with respect to thecoefficients of the output neuron. Applying the chain rule, weget:     E tvE teeeeu tu trrvj yyuuj     (10)        E tveeeu t u t hjyyuj       1 1 (11)In (11) it is used the well known relation:  u trereer rr  1112     u tree eu t u trr r    1111 (12)Let us define:    1 1   e u t u ty (13)Then: E tvheejjyu  1 (14)In the equation (14), it appears the partial derivativeeeyu,which can be interpreted as some kind of “equivalent gain”of the process. Further we will make some considerationsabout that term.The partial derivative of function E(t) with respect to theweighting coefficients wji, can be obtained applying again thechain rule:     E twE teeeeu tu trrhhSSji yyuujjj      jjiw(15)        E tweeeu t u t v h h xjiyyuj j j i      1 1   E twv h h xeejij j j iyu  1 1 (16)Let us define:  j j j jv h h2 1 1  (17)and then: E tvxeejj iyu  2 (18)Using equations (14) and (18), the adjustments ofweighting coefficients vj , wji can be made by means of theexpressions:   v t v teehj jyuj  11 (19)   w t w teexji jiyuj i  12 (20)where  is the so-called learning coefficient andeeyuis the“equivalent gain” of the plant. Themain obstacle to apply theadjustment equations (19) and (20) is that in general the plantequivalent gaineeyuis unknown. However, in the abovementioned paper by Cui and Shin [4], it is demonstrated thatit is only required to know the sign of that term to ensure theconvergence of the weighting coefficients, because themagni-tude can be incorporated in the learning coefficient  if the50 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No. 1/2004 Czech Technical University in Praguee t( )e t( 1)e t( 2)u(t)vjwjiFig. 2.: Neural net controller structurenon-restrictive conditioneeyu  is accomplished. Besides,the sign ofeeyucan be easily estimated by means of a verysimple auxiliary experiment, by instance, to apply a step func-tion at the process input.The assumption that the sign of the gain remains constantin a neighbourhood of the operation point of the process isnot very strong and it is accomplished in most practical cases.Finally, in the worst of cases, real time estimation of the gainsign could be incorporated, without great difficulties, in thecontrol algorithm.Having in mind the above considerations, the equations(19) and (20) could be written as follows:   v t v teehj jyuj  11sign (21)   w t w teexji jiyuj i  12sign (22)The right value of learning coefficient  can be experi-mentally determined from the observation of closed loopsystem performance when some changes are made in thecontrolled variable set-point.The equations (21) and (22) for the proposed neuralcontroller structure, may be interpreted as the regulatoradaptation equations instead of training equations as it is nor-mally done. Indeed, the simulation study done until nowusing the scheme shown in Fig. 1 and permanently adjustingthe weighting coefficients vj and wji by means of (21) and (22)allowed us to arrive to the next conclusions: It is not in general required a previous training of the netand once the control loop is closed, the weighting coef-ficients self-adjust in a few control periods, carrying theregulation error ey(t) to zero. The dynamical performance of the closed loop dependsexclusively on the learning coefficient magnitude, corre-sponding to higher values of , a faster response that caneven present a considerable over-shoot. Diminishing thevalue of , the system is damped up to the point in whichthe desired response is obtained. The system, however,keeps the stability for a wide range of  values. It is very convenient to use a variable learning coefficient,using an expression as:      abs e y (23)In this way a small basic value of  can be used, forinstance  01. , and the effective value  is incrementeddepending of the regulation error magnitude. The rightvalue of  can be tuned experimentally without troubles.The use of equation (23) gives the system the possibility topresent a fast response when the errors are big and then to goslowly to the reference value. That behaviour is, indeed, veryconvenient in practice.4 Some simulation resultsAlthough the above described algorithm has been testedinmany examples, here we will only show the results obtainedin two simulated cases corresponding to non-linear processeswhich can be described by means of the equation:    y tK eT s T su tT sd 1 2221 1(24)The simulation was carried out on a real time environ-ment provided by the CPG System (Aguado, 1992) so that theobtained results are very close to those that could be expectedin a real process.In Fig. 3 it is represented the closed loop behavior corre-sponding to the next process and regulator parameters:T T Td   5 3 51 2s s s, , , (25)  06 04. , . . (26)As can be seen, a practically perfect response is obtainedwhen positive and negative step changes in the referenceoutput value are applied. Given that the time constants arerelatively small, the values of  and  can be chosen relativelybig without producing positive or negative over-shoots. Thistype of performance is observed in general, it means that forfast dynamics processes, it is possible and convenient a fastlearning of the net. We have observed that for time constantsin the order of milliseconds, values of  of 5 and more canbe used.©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 51Czech Technical University in Prague Acta Polytechnica Vol. 44  No. 1/2004Fig 3.: Closed loop behaviour for process in equations (25)In Fig. 4, it is represented the close loop performance forthe next constants values:T T Td   60 10 201 2s s s, , , (27)  005 020. , . . (28)As can be observed, we have a large time-delay non linearsystem which can be hardly controlled by a conventionaladaptive algorithm, for instance a self-tuning PID. However,with the adaptive neural regulator, it is obtained a responsethat can be considered as very good, given the process charac-teristics. Notice that in this case, the values of  and  areconsiderably smaller, as expected, given that the processdynamics is much slower.5 ConclusionsThe self-tuning control algorithm based on a neural netpresented in this paper, promise to be a very interesting op-tion for the control of processes with a difficult dynamics thatcould not be adequately controlled with PID regulators, evenin their self-tuning versions, as it was shown in the above pre-sented simulation cases. The class of processes in which thealgorithm could be applied is very wide and it includes mostof the cases that can appear in practice. In the near future, weplan to apply the algorithm to some real laboratory processesand to extend the obtained results to the case of multivariableand multiconnected systems.References[1] Aguado A.: Controlador de Propósito General. Memoriasdel V Congreso Latinoamericano de Control Automá-tico, La Habana, Cuba, 1992.[2] Aguado A., del Pozo A.: Esquema de Control Combinadousando Redes Neuronales. Memorias de CIMAF 97, LaHabana, Cuba, 1997[3] Bhat N., McAvoy T. J.: “Use of Neural Nets for DynamicModelling and Control of Chemical Process Systems.”Computers on Chem. Engineering, Vol. 14 (1990), No. 4/5,p. 573–583.[4] Cui X., Shin K. G.: “Direct Control and CoordinationUsing Neural Networks.” IEEE Transactions on Systems,Man and Cybernetics, Vol. 23 (1992), No. 3, p. 686–697.[5] Narendra K. S., Parthasaraty K.: “Identification andControl of Dynamical Systems using Neural Networks.”IEEE Transactions on Neural Networks, Vol. 1 (1990),No. 1.M C. Alfonso Noriega Poncephone: 01 (442) 1921264fax: 01 (442) 1921264 ext. 103e-mail: anoriega@uaq.mxDr. Alberto Aguado BeharM. C. Antonio Ordaz HernándezDr. Vladimir Rauch SitarUniversidad Autónoma de QuerétaroCentro UniversitarioSantiago de QuerétaroQro. C. P. 76010, México52 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No. 1/2004 Czech Technical University in PragueFig. 4.: Closed loop behaviour for process in equations (27)

Neural Networks for Self-tuning Control Systems

In this paper, we presented a self-tuning control algorithm based on a three layers perceptron type neural network. The proposed algorithm is advantageous in the sense that practically a previous training of the net is not required and some changes in the set-point are generally enough to adjust the learning coefficient. Optionally, it is possible to introduce a self-tuning mechanism of the learning coefficient although by the moment it is not possible to give final conclusions about this possibility. The proposed algorithm has the special feature that the regulation error instead of the net output error is retropropagated for the weighting coefficients modifications

Neural Networks for Self-tuning Control Systems

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