Indirect Neural Control: A Robustness Analysis Against Perturbations

Transcription

1 Indiret Neural Control: A Robustne Analysis Against Perturbations ANDRÉ LAURINDO MAITELLI OSCAR GABRIEL FILHO Federal Uversity of Rio Grande do Norte Potiguar Uversity Natal/RN, BRAZIL, ZIP Code Natal/RN, BRAZIL, ZIP Code Abstrat: - In the area of robust intelligent ontrol one of the fous is on ontrollers development that an maintain good performane even if there exists a poor model of the plant and if there are some plant parameter variations. The aim of this paper is an effiient addre to this question, more speifially to the indiret approah using neural networs with on-line traing, in present a riterion to evaluate the ontrol system robustne against perturbations aused by modelling errors and plant parameter variations, sine onvergene and stability of neural networs are aured. The robustne was tested using omputer simulations applied to ontrol a nonlinear system obtained on tehal literature. Key-Words: - Robust Intelligent Control, Neural Control, Artifiial Neural Networs, Robustne Introdution Control systems based on Artifiial Neural Networs - ANN s have been developed by several researhers in all the world, eah one emphasizing an implementation aspet that mae it very attrative under some pratial viewpoint. However only a few of them are worried about fundamental problems related with robustne analysis. To overome these problems it is presented an indiret ontrol method using ANN s - the Indiret Neural Control system, in whih the main purpose hereafter is ahieving how muh this methodology is robust with respet to neural networs learng errors and plant parameter variations. The outline of this paper is: Setion is devoted to a brief presentation of the ANN s. Setion 3 presents a struture for modelling both the plant and the ontroller. Setion 4 details an indiret ontrol sheme using ANN s, alled Indiret Neural Control. Setion 5 presents some onepts related with robustne, ritial perturbations and stability margin, whih is the main ontribution of this paper. To validate these onepts, Setion 6 presents the omputer simulation results of a nonlinear plant ontrol, and finally, Setion 7 presents the onlusions. In this paper, the ontrol system and learng algorithm are based on disrete-time approah. Artifiial Neural Networs ANN s The ANN s have the purpose of simulate the human nervous system behaviour through software and/or hardware, to benefit of the powerful intelligent systems that human life is endowed [. The great development ourred on miroproeor tehnology of late years has made propitious an effiient software development able to implement the ANN s, i.e., to implement the mathematial elements [, [3 whih believes the human intelligene is based on. The neural networs are omposed of many proeing elements onneted eah one to another of some manner, in suh a fashion that maing poible its parallel operation. These elements are based on nervous human system, and are alled neurons. The manner how the neurons are onneted defines the networ arhiteture. This paper uses a multilayer arrangement depited in Fig. as follows Fig.: Artifiial Neural Networ (3-Layers) In forward diretion, the first layer is the input layer only, the intermediary layers are the hidden layers, and the last layer is the output layer.

2 Aording [4, the multilayer pereptrons are good to learng mathematial relationships from a set of input-output data, therefore being onsidered uversal approximators.. Error Bapropagation Algorithm Among the algorithms used to perform supervised learng, the bapropagation algorithm has emerged as the most widely and suefully algorithm for designg multilayer feedforward networs [4, where the learng proe rests on synapti weight adjustments so that the differene between the desired output and the urrent output is a mimum aeptable value performed on all traing points. The adjustments of the synapti weights are omputed in baward diretion, aording the following equations: ( ) w w α w α w [ [ [ [ E w η w E P K [ e p, p e p, o p, z p, () () (3) (4) where is the traing yle (or epoh), η is the learng rate, α is the momentum onstant, E is the ost funtion (or learng global error), p,,..., P input-output learng data,,,..., K neural networ outputs, e is the learng loal error, o and z are the desired and urrent output for the output layer neuron, respetively. Note that z is omputed in the forward diretion. The equation used to adjust the w, j synapti weights that onnet the j-neurons of the hidden layer to the -neurons of the output layer, suh that E is le than or equal to ε 0 a beforehand given tolerane, i.e., E ε, is w [ η( o z ) j f o WY, y j (5) j with j,,..., J and,,..., K, where f o (). is the derivative of the output layer neurons ativation funtion. The equation used to adjust the synapti weights from i-neurons of the input w ji, layer to the j-neurons of the hidden layer, is [ w j, i η f h [ WX xi δ w, j (6) i [ with i,,..., I and j,,..., J, where f h (). is the derivative of the hidden layer neurons ativation funtion and δ o z f WY ) is ( ) o ( j the error signal ba propagated from -neuron. The bapropagation algorithm an suffer onvergene problems [5. To overome these problems, are used the η-adaptive and the normalized momentum approahes [, [3. 3 Modelling Using Neural Networs In external plant model, the inputs and the plant outputs are represented by its derivatives, where the derivative orders will establish the plant dynamial behaviour. The symbols used to reogze the input and plant output orders are defined as follows: n y output order, n u input order, n plant order ( output order, that is, n n y ) and d plant transport delay (d n y - n u ). Taing aount the disrete-time nonlinear model (here is a time domain) using ANN s given by y ( d) g[ ϕ ( dn,, ), W e ( d) (7) where g is a mathematis funtion hosen by the designer to model the plant dynami behaviour, ϕ is the regreion vetor, e is the estimation error, i.e., e ( d) y ( d) y $( d), and W is the matrix of neural networ adjustable parameters (synapti weights), from what we immediately dedue that the neural estimate of the plant output is given by the following equation y $( d) g[ ϕ ( dn,, ), W (8) The physial systems identifiation using ANN s an be done employing one of the several model strutures presented in [6. In this paper, it will be adopted the model nown as Neural Networ Autoregreive with Exogeneous Inputs - NNARX explained as follows. 3. Neural Networ AutoRegreive with exogeneous inputs - NNARX Model The extended nonlinear NNARX model is given by the equation (8), in whih the regreion vetor ϕ is defined as ϕ( dn,, ) [ y ( d ),..., y ( d n) u ( ), u ( ),..., u ( d n), (9)

3 where the - augmented omponent is the input to the additional weight that is made equal to the threshold. The NNARX arhiteture is depited in Fig. below. where J (without hat) is the number of neurons in J n the hidden layer, NET WY, d NET hj WX o j i and the indexes i and j refer to the input and hidden neurons, respetively. It is important to point out that the synapti weights indiated by w h, j,n d orrespond to the onnetion of the input u(-), that oupied the position n-d enumerated from top to bottom, with the j-neurons of the hidden layer. The underwritten index that appears in (0) and () is due to the neural identifier poeing only one neuron in the output layer. Fig.: NNARX arhiteture This model is BIBO (Bounded Input Bounded Output) stable beause it does not have feedba of the estimate output. The absene of stability related problems maes the NNARX model the preferred hoie when the system is determisti or the noise level is not too sigfiant [6. 3. Estimate Jaobian Computing The omputing of the plant estimate Jaobian is done after eah neural identifier traing, aording to the hosen neural networ arhiteture. In this wor, we use only the followings: Neural identifier without hidden layer (only input and output layers). The plant estimate Jaobian is given by 4 Indiret Neural Control Sheme Up to this setion, the purpose was to present some basi nowledge about multilayer neural networs and the used struture for nonlinear modelling applied to unnown plant dynamis identifiation, to aemble a theoretial underground for hereafter formulation of the indiret ontrol sheme based on neural networs. The term indiret arises beause the ontroller parameters adjustment is based on estimate plant model instead on nominal plant model, probably due to the following reasons: I) the ontroller impoibility to diret attains the real plant parameters values, and/or II) the diffiult to ahieve aeptable plant parameters values, resulting from nonlinear unnown dynamis ourrene. It is used an Indiret Neural Control Sheme that is basially formed with a neural ontroller (n) and a neural identifier () [, [3, [7, [8, [9, both arranged as depited in the following figure: J ˆ( d ) f ( NET ). w (0) ' o o,n d where n d i NET o WX and i is the input layer index, being w, n-d the synapti weight that onnets the input u(-) to the plant estimate output y $( d ). It is important to point out that we are using the most reent nown input-estimate output pair, [ u( ), yˆ( d ), to ompute the plant estimate Jaobian. Neural identifier with just one hidden layer. The plant estimate Jaobian is given by d ) J j f ( NET ' o o ). f ( NET ' h hj ). w h, j,n d. w o,, j () Fig.3: Indiret Neural Control Sheme Firstly the neural identifier is traing, in suh manner to mimize the error e as shown in Fig.3. In sequene, it is performed the ontroller traing oupled with the neural identifier, being used now the error e n instead, and getting the identifier

4 synapti weights with fixed values, what were attained after the last neural identifier traing stage. The asteris that appears in means that the identifier remains with onstant synapti weights during all ontroller traing steps. 5 Robustne Analysis The robustne analysis developed hereafter is with respet to the modelling error that ould happen in neural networs learng proe, both the identifier and the ontroller in asade with a fixed identifier, and a plant parameter variations. The robustne analysis is an important ontribution of this paper, beause it addrees a hallenger question that beomes very attrative of late years. Theorem 5. (Control Error Robustne): Consider the Indiret Neural Control of a plant, BIBO stable, given by its input-output data [u(),y(d). If Ω is the ontrol errors set, i.e., Ω ( d) { ω ( d)}, where an affirm that ω ( d) y _ Ref ( d ) yˆ( d), then we sup M u( ) 0, > ε ε n Ω ( d M) where {.} is a set, is suh that u ( ) 0, >, M is the settling horizon beginng from the instant d and ε, ε 0 n are the beforehand toleranes hosen for neural identifier and neural ontroller traing, respetively. Proof: Turng ba to (3), that gives the learng ost funtion, and onform it for neural ontroller (n) oupled with a single output neural identifier (), the latter being maintained with fixed synapti weights during all ontroller traing steps, we have E P n ( d ) [ en ( d p ) p whose development gives E n ( d) { [ e ( d n [ en ( d [ e ( d P } n... Using a onservative proedure in taing aount only the most reent neural identifier traing inputoutput pair (p), we obtain e n ( d) En ( d) Aording to the sheme shown in Fig.3, we obtain e ( d) y _ Ref ( d) yˆ( ), what gives n d Sine have y _ Ref ( d) yˆ( d) E ( d) () n yˆ ( d) y( d) e ( d), from () we y _ Ref ( d) d) e En ( d) y _ Ref ( d) d) E ( d) e ( d) n ( d) (3) Auming that the estimate error e ( d) is not available, beause it just depends on unnown output y ( d), it is neeary to dispose the estimate error in reursive form, and so goes ba if d. Without generalization lo, we an tae e ( d) e ( d ) e ( d). In this ase, we have y _ Ref ( d) d) E [ e ( d ) e ( ) ( d) d n (4) Now applying the absolute value operator in (4), and nowing that the ontrol error is given by ω ( d) y _ Ref ( d) d), it beomes y _ Ref ( d) d) E E [ e ( d ) e ( ) ( d) d n ω ( d) n [ e ( d ) e ( ) ( d) d Next, taing the triangular inequality and reordering, we get ω ( d) [ e ( d ) e ( d) E n ( d) (5) In (5), the term orrespondent to estimate error hange, e ( d), tends to be null as soon as the

5 time rises from d instant, where is the time when the settled ontrol signal beomes onstant forever, i.e., u ( ) 0, >, there exists a positive integer M, suh that ω ( d M ) e ( d M ) e ( d M ) E n ( d M ) whose limit, as soon as M tends to infity, is given by lim u( M ) 0, > ω ( d M ) e ( d M ) E n ( d M ) (6) Considering that the identifier is a single output neural networ, from (3) it beomes E P ( d M ) [ e ( d M p) { p [ e ( d M [ e ( d M... [ e ( d M P } where E is the neural identifier ost funtion, and P is the number of traing input-output data. Now, using a onservative reasong in taing aount only the most reent neural identifier traing inputoutput pair (p), and nowing that the onvergene ondition is E ( d M ) ε, suh that ε 0 is the beforehand tolerane hosen for neural identifier traing, we obtain e ( d M ) E ε ( d M (7) Now taing aount (7), and also onsidering that En ( d M) ε n, with ε n 0, it omes from (6) that lim u( M ) 0, > ( d M ) ε ε n ω (8) Finally, using the supremum operator into ontrol errors set given by (8), we obtain sup Ω ( d M ) ε ε n u( M ) 0, > (9) where M is the settling horizon of the plant beginng from instant d and ε, ε n 0 are the beforehand toleranes hosen for neural identifier and neural ontroller traing, respetively. Defition 5.: The Critial Perturbation CP of the Indiret Neural Control system is the maximum deviation that an our on plant output without lo of stability. Theorem 5.3 (Critial Perturbation): The Critial Perturbation CP of the Indiret Neural Control system is given by CP u( ). d ) ε ε n (0) where u() is the poible variation of the ontrol signal aording atuator range, Jˆ ( d ) is the plant estimate Jaobian and ε, ε 0 n are the beforehand toleranes hosen to neural identifier and neural ontroller traing, respetively. Proof: The traing of the neural ontroller is performed oupled with the neural identifier, the latter taen with fixed synapti weights during all the ontroller traing proe. In this ase, the plant estimate Jaobian an be given by [0, [ y _ Ref ( d) yˆ( d ) J ˆ( d ) () u( ) u( ) Mapulating the equation above, and taing aount the Fig.3, in whih it is immediately seen that yˆ ( d ) y( d ) e ( d ), then we give [ u( ) u( ). d ) y _ Ref ( d) [ y( d ) e ( d ) u( ). d ) y _ Ref ( d) d ) e ( d ) Now, we suppose that the plant output is settled on steady-state, y, from instant d-, and at a future

6 instant the plant suffer a perturbation that ause an abrupt variation,, that is y u( ). d ) y _ Ref ( d) ( y y y ) e ( d ) u( ). d ) [ y _ Ref ( d) y () e ( d ) Taing the absolute value of equation (), and applying the triangular inequality, we give y u( ). d ) [ y _ Ref ( d) y e ( d ) Applying Theorem 5., and also nowing that e ( d ) ε, where 0 is a beforehand hosen tolerane for neural identifier onvergene, we have y y u( ). d ) ε ε ε u( ). d ) ε n n ε ε (3) It means that the plant output variation an not exeed the value omputed by equation (3), under the penalty to lose the stability. Using Defition 5., we finally onlude that CP u( ). d ) ε ε n. Defition 5.4: The Stability Margin SM of the Indiret Neural Control system is the fator that an be multiplied by the steady-state plant output without lo of stability. Theorem 5.5 (Stability Margin): The Stability Margin - SM of the Indiret Neural Control system is given by CP SM. (4) y Proof: Using Defition 5.4, it omes that SM. y y CP (5) Mapulating (5), finally we obtain y CP SM y CP y 6 Computer Simulation Results To evaluate the robustne riteria presented in this wor, it was performed the indiret neural ontrol of a nonlinear plant through omputer simulations. For this purpose, we used a neural identifier (4/ linear ativation funtion) and a neural ontroller (4/8/ tangent hyperboli and linear ativation funtions, in this order from input to output layer), whose learng rates were adjusted using the η-adaptive method, subjet to the stability aurane [. The system to be ontrolled was obtained from [, and it is a seond order nonlinear plant with utary delay, whose equation is the following: y( ) 0. os[0.8( y( ) y( )) 0.4sin[0.8( y( ) y( )) u( ) u( ) ( u( ) u( )) 0.[9 y( ) y( ) os( y( )) The set-point is y_ref 0.8, for In the instant 5, they were applied some perturbations in the plant, to evaluate the robustne of the Indiret Neural Control system. The simulations presented the following results: st ase: Perturbation of y 0. 5y Before perturbation be applied, the plant was stabilized at y 0.8, when in the instant 5 it suffer a perturbation equivalent to 50% of the steady-state output value. The plant input signal an be settled from u min -0.5 to u max 0.. One time instant before the ourrene of the perturbation, the value of the input signal was u(50) -0.3 and the estimate value of the plant Jaobian was Jˆ (50) Using (0) and (4), we obtain CP {[( 0.5) (0.)/ } %.5456 (0.0447)

7 SM (0.675) /(0.8).8439 Based on CP and SM values omputed as above, it is seen to ome that the system ontrol has the property to return to the stable ondition, one the applied perturbation (50%) was le than the ritial perturbation (67.5%). The results are shown in the figure below: Set-Point Perturbation K Fig.4: Plant output (perturbation0.5y ) The results shown in figure 4 onfirmed the Indiret Neural Control robust behaviour, whose values are reorded in Table. (instant) Table : Error Limit versus Control Error ε, ε n sup Ω (tolerane) (error limit) ω () (ontrol error) 50.0E E-03.76E E E E-03 nd ase: Perturbation of y 0. 8y The basi differene with respet to the first ase is that now it is applied a perturbation equivalent to 80% of the steady-state output value. One time instant before ourrene of the perturbation, the value of the input signal was u(50) -0.3 and the estimate value of the plant Jaobian was Jˆ (50).940. Using (0) and (4), we obtain CP {[( 0.5) (0.)/ } % SM (0.83) /(0.8) (0.0447) Based on CP and SM values omputed as above, it is seen to ome that the system ontrol has the property to return to the stable ondition, one the applied perturbation (80%) was le than the ritial perturbation (8.3%). The results are shown in the figure below: Set-Point Perturbation K Fig.5: Plant output (perturbation0.8y ) The results shown in figure 5 onfirmed the Indiret Neural Control robust behaviour, whose values are reorded in Table. (instant) Table : Error Limit versus Control Error ε, ε n sup Ω (tolerane) (error limit) ω () (ontrol error) 50.0E E E E E-03.63E-03 3 rd ase: Perturbation of y. 0y In this last ase, the differene with respet to the formers is that now it is applied a perturbation equivalent to 00% of the steady-state output value. One time instant before ourrene of the perturbation, the value of the input signal was u(50) -0.3 and the estimate value of the plant Jaobian was Jˆ (50) Using (0) and (4), we obtain CP {[( 0.5) (0.)/ } % SM (0.675) /(0.8) (0.0447) Based on CP and SM values omputed as above, it is seen to ome that the system ontrol does not have the property to return to the stable ondition, one the applied perturbation (00%) was greater than the ritial perturbation (74.6%).

8 The results are shown in the figure below: Set-Point Perturbation K Fig.6: Plant output (perturbation.0y ) The results shown in figure 6 onfirmed the Indiret Neural Control la of robustne, whose values are reorded in Table 3. (instant) Table 3: Error Limit versus Control Error ε, ε n sup Ω (tolerane) (error limit) ω () (ontrol error) 50.0E E E E E-03 7 Conlusions Through the analysis of the simulation results putting them on a parallel with the theoretial omputed values, we an onlude that the values obtained for Critial Perturbation and the Stability Margin are onservatives, i.e., their values are le than the true values due to approximations made on dedue them. Consequently, the onlusion goes to meet a safety proedure. In spite of the onservative value omputed for the Stability Margin, it was shown that the Indiret Neural system is very robust, beause it an absorber relative high perturbations omparing to the steadystate output values that existing immediately before their ourrene. This means that it is apable to support perturbations aused by modelling errors and plant parameter variations due, e.g., to the unnown dynamis and the aging of the physial omponents. Finally, we onlude that the new methods for evaluation of the neural robustne presented in this wor produed good results, and ould be used as a tool for Indiret Neural Control design. Referenes: [ Rumelhart, D.E., MClelland, J.L. e The PDP Group. Parallel Distributed Proeing: Explorations in the Mirostruture of Cogtion, Vols. e.the MIT Pre. Cambridge, Maahusetts. USA [ Gabriel, O. Fo. A Neural Adaptive Control Sheme with On-Line Traing. Master Diertation. Federal Uversity of Rio Grande do Norte. Natal/RN. Brazil [3 Maitelli, A.L. e Gabriel Fo, O. A Neural Adaptive Control Sheme with On-Line Traing. In: Pro. of the 7 o Latin-Amerian Congre of Automati Control - LACC - IFAC, Volumen, pp Buenos Aires. Argentina [4 Hayin, S. Neural Networs. Artmed Editora Ltda. Porto Alegre/RS. Brazil [5 Pansalar, V.V. e Sastry, P.S. Analysis of the Ba-Propagation Algorithm with Momentum. IEEE Transations on Neural Networs, Vol. 5, No. 3, pp USA [6 Nφrgaard, M., Rvn, O., Poulse, N.K. e Hansen, L.K. Neural Networs for Modelling and Control of Dynami Systems. Springer-Verlag London Limited. London. England. 00. [7 Tanomaru, J. e Omatu, S. Proe Control by On- Line Trained Neural Controllers. IEEE Transations on Industrial Eletros, Vol. 39, No. 6. USA. 99. [8 Maitelli, A.L. e Gabriel, O. Fo. A Neural Adaptive Controller Applied to a Nonlinear Plant. In: Pro. of the III Brazilian Congre of Neural Networs, pp Florianópolis/SC. Brazil [9 Maitelli, A.L. e Gabriel, O. Fo. Stability and Robustne Analysis Applied to a Indiret Neural Control System. Controle&Automação Journal (still being reviewed). Brazil [0 Maitelli, A.L. e Gabriel, O. Fo. Indiret Hybrid Controller Based on Neural Networs Part I: Development and Implementation. In: Pro. of the 6 o Intelligent Automation Brazilian Symposium, pp Bauru, Brazil [ Adetona, O., Sathananthan, S. e Keel, L. H. Robust Nonlinear Adaptive Control Using Neural Networs. In: Pro of the Amerian Control Conferene, pp Arlington, VA. USA. 00. [ Ng, G.W. Appliation of Neural Networs to Adaptive Control of Nonlinear Systems. Researh Studies Pre Ltd. London. England. 997.