Adaptive Inverse Control Using SOM based Multiple Models

Transcription

1 Adaptive Invese Contol Using SOM based Multiple Models Geetha K. hampi, Jose C. Pinipe, JeongHo Cho, Deniz Edogmus, Mak A. Motte 2 Computational NeuoEngineeing Laboatoy, Univesity of Floida, Gainesville, FL NASA Langley Reseah Cente, Hampton, Viginia, Abstat: his pape addesses the poblem of modeling and ontolling non-linea plants by utilizing a self-oganizing map (SOM). he poposed method uses multiple models of the non-linea plant fo identifiation, as well as a multi-model ontolle. he mathematial fomulation of the ontolle algoithm and the swithing stategy ae disussed. Simulation and expeimental studies ae inluded fo a vaiety of non-linea systems to demonstate the pefomane of the poposed stategy. Copyight Contolo 22 Keywods: SOM, PE, Contolle, Non-linea, Identifiation.. INRODUCION It has been widely believed that most of the dynamial systems we enounte in patie exhibit non-linea behavio. he lineaization of these nonlinea systems is done based on the assumption that the plant nomally opeates nea the equilibium. But in many patial ases the system may be equied to opeate in the state spae whee the linea appoximations ae no longe valid. his gave ise to the notion of using multiple models to epesent the system dynamis ove the entie opeating egime. System identifiation using multiple model method has eeived onsideable attention in eent yeas due to its divide and onque appoah. In this appoah, a set of models is designed to epesent the possible system behavio. he onept of multiple models and swithing between the models has been an aea of inteest in Contol heoy. Multiple Kalman filte-based models (Lainiotis, 976) wee studied in the past to impove the auay in state estimation and ontol poblems. In eent yeas (Naenda et al, 997, 997, 995) multiple models using neual netwoks have been used with swithing between the models. Pinipe (Pinipe et al, 998) have suessfully modeled a haoti time seies using multiple models and applied it to the set point egulation of a NASA Langley wind tunnel duing the aeodynami testing of model aiaft (Motte, 997). Inspied by this appoah, we popose a method using self-oganizing map (SOM) (Kohonen, 995) to identify the plant at diffeent segments of the state spae tajetoy, giving ise to the onept of multiple models stutued by an SOM. Hee the global dynamis is appoximated by a peset numbe of loal linea models, whih ae onuently deived though ompetition using Kohonen s selfoganizing map (SOM). he loal models ae deived fom the weights of the SOM. At any time instant, the model epesenting the plant dynamis is hosen by the SOM depending on the state of the system. aking based on invese ontol onstitutes one of the ative aeas of eseah in ontol theoy. he basi objetive of adaptive invese ontol is to detemine the ontol input suh that the system output follows a speified tajetoy. Sine we identify the plant using multiple models, it is neessay to assoiate these models with a oesponding ontolle. How we do this in the pesene of non-lineaity poses a hallenge that will be addessed in this pape. 2. PROBLEM SAEMEN Conside a dynamial system epesented by the state equation

2 : n ) f ( () whee the veto sequene, denotes the delayed input n ) [ u ( n ) n M )] and the veto is the delayed output sequene, y ( n )... n N) with u (, y (, f :, whee f an be any non-linea funtion. Qualitatively, the objetive of the invese ontol poblem is to detemine an input ontol law u () suh that the output ) behaves in some desied fashion. As seen, the design of the ontolle has two phases: fist the identifiation of the plant dynamis and seond is the detemination of the appopiate ontol ation needed to meet the oveall speifiations. If f is unknown and we have only the input-output data, then it is an adaptive ontol poblem. Assuming that the plant is BIBO stable, it an be identified fist and then the ontolle an be designed, although in the stit sense, the adaptive ontol poblem is one in whih the identifiation and ontol ae aied out simultaneously. Designing the ontolle fo the system desibed by () is diffiult even when f is a known non-linea system. When n is unknown, the task is to appoximate f so as to design the ontolle. hee ae two altenative appoahes fo this; one is to appoximate f by a single non-linea model fˆ, whih is alled the global modeling, the most ommon being the polynomial fitting of the tajetoy. Neual netwoks an be suessfully applied to this poblem due to thei univesal mapping apability (Haykin, 994). An altenative appoah is to divide the state spae into loal egions and model individually the loal dynamis in eah egion, i.e., to deompose f into a family of models fˆ with... N, eah fitting only the neighbos of the pesent point in the eonstution spae. So the oveall model is a onatenation of loal maps (Pinipe et al, 998). fˆ( fˆ ( (2),.., N Eah loal model fˆ an be potentially simple, even linea, but the paametes will be hanging aoss the state spae. he identifiation step is aied out by the SOM, whih will povide a odebook epesentation of the plant dynamis and oganizes the diffeent dynami egimes in topologial neighbohoods. he development of loal models based on SOM is elaboated next. 3. DYNAMICAL MODELING USING SOM Sine the taining of the stati SOM is well known we do not elaboate the manne in whih an SOM is tained (Kohonen, 995, Haykin, 994). he steps to develop the loal linea models based on the SOM has thee steps: m f ) Reonstution of the state spae fom the input signal (embedding) 2) aining of the SOM 3) Estimation of the loal linea models So the fist step is to eate an embedding of the input as well as the output time seies, the embedding dimension is detemined by the ode of the system. Aoding to akens Embedding theoem, the dimension N 2D, whee D is the dimension of the attato. his delayed input and output time seies is fed to the SOM and the SOM is tained. Next step is to utilize the SOM fo dynamial modeling hanessing its powe fo data epesentation, spae disetization and topologial pesevation (Pinipe et al, 998). he loal linea models an be deived fom the tained SOM by eating an exta laye of linea PEs (Poessing Elements) fo eah of the SOM PEs. hus the th SOM PE has an assoiated linea model fˆ, whih epesents the linea appoximation to the loal dynamis. fˆ ( a a a ) (3) 2 n Whee u ( is the oveall SOM input, n ) [ ] whih is nothing but the onatenated vesion of the delayed inputs and outputs. he paamete veto a an be estimated by least squaes (LS) algoithm using the points in the loal egion. he poedue fo building the loal models is as follows: ) Fom input-output data, fom a odebook of pais { n ),[ ]} 2) Selet pais fom the odebook within a neighbohood of size N L ( N L N 2, whee N is the numbe of inputs being fed to the SOM) enteed at the uent winning PE. 3) Fit the loal model n ) fˆ ( to the seleted pais in the odebook 4) Apply the loal models to obtain yˆ ( n ) fˆ ( In shot the loal dynami model woks as follows: the uent input n ) [ ] is plaed at the input of the SOM and the winne-take-all opeation will selet the PE that best epesents the uent input. his winne ativates a single linea PE that ontains the weights of the loal linea model and estimates the esponse of the system fo this input as yˆ ( n ) a. he simulation esults given in the next setion desibe how auately this goup of linea models emulates the system dynamis. 4. CASE SUDIES-DYNAMICAL MODELING he idea pesented above on loal modelling is best illustated by onsideing speifi examples. Hee we onside thee examples in whih a non-linea dynamial system is epesented by SOM based loal models. he output obtained fom the models is

3 ompaed with the plant output and is plotted with the identifiation eo. 4. Example In this example a pieewise linea but globally nonlinea system is taken. Loal linea models ae deived fo this system desibed by, 2 if y ( (4) 2( ) if Note that this system is non-invetible. We used a SOM of 7x7 to solve this poblem. Figue. shows the esponse of the system along with the esponse of the SOM based multiple models..9.7 system identifiation of the nonlinea plant SOM based Atual pevious value of the plant output. As disussed in the pevious setion, the modeling is aied out using least squae fit and we obtain 484 models fo the plant ove the whole dynami ange. So the models ae desibed by the equations y ( k ) a k) a ) (6) ˆ k whee, a, a denote the oeffiients of the weight th veto fo the model and k), k) epesent the th system output and input espetively at the k time instant. In ode to test these models, we will give a step input as shown in figue 2(a) with vaying amplitude levels thus making the plant swith egimes. he output fom the models is shown in figue 2(b), whih lealy follows the tajetoy of the plant theeby poving the veaity of the use of multiple models fo system identifiation.. 8 input to the sysem the plant and the model esponse fo a step input output pedited.2 input Figue. Atual and identified esponse of example Hee an input of magnitude less than one is applied to the system and the SOM will selet the winne that best epesents the input, whih in tun selets the loal model. Obseve that although the system in the ange u is not linea, the SOM swithes between the oesponding models based on the input. 4.2 Example 2 We will now onside a linea system with vaying oeffiients. he system is desibed by a H ( z), whih is a fist ode linea system b z with oeffiients a and b that vay aoding to the input magnitude, i.e.,, u, - z, u ; - H ( z).2z (5), u - z, u - z We assume that the input to the system anges between and and we know in advane the ode of the system though the paametes ae unknown. A SOM of size 22x22 is tained with values unifomly distibuted in [,] fo both dimensions. So eah weight veto has two elements, one oesponding to the pesent value of the input and othe fo the Example We will now take a non-linea system desibed by the equation, n ) sin[ ] (5 os[ ]) (7) L whee L is the nomalization onstant set to 6.3. We will again use a 22x22 SOM. he system is fed with a andom input. he system esponse is plotted along with the model esponse in figue 3. As an be seen the models pedit the system output petty well. Figue 2. (a) Step input fo the plant (b) Response of the plant and the model output esponse of the system fo a andom input model Figue 3. Random input esponse of the plant and the model

4 5. DESIGN OF SWICHING CONROLLERS he geneal ahitetue fo the ontol of a plant using multiple models is shown in figue 4. he unknown system to be ontolled is SISO and has input u and output y. As stated in setion 2 ou pinipal objetive is to detemine a ontol input k), whih will esult in the output k) of the plant in () taking a speified sequene y * ( k) with suffiient auay (Widow, 996). he system has N identifiation models denoted by, in paallel. Coesponding to eah model ontolle C N M M, a is designed suh that C ahieves the ontol objetive fo M. So at evey instant one of the models is seleted by the SOM as disussed in the pevious setion, and the oesponding ontolle is used to ontol the system. his is the swithing pat of the system. We will be mainly onened with the issues of leaning a ontolle fom a multiple model paadigm, using gadient desent leaning. his means that the output plant eo must be popagated though the multiple models to adapt the paametes of the ontolle. Sine the multiple models ae stutued by the SOM, we have to deive the dual of the SOM fo sensitivity popagation. Sine the SOM input/output map is a disontinuous poess, it may appea that it is impossible to obtain its dual system. Albeit stitly speaking this seems tue, patially it is possible to tansfe sensitivities though a SOM, povided sample-by-sample estimates ae used like in the LMS algoithm. In the LMS we will be popagating instantaneous eos and so the algoithm will not see the disontinuities. With this bakgound, we an appoah the design of the ontolle in two ways.. Assign one ontolle fo eah of the models and design them independent of the othes. his is possible beause thee is a model oesponding to eah pieewise linea egion of the state-spae. 2. Assign a ontolle fo a goup of models, i.e., one ontolle being able to ontol a luste of models, whih is simila to obust ontol design. Hee, we design the ontolle using the fist method, i.e., one ontolle pe model. o design C, the linea model M is taken and the ontolle is designed adaptively using LMS algoithm (Haykin, 994, Widow 985). Sine the models ae linea, the ontolle is also linea. he blok diagam assoiated with the design of the ontolle is shown in the figue 5. Let the desied output be d( and the ontolle weight veto be is given by, w (. hen the ontolle output w ( d( (8) whee d ( n ) [ d( n )] input., the oveall ontolle W y Model S O M W n Model 2 y n y 2 Identifiation Eos Model N DL DL k) Plant Contol Eo DL DL u Contolle d=desied esponse u 2 Contolle 2 u n Contolle N Figue 4. Geneal ahitetue fo multiple ontolles

5 d( DL Contolle C i DL Model M i e( 6. CASE SUDIES-CONROLLER he poposed stategy fo ontolle design though loal modeling is tested fo the examples we used in setion Example he oesponding SOM output is the winne PE weight, w * (, whih is applied to the model M i (with paamete veto a i ) to get a esponse yˆ ( n ). he ontolle weight veto is adapted using simple LMS ule, whih utilizes the instantaneous gadient. he instantaneous gadient is defined as J ( e 2 ( (9) 2 So the weight veto update equation is J ( w ( n ) w ( w ( We an ompute this gadient by using hain ule. J ( e( e( w ( w ( plant yˆ( e( w ( e( a w ( e( a d( i i Figue 5. Single ontolle design () he adaptation poess is epeated fo all the models esulting in a model ontolle pai { M i, Ci} assoiated with evey SOM PE. As mentioned befoe the natual way to deide when and to whih ontolle one should swith is to detemine the model that best desibes the plant. his will be deided by SOM in aodane with the pesent state of the system. In a nutshell, the poedue fo ontolling the non-linea plant is as follows: he pesent state of the system [ n ) ] is applied to the SOM to get the winning PE. Now this PE will give the assoiated loal model epesenting the pesent dynamis of the system and will fie the oesponding ontolle. he ontolle output is then fed to the system, whih is foed to tak the efeene tajetoy. As an be expeted an eo in the identifiation will lead to an eo in the ontol. Conside the non-linea system desibed by (4). Contolles ae designed fo evey model as desibed in the pevious setion. he whole system with the multiple ontolles is evaluated in a tajetoytaking poblem. Figue 6 shows the desied tajetoy and the system output. Note that the system output taks the tajetoy with almost zeo eo and this lealy indiates that the ontolles ae swithing auately. output.9.7 ajetoy taking desied plant output.7.2. ajetoy taking Figue 6. ajetoy taking fo example 6. It an be onluded fom this example that though the design of ontolle design fo this non-invetible system may seen to be an impossible task, it is possible to design the invese if one knows at whih patiula pat of the spae the system is opeating. his is possible due to the loal natue of the models desibing the plant. 6.2 Example 2 Conside the fist ode linea plant desibed by (5). Hee ou objetive is to study how the diffeent models pefom in a ontol ontext. he system is fed with many diffeent test signals and the system is found to tak them petty well, poving the multiple ontolle s ability to tak the dynamis of the plant. Figue 7 shows the plots fo tajetoy taking the taking of the system fo a sine wave desied the taking of the system fo a smooth wavefom desied plant Figue 7. ajetoy taking fo example Example 3 We will finally onside the non-linea system given by (7). he pevious examples showed piee-wise linea plant dynamis. Howeve, the plant in (7) is inheently non-linea. Again, the ontolles ae designed fo evey loal linea model and the plant is tested with vaious tajetoies. Figue 8 shows the desied

6 plots of the tajetoy taking. It is obvious that the multiple model-based ontolles do well even when the system is ompletely non-linea taking of the system fo sum of two sine waves CONCLUSIONS Modeling of non-linea systems has a vaiety of appliations. In this pape we onsideed the nonlinea system modeling in the ontol poblem senaio. A global o a set of loal models an be used to identify the non-linea system. In this pape the modeling is based on SOM whih woks not only as an enhaned lusteing algoithm peseving the neighbohoods of the eonstution spae dynamis, but also as an identifie of the loal dynamis. he SOM is tained globally to luste the dynamis. he disontinuities that one may enounte aoss the boundaies ae minimized due to the inheent natue of least squaes we used in the deivation of loal models. his appoah of utilizing SOM fo nonlinea system identifiation is elevant fo designing ontolles. he models tun out to be eally simple in stutue and so ae the ontolles. he method is tested fo a vaiety of ontol poblem yielding satisfatoy pefomane. As an added advantage this method an be used to identify the unstable modes of a system. Wok in this egad is still unde pogess. In this pape we used a swithing stategy depending on the values of the inputs to the SOM o the pesent state of the system. Instead what we an do is to evaluate a pefomane index and swith the models aodingly to this index. he quantization eo that is inheent in SOM will intodue some eo in the identifiation step. his will give ise to an additional eo in the ontolle design. We an estimate an uppe bound on this quantization eo, and an detemine the maximum ontol eo taking of the system fo a smooth signal Figue 8. ajetoy taking fo example 6.3 K. S. Naenda, S. Mukhopadhyay, (997) Adaptive Contol using Neual Netwoks and Appoximate Models, IEEE ansations on Neual Netwoks, , Vol. 8, No. 3, May 997. K. S. Naenda, J. Balakishnan, (997) Adaptive Contol using Multiple Models, IEEE ansations on Automati Contol, 7-87, Vol. 42, No. 2, Febuay 997. K. S. Naenda, J. Balakishnan, M. K. Ciliz, (995) Adaptive and Leaning using Multiple Models, Swithing and uning, IEEE Contol Systems, 37-5, June 995. J. C. Pinipe, L. Wang, M. A. Motte, (998) Loal Dynami Modeling with Self-Oganizing Maps and Appliations to Nonlinea System Identifiation and Contol, Poeedings of IEEE, Vol. 86, No., Novembe 998. B. Widow, S. D. Steans (985), Adaptive Signal Poessing, Pentie-Hall, Englewood Cliffs, NJ. M. Motte (997), Contol of the NASA tansoni wind tunnel the self-oganizing featue map, Ph.D. dissetation, Univ. of Floida, Gainesville, De B. Widow, E. Walah, (996), Adaptive Invese Contol, Pentie Hall Infomation & System Siene Seies, NJ S. Haykin (994), Neual Netwoks: A Compehensive Foundation, Mamillan, New Yok. Aknowledgements: his wok was patially suppoted by Auate Automation Copoation unde gant #463. REFERENCES D. G. Lainiotis, (976) Patitioning: A Unifying Famewok fo Adaptive Systems, Poeedings of the IEEE, 64:26-43,82-97, August 976.