A Novel Computationally Intelligent Architecture for Identification and Control of Nonlinear Systems

Transcription

1 A Novel Computationally Intelligent Arcitecture or Identiication and Control o Nonlinear Systems M. Önder Ee, Oyay Kayna and Imre J. Rudas, Mecatronics Researc and Application Center, Bogaziçi University, Bebe, 885, Istanbul, urey Dept. o Inormation ecnology, Bani Donat Polytecnic, H-8, Budapest, Nepszinaz u.8, Hungary Abstract - In tis study, a novel metod or identiication and control o nonlinear systems is developed. e metod proposed realizes te dynamics o a system by employing te Runge-Kutta metod at te upper level. e intermediate level o te strategy constructs te arcitecture utilizing an adaptive neuro uzzy inerence system. e overall system is able to imitate te beavior o a complex dynamic system it a e rules or to control te system it ig accuracy. e proposed metod as been applied to a to degrees o reedom direct drive SCARA robot.. Introduction Identiication and control o nonlinear systems ave been studied by many researcers and e ave, during te last decade, itnessed distinguised solutions or speciic problems. e approaces integrating intelligence and smart numerical analysis are especially orty o attention in tis respect as tey ave resulted in ybrid arcitectures capable o acieving ig perormance. e success o intelligent control systems is commonly attributed to constituents o te metodology providing intelligence. Neural Netors (NN and Fuzzy Inerence Systems (FIS are to o tese constituents leading to te usion o verbal processing o data togeter it small scale brain-lie activity provided by artiicial neuron models. Various arcitectures o NN and FIS ave been studied or years. In [], Narendra and Partasaraty ave son tat NN arcitectures can easily be used or system identiication and control purposes. In teir aard inning paper, te system nonlinearities ave been assumed to be represented by NN models. is as resulted in an easier construction o control signals and led to a robust closed loop system in te qualitative sense. On te oter and, uzzy systems ave been son to be able to substitute te uman controller it teir property o representing te actions on te basis o linguistic variables. In [] and [], various metods ave been taen into consideration or identiication o a direct drive and an antropoid robotic manipulator aving to and tree degrees o reedom respectively. In tis study, Adaptive Neuro Fuzzy Inerence Systems (ANFIS ave been cosen as te core o te approac. A computationally intelligent arcitecture is acieved by integrating te ANFIS structure it te Runge-Kutta metod. e original orm o ANFIS arcitecture as explained analytically in [] and as dran a great interest due to its extensive design lexibility. e matematical analysis o ANFIS arcitecture clearly implies tat many FIS models can be realized by ANFIS arcitecture by appropriately setting te parameters. e upper level o te arcitecture introduced in tis paper employs te Runge-Kutta metod, ic is a poerul ay o solving te beavior o systems modeled by ordinary dierential equations. In [5], te metod is combined it radial basis unction neural netors. Wang and Lin [5], as son tat te metod is igly sucessul in estimating te beavior o a system in te long run. In [] and [], Ee and Kayna ave combined te metod it ordinary Feedorard Neural Netors (FNN and realized te identiication o bot a to do direct drive SCARA and a tree degrees o reedom antropoid robotic manipulator it on-line tuning o te neural netor stage parameters. is paper replaces te ordinary FNN arcitecture it ANFIS and uses te resulting arcitecture or te identiication and control o a to degrees o reedon direct drive SCARA manipulator. e parameter update mecanism operates on-line. e organization o tis paper is as ollos. e next section describes te plant to be identiied and controlled. e tird section introduces ANFIS structure ic unctions as a single stage o te proposed arcitecture. e ourt section explains o ANFIS arcitecture could be embedded into te Runge-Kutta metod. e evaluation o te equivalent output error rom plant outputs to controller outputs is briely explained, te equivalent stage errors or ANFIS structure are evaluated and te update mecanism is derived. e it section discusses te simulation results and inally te sixt section maes te concluding remars on te basis o te results obtained.. o DOF Direct Drive SCARA Robot Dynamics Robotic manipulators are appropriate candidates or perormance evaluation o computationally intelligent identiication and control metods because te coupled nonlinear equations and ambiguities on te riction related dynamics inevitably require te use o lexible control arcitectures. is necessity becomes more apparent i ig tracing precision is sougt. e general orm o te dynamics o te manipulator considered in tis paper is given by ( and te nominal values o te parameters are summarized in able in standard units. e pysical vie o te manipulator is illustrated in Fig.. M ( θ θ V ( θ, θ τ ( t ( By assuming te angular positions and angular velocities as state variables o te system, a total o our st order dierential equations are obtained. e state varying inertia matrix and coriolis terms are given in ( and ( respectively.

2 p p cos( θ p p cos( θ M( θ ( p p cos( θ p θ (θ θ p sin( θ V ( θ, θ ( θ p sin( θ ere p.857, p.8, p.. able. Manipulator Parameters Motor Rotor Inertia.7 I Payload Mass. Mp Arm Inertia. I Arm lengt.59 L Motor Rotor Inertia.75 I Arm lengt. L Motor Stator Inertia. IC Arm CG. L Arm inertia. I Arm CG. L Payload Inertia. IP Axis Friction 5. F Motor Mass 7. M Axis Friction. F Arm Mass 9.78 M orque Limit 5. Motor Mass. M orque Limit 9. Arm Mass.5 M ( px q y r ( p x q y r (7 (8 x y Α Α Β Π Π Β Ν Ν x y x y Figure. ANFIS Arcitecture e ANFIS output is clearly a linear unction o te adustable deuzziier parameters. At te adustment o [p q r] vector, gradient descent metod is applied. For te identiication o to do manipulator considered in tis paper, te uzziier possesses six inputs, te rule base contains only ive rules and te deuzziier as our outputs. In te case o te proposed structure being used as a controller, it uses te same number o inputs and rules but to outputs.. Integration o Runge-Kutta Metod and ANFIS Arcitecture Figure. Pysical Vie o te Direct Drive SCARA. Adaptive Neuro Fuzzy Inerence Systems Adaptive Neuro-Fuzzy Inerence Systems are realized by an appropriate combination o neural and uzzy systems. is ybrid combination enables to utilize bot te verbal and te numeric poer o intelligent systems. As is non rom te teory o uzzy systems, dierent uzziication and deuzziication mecanisms it dierent rule base structures can result in various solutions to a given tas. is paper considers te ANFIS structure it irst order Sugeno model containing ive rules. Gaussian membersip unctions it product inerence rule are used at te uzziication level. Fuzziier outputs te iring strengts or eac rule. e vector o te iring strengts is normalized. e resulting vector is deuzziied by utilizing te irst order Sugeno model. e procedure is briely ormulated in ( troug (8 or te ANFIS arcitecture illustrated in Fig., it te construction o a simple rule base being as ollos: IF x is A and y is B HEN p xq yr IF x is A and y is B HEN p xq yr µ A ( x µ ( ( B y µ ( ( y A x µ B (5 ( Runge-Kutta metod is a poerul ay o solving te beavior o a dynamic system i te system is caracterized by ordinary dierential equations. In [5], te proposed metod is applied to several problems and it is seen tat te metod is successul in estimating te system states given long enoug time. It sould be empasized tat te ANFIS arcitecture realizes te canging rates o te system states instead o te [x(,τ(] [x(] mapping tat is aimed at it conventional neural identiiers. As non, te irst order discretization brings large approximation errors ereas Runge-Kutta realizes a muc better solution. ereore, te integration o Runge-Kutta and ANFIS is expected to bring about a muc better perormance tan tose obtained in [], [] and [5]. Wang and Lin [5] ave studied tis concept it radial basis unction neural netors (RBFNN and trained te arcitecture or observed data ereas an on-line approac is adopted in tis paper. e proposed arcitecture is illustrated in Fig.. In tis igure, denotes te Runge-Kutta integration stepsize, ic as been set to.5 ms or all simulations. Robot dynamics o ( can be stated more compactly as in (9. e vector unction is realized by te ANFIS structures depicted in Fig.. For a ourt order Runge- Kutta approximation, te overall sceme is comprised o an ANFIS bloc being repeatedly connected our times it te son stage gains. e update mecanism is based on te error bacpropagation. e derivation or ANFIS based identiication sceme is given in ( troug ( ere N represents te ANFIS stages o te arcitecture and is a generic parameter o ANFIS. x ( x, τ (9 x ( i x( i ( (

3 N x; N( x ; ( ( N( x ; N( x; ( N( x ; N( x; ( N x ; N( x ; ( ( ( ( t tan( ( r ( t tan t ( e second part o tis study analyzes te control o te manipulator. e control arcitecture is illustrated in Fig. 5, in ic te son controller as te structure based on an integration as depicted in Fig.. Figure clariies o te error bacpropagation rule is applied. ere are to pats to be considered in tis propagation. e irst is te direct connection to te output summation, te oter is troug te ANFIS stages o te arcitecture. ereore, eac derivation, except te irst one, ill concern to terms. e rule is summarized belo or te ourt order Runge-Kutta approximation. r(t δ e Controller τ(t FNN ROBO e(t N( x ; (5 x ( x x (7 x x (8 x η ( i ( d ( i x ( i (9 Figure 5. Control o Robotic Manipulator δ e 5 r(t Controller τ(t In (9, η represents te learning rate and d(i represents te measured state vector o te plant at time index i. e identiication sceme is illustrated in Fig.. In tis part o te study, te robot manipulator is ept under an external control loop ile te identiier perormance is being tested. Simulation results obtained are son in Figs. 7 and 8, te reerence signal o bot axes being as described by (. τ(t Identiier ROBO Figure. Identiication Sceme e(t x(t Figure. Controller Structure o acieve an appropriate control signal, te error in te applied control signal must be determined. is is acieved troug te utilization o a pre-trained ordinary FNN. Since te desired and actual values o te state vector are measurable quantities, te output error can be evaluated and can be propagated bac troug te FNN model until te controller outputs are reaced as son in Fig. 5. is is summarized as ollos.,p p,p,p δ ( d x Ψ (S ( δ,p #neurons δ,p Ψ (S,p ( / / / x( / / / x( τ( Figure. Runge-Kutta ANFIS Hybrid Arcitecture

4 Equation ( describes te delta values or te output layer. e term x,p denotes te t entry o p t pattern in FNN response, d p denotes te t entry o p t target vector. In (, delta values or te idden layers are ormulated. In tis equation S,p, Ψ and δ,p denote te evaluated eigted sum, neuronal activation unction and te delta value attaced to t neuron in te ( t layer respectively. For on-line adustment mecanism, p corresponds to te time index. For updating te parameters o controller arcitecture, te same sensitivity derivatives as given in (5 troug (8 are used. Hoever, in te updating rule described by (9, an equivalent error δ e is used ic is obtained by propagating te output error bac troug te FNN. e next step in ormulation o te sceme is te derivation o te parameter update or te controller itsel. Since te plant to be controlled is a MIMO one it to inputs and our outputs, te ANFIS arcitecture o te controller is muc more complicated tan tat illustrated in Fig.. No, te matrix multiplying te vector o normalized iring strengts is termed WR instead o using p, q, r terms. Having tis in mind, te sensitivity derivatives o a single ANFIS stage can be evaluated as ollos. τ s s s ( τ τ ere s is te s t output and τ is te t input o ANFIS stages. Wit tese deinitions, te s t output o ANFIS can be reexpressed as given by (. ( WR τ ( s s a In above, τ a denotes te augmented input. is is clear rom (8 ere r terms are multiplied by unity. ereore, in te inal evaluations, tis is taen into consideration and te related matrices are diminised. τ s a τ ( WRs (5 s ( WRsτ a ( RULES i δ (, i ci τ i σ i (7 I tese terms are combined as ormulated by (, eac ANFIS arcitecture ill propagate te equivalent stage errors as imposed by te arcitecture o Runge-Kutta metod. In (7, c i and σ i are te center and idt values or te i t rules s t membersip unction, δ(,i denotes te Kronecer delta unction. e controller can no be ormulated as ollos. τ ( i τ ( i ( λδ (8 e In (8, δ e is te equivalent control error provided by FNN identiier, λ is a constant and during te simulations tis as been tuned to 5. e last term acts as a correcting term. Wen implementing tis controller, it sould be empasized tat te controller inputs are composed o its previous outputs and te state tracing errors. Simulation results or te control problem are illustrated in Figs. 9 and it te reerence signal deined as (. 5. Discussion o te Simulation Results o sets o simulation studies ave been carried out in tis study. In te irst part, te identiication o te robotic manipulator is considered. For te identiication problem, te identiier possesses six inputs and our outputs. e Runge-Kutta ANFIS arcitecture is supposed to estimate te state vector o te plant. Wile te estimation process is going on, te manipulator is ept under an external control loop. As te positional reerence signal, a smoot pulse, ic is deined by (, as been cosen or bot axes. e second part o te paper concerns te control o manipulator. In tis part, an additional FNN identiier is used to obtain te equivalent error on applied controls. is is a idely used approac because in general, one as te observed data instead o te exact model o te plant. ereore te training o suc a FNN model or equivalent control error evaluation is a reasonable approac. is error measure is ten used to adust te parameters o ANFIS stages by appropriately propagating it bac troug te ANFIS structure. It must be empasized tat te ANFIS structures embedded into te Runge-Kutta metod realize te same vector unction. ereore once te error bacpropagation troug all our stages is completed, te update rule applies in a cumulative manner. In Fig. 7, te reerence position and velocities are illustrated in te top and bottom ros respectively. Figure 8 sos te perormance o Runge-Kutta ANFIS integration in identiication o te manipulator model. e simulation results obtained it an FNN identiier and controller are given in Figs. 9 and. e signal, described by ( and son in Fig. 7 is used as te reerence state traectory. e state tracing errors are depicted in Fig. 9 and it is seen tat te controller as good tracing capability as indicated by te bounds o tracing errors. In te top ro o Fig., te produced control signals are illustrated. e bottom ro sos te equivalent delta values evaluated during operation. In all simulations, te rule base o ANFIS contains only ive rules. e on-line operation and te e number o rules required mae te proposed metod attractive.. Conclusions A novel computationally intelligent metod is developed in tis paper. e perormance o te proposed approac as been tested on a to degres o reedom direct drive robotic manipulator. e results obtained indicate tat te proposed arcitecture is a

5 good candidate bot or identiication and control purposes.. Base Position Error. Elbo Position Error e main advantage o te metod is te compactness o te rulebase and on-line adustment o parameters. e only drabac is te computational cost. e or is in progress or reducing te computational requirements o te arcitecture. 7. Acnoledgments Base Velocity Error Elbo Velocity Error is or is supported in part by a grant o Foundation or Promotion o Advanced Automation ecnology. Reerences [] Narendra, K. S. and K. Partasaraty, Identiication and Control o Dynamical Systems Using Neural Netors, IEEE ransactions on Neural Netors, vol., no., pp. -7, Marc 99. [] Ee, M. O. and O. Kayna, A Comparative Study o Sot Computing Metodologies in Identiication o Robotic Manipulators, Proc. rd Int. Con. on Advanced Mecatronics, ICAM 98, - August, vol., pp. -, Oayama, Japan, 998. [] Ee, M. O. and O. Kayna, A Comparative Study o Neural Netor Structures in Identiication o Nonlinear Systems, Int. Journal o Mecatronics, (accepted or publication 998. [] Jang, J.-S. R., C.-. Sun and E. Mizutani, Neuro- Fuzzy and Sot Computing, PR Prentice Hall, 997. [5] Wang, Y-J. and C-. Lin Runge-Kutta Neural Netor or Identiication o Dynamical Systems in Hig Accuracy, IEEE ransactions on Neural Netors, vol. 9., no., pp. 9-7, Marc Figure 8. Estimation Errors in Identiication o Robotic Manipulator Base Position Error 8 Base Velocity Error x - 8 x - Elbo Pos. Err. Elbo Vel. Err. 8 Figure 9. State racing Errors in Control o Robotic Manipulator.5 Re. Base Position.5 Re. Elbo Position Produced Base orque Produced Elbo orque Re. Base Velocity Re. Elbo Velocity 8 x - 8 Base Delta Elbo Delta x Figure 7. Reerence Position and Velocity Proiles Figure. Produced Control Signal on te top ro and Equivalent Deltas on te bottom ro or Control o Robotic Manipulator