Neural Network PID Algorithm for a Class of Discrete-Time Nonlinear Systems

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1 Neural Network PID Algorthm for a Class of Dscrete-Tme Nonlnear Systems Hufang Kong ", Yao Fang Hefe Unversty of Technology, Hefe, P.R.Chna konghufang@6.com Abstract The control of nonlnear system s the hotspot n the control feld. The paper proposes an algorthm to solve the trackng and robustness problem for the dscrete-tme nonlnear system. The completed control algorthm contans three parts. Frst, the dynamc lnearzaton model of nonlnear system s desgned based on Model Free Adaptve Control, whose model parameters are calculated by the nput and output data of system. Second, the model error s estmated usng the Quas-sldng mode control algorthm, hence, the whole model of system s estmated. Fnally, the neural network PID controller s desgned to get the optmal control law. The convergence and BIBO stablty of the control system s proved by the Lyapunov functon. The smulaton results n the lnear and nonlnear system valdate the effectveness and robustness of the algorthm. The robustness effort of Quas-sldng mode control algorthm n nonlnear system s also verfed n the paper. Keywords dynamc lnearzaton model, quas-sldng mode control, nonlnear system, neural network Introducton Wth ncreasng demands on the mprovement of the precson control, the control problem of nonlnear system s attractng more and more attenton[]. Especally, the robustness problem of nonlnear control system model has gradually become an mportant topc n control theory. In general, control algorthms of nonlnear system manly contan the neural network control, fuzzy logc control, the sldng mode control and model free adaptve control (MFAC). From unversal approxmaton theory, a sngle hdden layer neural network can approxmate any nonlnear functon to any prescrbed accuracy f suffcent hdden neurons are provded[]. However, the tranng examples are usually much larger than the hdden nodes, and selectng the approprate number of hdden nodes s the key factor whch determnng the error of the predcton model. And the algorthm of the control system can t ensure the global optmal value of weght[]. The fuzzy logc system can also approxmate any nonlnear functon to any prescrbed accuracy, and t has the advantages n dealng wth the tme-delay, tme-varyng, mult nput sngle output nonlnear system[4]. Especally, the combnaton between fuzzy control and PID JOE Vol. 4, No., 08 0

2 control algorthm, the objectve of mproved control algorthm s to elmnate the steady state error, and the algorthm can also mprove the control accuracy and statonary performance. However, the establshment of fuzzy rules s lack of the systemc approach, and t has to rely on the engneerng experments[5]. Sldng mode varable structure control (SMC) has good robustness to perturbaton and external dsturbance n sldng mode, so t becomes one of the effectve methods to deal wth nonlnear systems wth varous uncertantes[6]. However, the varable structure control wll be the quas-sldng mode control for nonlnear system. Especally, wth the applcaton of the computer, the propertes, stablty and reachng condtons of sldng mode varable structure are changed; and varable structure control method of contnuous tme systems can not be drectly appled to the dscrete tme system. Hence, the quassldng mode control has mportant theoretcal value and practcal sgnfcance to study the varable structure control method for dscrete-tme nonlnear systems. In recent years, the theory and desgn of varable structure control for dscrete tme systems are gradually ncreasng, and adaptve varable structure control method attracts many scholars because of ts advantages of adaptve control and varable structure control[7]. And, n order to solve the dsturbance problem n the system, the upper bound of uncertanty s adopted; t can also express that the dsturbance problem s solved by the worst stuaton. It s no doubt that the dynamc performance and robustness of system can be worse. However, the boundary parameters are dffcult to obtan n real control system. It s all known that the modelng of nonlnear systems s tmeconsumng and laborous, and t s very dffcult to buld relable models for many nonlnear systems, and the cost of modelng s very hgh. The Model Free Adaptve Control (MFAC) s proposed to buld the dynamc lnearzaton model of nonlnear system, the algorthm s based on the theory of pseudo-partal-dervatve (PPD), and the controller just uses the nput and output nformaton[8-9]. The paper combnes the dynamc lnearzaton deology of MFAC and quassldng mode control algorthm to buld a precse model and the neural network s desgned to obtan the optmal control law based on the above model. The control algorthm has the followng advantages, () a better robustness; () a better trackng performance of system. System dentfcaton and control law of the nonlnear system. Dynamc lnearzaton of the nonlnear system Consderng the followng unknown dscrete-tme SISO system yk ( + ) = ( yk ( ),..., yk ( " n), uk ( ),..., uk ( " n)) () y where "() s an unknown nonlnear dfference equaton representng the plant dynamcs, uk ( )" and yk ( )" are measurable scalar nput and output, n y u y and n u s the unknown order of the system. The system () s the normal NARMA model. u 04

3 The followng assumptons are made about the model (): A: The system model s observable and controllable. Ths s the basc requrement for desgnng controller parameters. A: The partal dervatve of () whch s wth respect to the nput parameters u ( s contnuous. Assumpton A s a typcal condton of control system desgn for general nonlnear system. A: The system generalzed Lpschtz, ths s yk ( + ) yk ( + ) " buk ( ) uk ( ) for any and k, k k, k, k" 0 and uk ( ) uk ( ).Assumpton A poses a lmtaton on the rate of change of the system output permssble before the control law to be formulated s applcable. Theorem For the nonlnear system () satsfyng the assumptons A-A, then there must exst a tme-varyng parameter (, called pseudo-partal-dervatve (PPD) [0]. If " u ( k ) 0, the system () can be descrbed as the followng model. yk ( + ) = yk ( ) + ( " uk ( ) () Where, yk ( ) s the estmaton output of system model, "( k ) b and b s the postve constant. Remark: In order to make the condton " u ( 0 n () be satsfed, and meanwhle to make the parameter estmaton algorthm has stronger ablty n trackng tme-varyng parameter, a reset algorthm has been added nto ths MFAC scheme as follows f ˆ "( k ) # or sgn ( ˆ( ) " sgn( ˆ( )) or $ u( k # ) " then ˆ ( k ) = ˆ(). Where s a small postve constant, and ˆ( ) s the ntal value of ˆ ( k ). Consderng ( s a tme-varyng parameter, only the nput and output data can be used to get the optmal soluton. The objectve functon can be desgned n the followng way ˆ J( ( ) = yk ( )" yk ( " ) "( k-) # uk ( -) + µ ( "( k" ) () ˆ where µ > 0 s weght factor, and "( k ) s the estmated value of the delay tme. Gettng the partal dervatve about the (,we can get the optmal estmate value ˆ ( ) ( ) ˆ # uk$ " k = "( k$ ) + ( yk ( ) $ yk ( $ ) $ ˆ "( k$ ) # uk ( -)) µ + # uk ( $ ) (4) where "(0,] s the step factor. JOE Vol. 4, No., 08 05

4 . Model error estmaton usng Quas-sldng mode algorthm In the last secton, the dynamc lnearzaton model (DL model) s desgned to represent the nonlnear system. But the model error stll exsts; n ths secton, we use the Quas-sldng mode control algorthm to estmate the model error. Defne the estmaton error of dynamc lnearzaton model e( e ( y( y( = (5) where y( and y( are the real output and dynamc lnearzaton model output [] respectvely. Defne the functon of sldng mode surface s( T s( = c E( = e( + c0 e( k ) (6) where reachng-law T T c = [,c 0], 0 c s a postve value. Adoptng dscrete-tme sldng mode ( s ( k + ) - s( )/ T = -qs( " sgn( s( ) (7) s ( k + ) = ( " qt ) s( " T sgn( s( ) (8) where > 0 > qt > 0 and > 0 T s samplng perod. Rewrtten (7) as T s( k + ) = ( qt ) s( " T sgn( s( ) = c E( k + ) (9) Accordng to the formula (6)-(9), we can get the estmaton of the model error ek+ ( ) at the next samplng tme: ek ( + ) = (" qtsk ) ( )" Tsgn( sk ( ))" cek ( ) (0) In summary, the dscrete-tme SISO model can be represented n the formula (): y(k +) = y(k +) + e (k +) () where y ( ) s the estmaton value of system model after the addton of Quas- ~ k sldng mode algorthm. ek+ ( ) and yk+ ( ) o represent the estmaton error and output of the dynamc lnearzaton model n the next sample tme, respectvely. The addton of QSMC s to predct the estmaton error n advance and feed back to the neural network controller, therefore, the system can converge n a shorter tme.. Control law desgn based on the artfcal neural network In order to get the optmal control law, we adopt the followng structure of the nonlnear system. NNC s a neural network controller. DL model represents the dynamc 06

5 lnearzaton model n form of (), and QSMC s the Quas-sldng mode controller n secton.. The structure of completed control algorthm s shown n the fgure. " y( k ) + ek () uk ( ) System yk ( ) NNC model yk+ ( ) + DL model " Learnng ek ( ) + " yk+ ( ) + + QSMC Fg.. the completed control structure of the nonlnear system The structure of NNC. The NNC (neural network controller) s represented by the followng structure, whch s the three-layer neural network. The frst layer of NNC s nput layer, the nput data X ( = [ e(, e( k ), e( k )], where e ( represents the trackng error between real output and expected output at sample tme k. where ( k ) y, ( e ( y ( y( = () y represent the desred output and real output respectvely. ek ( ) ek" ( ) ek" ( ) x ( x ( W( u( W( x ( W( Fg.. The nner structure of NNC The second layer s the hdden layer whch contans three neural unts; the response of a neural unt s as follows. $ x( = e( # x( = & e( = e( % e( k % ) " x( = & e( = e( % e( k % ) + e( k % ) () The output of neural network controller u = w x ( + w x ( + w x ( ) (4) ( k JOE Vol. 4, No., 08 07

6 where{ }, =,, w means the weght of neural network, when the approprate value of the w s selected, the controller usually can have a good performance. u ( = u( k " ) + u( k s the control law u( s varable rate of control law. ) at the samplng tme k. So the NNC has the structure of neural network PID controller. The weght of the neural network controller { w }, =,, can be traned by the optmal objectve functon J [ y ( ~ y ( ] c =. So w $ J c ( w ( = -" = " # $ w [ y ( ~ y ( ] ˆ( x ( % (5) where > 0 means the learnng rate of neural network controller, and =,,. ( = w ( k -) + w ( means the weght of neural network controller at the samplng tme k. The stablty and convergence of control algorthm In order to obtan the convergence and stablty of the controller of nonlnear system, another assumpton about the controlled system should be made. A4: The PPD satsfes " c ( > > 0 or " c ( < -, and s a small postve constant. Wthout the loss of generalty, t s assumed n ths paper. Theorem : The nonlnear system () satsfes Assumpton -, the estmaton algorthm of model adopts the formula (5-6). The control rate adopts formula (5). If the expected output of nonlnear system s bounded, and the estmaton of model s stable, and the control algorthm s stable. And the real output and nput s bounded. Proof ) Tme-varyng parameter ˆ( s bounded. Consderng the length of artcle, the paper don t gve the detaled proof, the detaled process s stated n the paper[0]. ) The convergence of the nonlnear system model Defne Lyapunov functon V ( = [ s( ] + (6) where s a small postve const to guarantee V ( whch s a postve functon. Accordng to the reachng-law of contnuous sldng mode control, the reachng law of dscrete-tme sldng mode control can be wrtten as []. " V ( = V ( k + ) V ( < 0 (7) 08

7 " V ( = V ( k + ) V ( = ( s( k + )) = [( qt ) s( # T sgn( s( )] ( s( ) (8) = qt ( qt ) # T( qt ) s( - ( s( ), hence "V 0. Snce > qt > 0,, q, T > 0 Accordng to the Lyapunov stablty theorem, the dentfcaton model s stable. On the promse of the gven ntal value, V (0) s bounded, usng the control law (4), the estmaton of model error e ( tends to zero, ths s lm y( ~ y ( = y( y( = 0. k#" ) The convergence of control algorthm Accordng to the approach ablty of neural network, the network can optmze the parameters of controller by the error back-propagaton way, so the output of controller can approach the desred output[]. Defne global Lyapunov functon V g ( = 0.5( e( ) (9) V g ( = 0.5( e( + e( ) " = e( e( + 0.5( 0.5( e( ) e( ) (0) { As we all know, e( s the functon about the } w, t also ( w e f ( w, w, ). Accordng to the prncple of calculus, we can get the followng expresson. # e( = = " = = " ˆ( & % e( # w = % w = = " '( ˆ( & ) %( y %( y( k " ) + ˆ( & $ # u( ) $ # w % w %(# u( ) $ # w % w [ y ( " ~ y ( ] ( " ~ y ( ) # w % w = ( x ( ) Due to the estmaton model s convergence, so the real output y ( tends to equal to the estmaton value ~ y ( k ). So we can get the followng concluson. () JOE Vol. 4, No., 08 09

8 $ e ( # " ( ˆ( % ) [ y ( " y( ] = & ( x ( ) () Accordng to the formula (0) and (), # V = " g ( = # e( ( e( # e( ) ˆ( [ " ] y ( y( ( x ( ) [ " 0.5 %( ˆ( $ ) %( $ ) () If ( ˆ( # ) = = = $ ( x ( ) ", hence " V g ( 0 ( x ( ), accordng to the prncple of Lyapunov stablty, the control system s stable. Overall, the maxmal trackng error of nonlnear system can be expressed Max ( e( ) = max y ( " y( <. In the range of allowable error, the control system has a global convergence. 4) The boundness of real output and nput Max ( e( ) = max y ( " y( <, e( s bounded, and the desred output s bounded. So the real output s bounded. Snce the tme-varyng parameter ( ) s u ( " k + Max( y( ) ) bounded, and, and also bounded. Clearly, the Theorem can be proved. 4 Smulaton results ] ˆ k, s a const. Hence ( u s In ths secton, the two smulatons are used to show the effectveness and robustness of the control algorthm.. The convergence and superorty of algorthm compared wth the tradtonal PID algorthm n lnear system.. The convergence and robustness of control system n nonlnear system. 4. The convergence and robustness of algorthm n lnear system The convergence of algorthm n lnear system. As we all known, the tradtonal PID algorthm s the most mportant control algorthm n lnear system. The output of the ncremental PID algorthm s uk = k ek + kek + k ek (4) ( ) p ( ) ( ) d ( ( )) 0

9 In order to verfy the superorty compared wth the tradtonal PID algorthm, the dscrete-tme lnear system s defned as follows yk ( ) = 0.8 yk ( ) 0.5 yk ( ) uk ( ) (5) where k s a postve nteger, and the desred output be wrtten as (6) y( = k< 40 (6).5 New algorthm Desred output Tradtonal pd algorthm The system output Sample tme k Fg.. The trackng performance compared wth tradtonal PID n lnear system The smulaton dagram s showed n fgure, as we can see, f the samplng tme s less than, the controller needs to collect the error nformaton of the trackng system, so the real output equal to zero; After three sample tme, the control system can be stable n the fnte tme; the real output can track the desred output perfectly and the error of trackng can be n allowable range. The smulaton results show the new algorthm can depress the overshoot compared wth the tradtonal PID algorthm, decrease the adjustment tme and get the satsfactory trackng performance. The robustness of algorthm n lnear system. The robustness of system s the mportant ndex n control systems, t refers to the perturbaton of system parameters or system structure, the system stll can mantan the characterstcs, lke convergence or stablty. In order to verfy the robustness of lnear system compared wth the tradtonal PID algorthm, the smulaton model s as follows, the parameters perturbaton happens at sample tme k = 0. " 0.8 yk ( ) 0.5 yk ( ) uk ( ) k < 0 yk ( ) = # $ 0.8 yk ( ) 0.8 yk ( ) uk ( ) k >= 0 (7) JOE Vol. 4, No., 08

10 .5 New algorthm Desred output Tradtonal pd algorthm The system output Sample tme k Fg. 4. The robustness of algorthm compared wth tradtonal PID algorthm n lnear system As we can see n the fgure 4, when the structure parameters of system change at the sample tme k = 0, the new algorthm stll has the better trackng performance, and the trackng error can be n the range of allowable at the fnte tme. And the adjustment tme and adjustment ampltude of new algorthm can be smaller durng the perod of the system parameters perturbaton compared wth tradtonal PID algorthm. 4. The convergence and robustness of nonlnear system The convergence and superorty compared wth the MFAC algorthm. MFAC (model free adaptve control), as we know, has the good performance n the nonlnear control system. So, n order to verfy the superorty, we compare wth the MFAC algorthm. The dscrete-tme nonlnear system y( k ) y ( + ) + ( y( k )) = ( u( (8) where u ( s the control varable of controller, y( and y ( k ) are the current and the one step delayed outputs, respectvely. And the desred output of the system s as follows: " 0.5sn( k 500) + 0.cos( k 500)... k < 000 # ( ) = $ > k >= 000 # & % 0... k >= 500 y k (9)

11 New algorthm Desred output MFAC algorthm The system output Sample tme k (a) the comparatve fgure between new algorthm and MFAC algorthm The system output New algorthm Desred output MFAC algorthm Sample tme k (b) Local enlarged fgure of smulaton results Fg. 5. The superorty of control system compared wth the MFAC algorthm As we can see from the fgure 5(a), the trackng error of system can be gnored n the range of error allowed except the sudden change of desred output. The smulaton result valdates the convergence of the algorthm and the stablty of the system dentfcaton n nonlnear system. As shown n the local enlarged fgure 5(b), the MFAC and new algorthm can track the desred output after the several step tmes. And the adjustment tme and adjustment ampltude of the new algorthm have a great mprovement compared wth the MFAC algorthm. The robustness of control system. In order to verfy the robustness of the algorthm, the paper adopts the same control parameters to valdate the robustness of the JOE Vol. 4, No., 08

12 new control algorthm; the trackng performance s shown n the fgure 6. The smulaton model s adopted (0), and the desred output s as the form of (9). yk ( " ) # + ( uk ( ))... k< 00 # + ( yk ( " )) # yk ( " ) yk ( ) = $ + ( uk ( ))...00 > k>= 00 # + ( yk ( " )) # 0.8 yk ( " ) # + ( uk ( ))... k>= 00 % + ( yk ( " )) (0) New algorthm Desred output MFAC The system output Sample tme k (a) The robustness analyss fgure n nonlnear system 0.6 New algorthm Desred output MFAC 0.5 The system output Sample tme k (b) The local enlarged fgure between new algorthm and MFAC algorthm Fg. 6. The robust analyss n the nonlnear system 4

13 As we can see n the fgure 6, when the structure parameters of system change at the samplng tme k = 00, k = 00, the algorthm stll has the better trackng performance, and the trackng error can be n the range of allowable at the fnte tme. From the local enlarged fgure 6(b), we can see that the adjustment tme and adjustment ampltude of the new algorthm s less than the MFAC algorthm. 5 Concluson In ths paper, a new dynamc lnearzaton model about the Quas-Sldng Mode algorthm was proposed for dentfcaton of complex nonlnear system. A neural network PID controller s desgned to get the optmzed control rate based on the above model. The advantages of algorthm are as follows:. better robustness. better trackng performance. less adjustment tme and ampltude. And the global stablty of the system s proved by the way of the Lyponou functon and Matlab smulaton. 6 Acknowledgment The work s supported by the Natonal Scence and technology support program (NO 04BAG06B0) and Fundamental Research Funds for the Central Unverstes (No 04HGCH000). 7 References [] Dng, Feng. "Herarchcal mult-nnovaton stochastc gradent algorthm for Hammersten nonlnear system modelng." Appled Mathematcal Modellng 7.4 (0): [] Ngam, Vvek Prakash, and Danel Graupe. "A neural-network-based detecton of eplepsy." Neurologcal Research 6. (004): [] Wang, Tong, Hujun Gao, and Janbn Qu. "A combned adaptve neural network and nonlnear model predctve control for multrate networked ndustral process control." IEEE Transactons on Neural Networks and Learnng Systems 7. (06): [4] L, Hongy, et al. "Control of nonlnear networked systems wth packet dropouts: nterval type- fuzzy model-based approach." IEEE Transactons on Cybernetcs 45. (05): [5] Lu, Yan-Jun, Shaocheng Tong, and CL Phlp Chen. "Adaptve fuzzy control va observer desgn for uncertan nonlnear systems wth unmodeled dynamcs."ieee Transactons on Fuzzy Systems. (0): JOE Vol. 4, No., 08 5

14 [6] Hou, Y-You, and Zhang-Ln Wan. "Robust Stablty for Nonlnear Systems wth Tme- Varyng Delay and Uncertantes va the Quas-Sldng Mode Control." Internatonal Journal of Antennas and Propagaton 04 (04). [7] Ma, Hafeng, Janhua Wu, and Zhenhua Xong. "Dscrete-tme sldng-mode control wth mproved quas-sldng-mode doman." IEEE Transactons on Industral Electroncs 6.0 (06): [8] Zhu, Yuanmng, and Zhongsheng Hou. "Data-drven MFAC for a class of dscrete-tme nonlnear systems wth RBFNN." IEEE transactons on neural networks and learnng systems 5.5 (04): [9] Ma, Png, et al. "Man steam temperature control system based on MFAC." Electrc Power Scence and Engneerng (006): 006. [0] Cheng, Zhhu, Zhongsheng Hou, and Shangta Jn. "MFAC-based balance control for freeway and auxlary road system wth mult-ntersectons." Control Conference (ASCC), 05 0th Asan. IEEE, 05. [] Han, Yueqao, Yonggu Kao, and Cunchen Gao. "Robust sldng mode control for uncertan dscrete sngular systems wth tme-varyng delays and external dsturbances." Automatca 75 (07): [] Sarpturk, S. Z., Yorgo Istefanopulos, and Okyay Kaynak. "On the stablty of dscrete-tme sldng mode control systems." IEEE Transactons on Automatc Control.0 (987): [] Mukhopadhyay, Snehass, and Kumpat S. Narendra. "Dsturbance rejecton n nonlnear systems usng neural networks." IEEE Transactons on Neural Networks 4. (99): [4] Chandrasena Premawardhena, N., ICT n the foregn language classroom n Sr Lanka: A journey through a decade. 0th World Conference on Computers n Educaton (WCCE 0), Ncolaus Coperncus Unversty, July -5 0, Torun, Poland.pp -4 8 Authors Hufang Kong s the Professor and the Drector of the Research Center of Automotve Electroncs and Control Technology n Hefe Unversty of Technology, No 9, Tunx Road, Hefe, Chna, Her research nterests ncludng Control Engneerng, Industral Automaton, and Dynamcs and Control. (konghufang@6.com). Yao Fang s a PH.D student n the Automotve Electroncs and Control Research Center n Hefe Unversty of Technology, Majorng n modelng and control of complex nonlnear systemsenergy management of hybrd electrc vehcle and the behavor decson of drvers. Artcle submtted October 07. Resubmtted December 07. Fnal acceptance 06 February 08. Fnal verson publshed as submtted by the authors. 6