Generation of Lyapunov Functions by Neural Networks

Transcription

1 WCE 28, July 2-4, 28, London, U.K. Genertion of Lypunov Functions by Neurl Networks Nvid Noroozi, Pknoosh Krimghee, Ftemeh Sfei, nd Hmed Jvdi Abstrct Lypunov function is generlly obtined bsed on tril nd error. Systems with high complex nonlinerities nd more dimensions re much more difficult to be delt with nd in mny cses it leds to dull one. This pper proposes two strightforwrd methods for determining or pproximting Lypunov function bsed on pproximtion theory nd the bilities of rtificil neurl networks. The potentil of the proposed methods re demonstrted by simultion exmples. Index Terms Approximtion theory, Lypunov function, nonliner system, neurl network. I. INTRODUCTION Stbility of nonliner dynmic systems plys n importnt role in systems theory nd engineering. There re severl pproches in the literture ddressing this problem. The most useful nd generl pproch for studying the stbility of nonliner control systems is the theory introduced in Alexndr Mikhilovich Lypunov. Qulittively, system is described s stble if strting the system somewhere ner its desired operting point implies tht it will sty round the point ever fter. Identifying Lypunov function (ssuming it exists) for n rbitrry nonliner dynmic system in order to demonstrte its stbility in the Lypunov sense is no trivil tsk. The vst mjority of existing methodologies fll in one of the following two ctegories:) methods which construct or serch for Lypunov function nd 2) methods which try to pproximte it [], [2], [3]. Usully methods of the first ctegory re only pplicble to some clsses of systems i.e. the given system is required to hve some desired chrcteristics, e.g. the nonliner system must hve polynomil vector field [4]. Methods of ltter ctegory usully do not hve strong mthemticl frmework. Reserchers developed myrid of techniques nd procedures to identify the Lypunov function. Among the significnt prior reserch works reported in the literture, we focus on methods tht construct or pproximte Lypunov function by neurl networks. Prokhorov in [] suggests Lypunov Mchine, which is specil-design rtificil neurl network, for pproximting Lypunov function. The uthor indictes tht the proposed lgorithm, the Lypunov Mchine, hs substntil computtionl complexity mong other issues to be resolved nd defers their resolution to N. Noroozi is with the Deprtment of Electricl Engineering, Shirz University, Shirz, Irn (corresponding uthor to provide e-mil: nvidnoroozi@gmil.com). P. Krimghee is with the Deprtment of Electricl Engineering, Shirz University, Shirz, Irn (e-mil: kghee@shirzu.c.ir). F. Sfei is with the Deprtment of Electricl Engineering, Shirz University, Shirz, Irn (e-mil: smne_sfei@yhoo.com). H. Jvdi is with the Deprtment of Electricl Engineering, Irn University of Science nd Technology, Tehrn, Irn (e-mil: hmed_jvdi@ee.ust.c.ir). future work. In [2] uthor suggests n lgorithm for pproximting Lypunov function when system is symptoticlly stble. In the other work, Genetic lgorithms re used in [5] for lerning neurl networks. Bsed on this ide, two different pproches were used in order to clculte the pproprite network s weights. Another recently reported studies re in [3],[6],[7]. In this pper, we propose two strightforwrd pproches for constructing or pproximting Lypunov function bsed on pproximtion theory nd the fetures of rtificil neurl networks. Our pproches re ble to use for both explined ctegories. Both pproches re bsed on using some stte trjectories of system for constructing or pproximting Lypunov function. The potentils of the proposed methods re demonstrted by simultion exmples. The reminder of this pper is orgnized s follows. In section 2, we introduce preliminries relted to Lypunov theory, pproximtion theory. In section 3, we propose two pproches for constructing or pproximting Lypunov function of symptoticlly stble systems. In section 4, we illustrte our methods by two exmples. Finlly, section 5 summrizes our ccomplishments. II. PRELIMINARIES In this section, we introduce nottion nd definitions, nd present some key results needed for developing the min results of this pper. Let R denote the set of rel numbers, R + denote the set of positive rel numbers,. denote norm on R n n, nd D R be n open set contining x =. Consider the nonliner dynmicl system given by & (2-) x = f ( x), x() = x n x( t) D R where continuous on D. is the stte vector, f() =, nd f(.) is Theorem 2-(Lypunov Theory)[8] Let x = be n equilibrium point for (2-). Let V : D R be continuously differentible function, such tht V() = nd V(x) > in D {} (2-2) V & ( x) in D {} (2-3) Then, x = is stble. Moreover, if V & ( x) < in D {} (2-4) then x = is symptoticlly stble. Theorem 2-2(Weierstrss Results)[9] Given Bnch spce X with elements f, norm f, nd sequence ISBN: WCE 28

2 WCE 28, July 2-4, 28, London, U.K. N Φ N = {ϕ i} i= X of bsis elements, f is sid to be pproximble by liner combintions of Φ N with respect to the norm. if for ech ε > there exists N such tht f PN < ε where N PN = θ iϕi ( x), for some θi R. (2-5) i= Theorem 2-3[9] Ech rel function f tht is continuous on D = [, b] is pproximble by lgebric polynomils with respect to the norm : ε >, M such tht if N > M polynomil p P N with f PN < ε for ll x D. III. METHOD Due to the lck of specific method to determine Lypunov function, this function is generlly obtined bsed on tril nd error. Systems with high complex nonlinerities nd more dimensions mke the humn s job more difficult nd in mny cses it leds to dull one. It will be productive, if we mke the computer engged in this tril nd error so tht with the ppliction of lgorithms nd intelligent systems, it will grdully lern from previous itertions nd lerning t ech stge it will come up with the nswer which is Lypunov function for given system. The following study is proposed two forwrd nd bckwrd intelligent lgorithms to generte Lypunov function. Bsed on theorem (2-2), set of bsis elements re cpble of uniform pproximtion of continuous functions over compct region D X. In this study bsed on pproximtion theory, we intend to determine or pproximte Lypunov function of n symptoticlly stble system. In the function pproximtion literture there re vrious brod clsses of function pproximtion problems. The clss of problems tht will be of interest herein the development of pproximtion to Lypunov function bsed on informtion relted to their input-output smples. However, the key issue is tht Lypunov function is ultimte gol to be chieved. So some ssumptions need to be considered which will be explined lter. A. Forwrd Method Let us consider the system (2-). Since Lypunov function is not ccessible now, the stte trjectories of the system re clculted for some initil conditions in both forwrd nd bckwrd lgorithms which will be introduced lter on. Then one positive rel number is ssigned to V(x) for ech component of clculted stte trjectory. The forwrd lgorithm is s follows Step ) Specify the form of the bsis function in chosen neurl network. Step 2) Select p points s initil conditions in D then clculte their stte trjectories (suppose tht ll those stte trjectories converge to the origin.). The stte trjectories re clculted using numericl methods nd re nmed s follows x j ( k ) = x kj j =, 2,, p where the index x j denotes jth stte trjectory, nd k denotes kth smple of stte trjectory x j. Also n shows the number of components of the stte vector x j. Moreover, the stte trjectories re clculted up to x j ( n ). Step 3) Consider n unknown cndidte Lypunov function V(x). Now for ech of p initil conditions, positive rel numbers re ssigned to V(x j ): { p} j,2,..., : V ( x j ) = α j, α j > This set of points with V()= re chosen s trining dt for network lerning. Note tht it is likely x j >> therefore the origin is outlying point in the trining dt nd in order to resolve this chllenge, some solutions will be suggested in this section. Step 4) Trin the network. Step 5) Exmine whether V & (x) is negtive definite or not. If yes, go to step 6. If not, x j nd its ssigned vlue V(x j ) re dded to the trining dt. Therefore we hve j {,2,..., p} : V ( x k + j ) = α k+ j, α k + j >, α k+ j α kj < nd return to step 4. Step 6) Finish. This lgorithm is so generl nd its success depends on vrying fctors including, structure, lerning method of network nd method of choosing lerning dt. We cn choose initil conditions bsed on the mount of sensitivity of stte trjectories in reltion to them [8]. A desired negtive definite function cn be used for choosing rte of decrese V(x) (i.e. V & (x) ). This implies tht the time derivtive of the network tries to mimic the negtive definite function. Also experimentl results show tht using network with pproximtors with globl influence functions is more effective. But forget tht given pproximtion structure either locl or globl is dependent on the function tht is being modeled. Studying different simultions show tht forwrd method cn pproximte (or determine) Lypunov function fter some itertions. However, we propose nother lgorithm to improve some chllenges of forwrd lgorithm. B. Bckwrd Method The bckwrd method is the reverse of forwrd method. Their min difference is relted to choice of trining dt. In this lgorithm p points in D s initil condition will be chosen then we clculte their stte trjectories. Now x= nd x j (n) re selected s the first trining dt. Then we trin the network. After the trining if the pproximted function stisfies in theorem (2-), we choose the trined network s Lypunov function. Otherwise, first we ssign V(x j (n-)) to x j (n-) then we dd it to the trining dt. Finlly we retrin the network. This procedure is continued until the pproximted function stisfies in theorem (2-). The second proposed lgorithm s follows ISBN: WCE 28

3 WCE 28, July 2-4, 28, London, U.K. Step ) Specify the form of the bsis function in chosen neurl network. Step 2) Select p points s initil conditions in D then clculte their stte trjectories (suppose tht ll those stte trjectories converge to the origin.). The stte trjectories re clculted using numericl methods nd re nmed s follows x j ( k ) = x kj j =, 2,, p where the index x j denotes jth stte trjectory, nd k denotes kth smple of stte trjectory x j. Also n shows the number of components of the stte vector x j. Moreover the stte trjectories re clculted up to x j ( n ) Step 3) Consider n unknown cndidte Lypunov function V(x). Consider negtive definite function s desired V & (x) from which clculte V(xnj). Step 4) Trin the network. Step 5) Exmine whether V & (x) is negtive definite or not. If yes, go to step 6. If not, xn- j nd its ssigned vlue V(x n- j ) re dded to the trining dt nd return to step 4. s cndidte for Lypunov function. V(x) = x x x x 2 (4-2) where i, i 3, is n unknown coefficient which is determined during lerning. Decresing rte of V(x k ) describes s follows: V(x k+ ) = V(x k ) b*rnd[,] (4-3) where rnd[,] is rndom vlue with uniform distribution nd b is lerning fctor, <b<. According to previous section, the forwrd lgorithm is implemented. After 25 itertions, we hve 2 3 =.87 =.44 =.58 The obtined Lypunov function nd its time derivtive re shown in figures 2 nd 3, respectively. Step 6) Finish. Both lgorithms could be unsuccessful bsed on some resons. The followings re the most importnt ones: ) The given system is unstble. 2) The number of initil conditions is not enough. 3) The network structure or its trining method is not suitble. IV. SIMULATION RESULTS In this section, we give two simultion exmples to demonstrte the effectiveness of proposed lgorithms. Both exmples re chosen from [8]. In these exmples the sigmoidl nd polynomil neurl network pproximtor re used. As we will see lter, the lgorithm success depends on loction nd number of initil conditions. A. Exmple Let us consider the following system Fig.. denotes the loction of initil conditions = x + x2 2 = ( x + x2 )sin( x) 3x2 (4-) The forwrd method is used in this exmple. As shown in Fig., 6 points re chosen s initil condition where x j ( ) =, j=, 2,..., 6. A-- Approximtion of Lypunov Function by Polynomil Neurl Network As we know, lgebric polynomils re universl pproximtors [9]. Also qudrtic function is n pproprite Lypunov cndidte for mny groups of dynmic system [5], [], []. Therefore polynomils cn pproximte or construct Lypunov function. So we choose qudrtic polynomil function with unknown coefficients Fig. 2. The constructed Lypunov function for the system (4-) ISBN: WCE 28

4 WCE 28, July 2-4, 28, London, U.K. Fig. 3. The time derivtive of the constructed Lypunov function Fig. 6. The pproximte Lypunov function trjectory for initil condition (,-) A-- Approximtion of Lypunov Function by Sigmoidl Neurl Network In this cse, the trining dt is chosen like previous section. The network is consisted of one hidden lyer nd nodes in it. After 25 itertions, the pproximtion of Lypunov function is obtined. Since obtining the closed form of the clculted function is difficult, if the number of initil conditions hve selected enough then the neurl network gives locl generliztion [9], therefore V(x) in the vicinity of the trining dt is pproximted correctly. Figure 4 indictes the pproximte Lypunov function. The behviors of V(x) for three stte trjectories of the system re shown in figures 5 through 7. Fig. 7. The pproximte Lypunov function trjectory for initil condition (-,) B. Exmple 2 In this exmple the bckwrd method nd polynomil neurl network re used to generte Lypunov function for the following system. Fig. 4. The pproximte Lypunov function for the system (4-) = x + sin( x3) 2 = sin( x3) (4-4) 3 = x + 2x2 sin( x3) Around the origin 26 points re selected s initil conditions such tht x. Consider the following qudrtic j = function s cndidte Lypunov function: V(x) = x x x x x x x x 2 x 3 (4-5) where i, i 6, is n unknown coefficient which is determined during lerning. The following negtive definite function is considered s desired time derivtive of Lypunov function V& d (x) Fig. 5. The pproximte Lypunov function trjectory for initil condition (-,) V& d ( x) = x T 5 x 2 After 2 itertions, we hve ISBN: WCE 28

5 WCE 28, July 2-4, 28, London, U.K. = = = = = = We cn esily check tht the trined network is Lypunov function for the system (4-4). Therefore the origin of the system is symptoticlly stble. V. CONCLUSION In this pper two lgorithms hve been proposed to construct Lypunov function bsed on pproximtion theory nd the bilities of rtificil neurl networks. As we know, neurl networks found wide ppliction in mny fields, both for function pproximtion nd pttern recognition. However, in this study with considering some ssumptions the network solved problem subject to few conditions which showed potentils of neurl network for solving inequlities. REFERENCES [] D.V. Prokhorov, A Lypunov mchine for stbility nlysis of nonliner systems, IEEE Interntionl Conference on Neurl Networks, Orlndo, FL, USA, Vol. 2, pp. 28-3, 994. [2] G. Serpen, Empiricl Approximtion for Lypunov Functions with Artificil Neurl Nets, Proc. of IJCNN, Montrel, Cnd, pp , July 3-Aug. 2, 25. [3] D. V. Prokhorov, L. Deldkmp, Appliction of SVM to Lypunov Function Approximtion, Proc. of the IJCNN, Wshington DC, Vol., pp , 999. [4] A. Ppchristodoulou, S. Prjn, On the Construction of Lypunov Functions Using Sum of Squres Decomposition, Proc. of IEEE Conference on Decision nd Control, 22. [5] V. Petridis, S. Petridis, Construction of Neurl Network Bsed Lypunov Functions, Proc. of the IJCNN, Vncouver, BC, Cnd, July 6-2, pp , 26. [6] D. V. Prokhorov, L. Deldkmp, Anlysing for Lypunov Stbility with Adptic Critics, Proc. of IEEE Conference on Systems, Mn nd Cybernetics, Sn Diego, pp , 998. [7] N. E. Brbnov, D. V. Prokhorov, A New method for Stbility Anlysis of Non-liner Discrete-Time Systems, IEEE Trnsctions on Automtic Control, Vol. 48, pp , 23. [8] H.K. Khlil, Nonliner Systems. 2nd ed. Prentice Hll. Englewood Cliffs, NJ. 996, ch. 3. [9] J. A. Frrell, M. M. Polycrpou, Adptive Approximtion Bsed Control: Unifying Neurl, Fuzzy nd Trditionl Adptive Approximtion Approches, John Wiley & Sons, Inc. 26, ch. 2. [] H. Bouzouche, N. B. Briek, On the Stbility Anlysis of Nonliner Systems using Polynomil Lypunov Functions, Mthemtics nd Computers in Simultion (27), to be ccepted. [] L. Vndenberghe, S. Boyd, A polynomil-time lgorithm for determining qudrtic Lypunov functions for nonliner systems, Proc. of the Europen Conference on Circuit Theory nd Design, pp.65-68, 993 ISBN: WCE 28