Optimal Control of Nonlinear Systems Using the Shifted Legendre Polynomials

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1 Optial Control of Nonlinear Systes Using Shifted Legendre Polynoi Rahan Hajohaadi 1, Mohaad Ali Vali 2, Mahoud Saavat 3 1- Departent of Electrical Engineering, Shahid Bahonar University of Keran, Keran, Iran. Eail: hajohaadi6@gail.co(corresponding author) 2- Departent of Maatics, Shahid Bahonar University of Keran, Keran, Iran. Eail: vali@ail.uk.ac.ir 3- Departent of Electrical Engineering, Shahid Bahonar University of Keran, Keran, Iran. Eail: saavat@ail.uk.ac.ir Received: Deceber 2013 Revised: March 201 Accepted: May 201 ABSTRACT A nuerical technique based on Legendre Polynoi for finding optial control of nonlinear systes with quadratic perforance index is presented. An operational atrix of integration and product atrix are introduced and are used to reduce nonlinear differential equations for solution of nonlinear algebraic equations. The optial solution fro two classes of first and second order nonlinear systes is considered. In case of second-order nonlinear systes, a new approach is introduced to find optial solution. In both cases, nuerical exaples are given and copared with to confir accuracy of. KEYWORDS: Non-linear systes; Legendre Polynoi; Optial Control; Nuerical Methods 1. INTRODUCTION Most of systes have nonlinear dynaics. So, study of se nonlinear systes are very iportant. Hence, any researchers and designers have showed an active interest in developent and applications of nonlinear systes [1-3]. Orthogonal functions and al series have received considerable attention in dealing with various probles of dynaical systes. Exaples are use of Wh functions (Chen Shih 1978 )[], blockpulse functions (Maleknejad and Shahrezaee 2005, wang and Li 2009) [5,6], Chebyshev (H. Jaddu and E. Shieura 1973) [7], series (Mouroutsos and Sparis 1985, Gulsu and Sezer 2006, S. Yalcinbas 2002 ) [8-], Fourier series (M.L. Nagurka, V. Yen 1990,Ardekani and Keyhani 1989, Ardekani, Saavat and Rahani 1991, Saavat and Rashidi 1995, Ebrahii, Saavat, Vali and Gharavisi 2007) [11-15]. The ain characteristic of technique is that it reduces se probles to those of solving a syste of algebraic equations; thus it greatly siplifies proble. In this paper, we use Legendre Polynoi to find optial control of nonlinear systes. For first tie in this paper, optial control of a particular class of second-order nonlinear systes using a new approach has been. By this nuerical technique, a difficult proble is reduced to straightforward nonlinear algebraic equations which can be solved by using a digital coputer. Nuerical exaples are given to show accuracy of technique. 2. PROPERTIES OF LEGENDER POLYNOMIALS 2.1. Legendre Polynoi The shifted Legendre,, where 0 are obtained fro[16], ,2,3, (1) 1, 2 1 (2) The orthogonal property is given by 0,, (3) 2.2. Function ation A function 0, can be approxi as: 33

2 ft f P t, () In practice we only consider a finite nuber of ters, that is ft f P t, (5) The shifted Legendre al f can be obtained by using f fτ P τd Equation (5) can be written in a atrix for as: ft F T Pt (6) Or ft Pt T F (7) F and Pt are 1 atrices which are given by: F T f f f f (8) Pt P P P P T (9) 2.3 The Operational Matrix of Integration Integration of vector Pt defined in Eq. (9) can be written as: Ps ds H Pt, () By using Eqs.(3) and (7) we have : fsds F T Ps ds F T HPt (11) By using Eqs.(7) and () we have : fsds Ps T F dspt T H T F (12) atrix H is obtained as follows[16]: H 1/2 1/ /6 0 1/6 0 t 0 1/ /1 1/ / The Product Operational Matrix The product operational atrix F can be defined as follows[17]: PtP T tf F P t, (13) PP T FFP, (1) To illustrate calculation procedures, we choose =3. Thus, we have: Ff f f T P(t) =P P P T, (15) P t 1, P t 2t 1 Using equation (1), we get:, P t 6t 6t1 P P P P P P P P P P f f f T FP P P T (16) P P P P P Finally, we have: F f f f f (17) 0 0 f 3. THE OPTIMAL CONTROL PROBLEM The ai of this section is to explain how we can use Legendre to find optial solution of first and second order nonlinear systes. In both cases results are copared with results of The first-order systes: Exaple 1: Consider optial control proble of fist-order nonlinear syste[12]: 2y t ٠ y t ut, y0, 0 t 1 (18) With respect to a quadratic perforance index: J y t u tdt (19) Integrating Eq. (18) fro zero to t and using Eqs. (6), (7), (), (11), (12) and (1) we have: ys ds yty0 Y T Pt Y T Pt, usds U T Ps ds ysds Y T Ps ds U T Psds Y T Psds U T HPt, Y T H Pt, for i = 0,1,2,...,1. In fact 3

3 y S ٠ ds y t y 0 Y T PtP T ty Y T Pt, Finally 2Y T YPt Y T PtY T Pt Y T Pt U T HPt, Eliinating Pt in equation (20) gives: 20 2Y T Y Y T U T H2Y T Y T 0 (21) Y T and Y T are 1 atrices which given by: Y T Y T For perforance index we have: J Y T PtP T ty U T PtP T tudt (22) J= Y T LY U T LU (23) L PtP T tdt (2) We now iniize equation (23) rel to equation (21),by using Lagrange ultiplier technique we get: J Jλ 2Y T Y Y T U T H2Y T Y T (25) λ is a 1 atrix as follows: λ λ λ λ λ λ The necessary conditions for finding iniu are: J 0, J 0 i 1,2,, 1 (26) J λ 0 (27) By using this technique Eq. (25) turns into a set of nonlinear algebraic equations which can be solved using Newton s iterative to obtain J. The approxi J and y0 in coparison with are given in Table 1. Table 1. 0, using in coparison with for =, =5 and =6 for exaple Exac t valu es of a of by al s a of by a of J by al s a of J by Exaple 2: Consider optial control proble of a different fistorder nonlinear syste[18]: y t y t ut, y0, 0 t 1 (28) With respect to a quadratic perforance index: J 0.5 y t u tdt (29) Integrating Eq. (28) fro zero to t and using Eqs. (6), (7), (), (11), (12) and (1) and eliinating Pt we have: Y T Y T Y T YH U T H (30) Using Newton s iterative explained in exaple 1, approxi J and y0 and coparison with are given in Table 2. Table 2. 0, using in coparison with for =, =5 and =6 for exaple Exac t value s of by by The second order systes: Exaple 3: J by Consider optial controlling proble of second-order nonlinear syste[12]: J by

4 yty t y t y t ut, y0, y 0 0 (31) Using sae perforance index as exaple 1. Let us use following assuption: y t Y T Pt (32) Integrating Eq. (32) fro zero to t: y t y 0 Y T HPt, (33) Integrating Eq. (33) fro zero to t: yt y0 Y T H Pt (3) Expanding y0 using Legendre : y0 f T Pt (35) f T is a 1 atrix given by: f T 0 0 Substituting Eq. (35) into Eq. (3) gives yt Y T H Pt f T Pt, (36) Expanding ut within using Legendre : ut U T Pt, (37) Substituting Eqs. (32), (36),(37) into Eq. (31) gives: Y T H PtPt T Yf T PtPt T YY T HPt Y T H PtPt T H T Yf T PtPt T f Y T H PtPt T ff T PtPt T H T Y U T Pt (38) Let us define: Y T H Z T (39) Substituting Eq. (39) and Eq. (1) into Eq. (38) and eliinating Pt gives: Z T Y S T Y f T Y Y T HZ T Z f T f Z T f f T Z U T 0 Therefore Eq. (0) turns into a set of nonlinear algebraic equations. By using Newton s iterative explained in exaple 1, we get Y T,, U T. In this case Y T is coefficients of y t. In order to find coefficients of yt, we should solve Eq. (36) n we get following equation: Coefficients of yt=y T H f T (1) Now this gives us iniu of J. The approxi J and y0 and coparing with are given in Table 3. Table 3. 0, using in coparison with for = 2, =3 and = for exaple Exa ct valu es of by by J by J by CONCLUSION In, using Legendre, nonlinear differential equations are reduced into a set of nonlinear algebraic equations, which can be solved using a digital coputer. Since operational atrix of integration and product atrix contain any zero entries, it gives coputational advantages when copared with or possible approxiations. Nuerical exaples are given to show accuracy and applicability of technique. Results show that Legendre have accurate approxiate values than. Finally, can be extended for optial control of nonlinear tie within varying systes. REFERENCES [1] S.P., Banks, Maatical ories of nonlinear systes, Prentice Hall, [2] J., Slotine, W., Li, Applied nonlinear control, Prentice Hall, [3] M., Saavat, A.K., Sedeegh and S.P., Banks, On approxiation of pseudo linear systes by linear tie varying systes, Int. J. Eng., Vol.17, No.1, pp , 200. [] W.L., Chen, Y.P., Shih, Paraeter estiation of bilinear systes via Wh functions, Journal of Franklin Institute, Vol 305, Issue 5, pp , [5] K., Maleknejad, M., Shahrezaee and H., Khatai, Nuerical solution of integral equations syste of second kind by block pulse functions, Applied Maatics and Coputation,, pp. 15 2, [6] X.T., Wang, Y., Min Li, Nuerical solutions of integro-differential systes by hybrid of general block-pulse functions and second Chebyshev, Applied Maatics and Coputation, pp ,

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