《序列二次規(guī)劃算法》word版

1、Legendre PolynomialsNamed in honor of Adrien-Marie (1752-1833) the mathematician, not Louis (1752-1797) the politician.IntroductionA polynomial is a finite sum of terms like akxk, where k is a positive integer or zero. There are sets of polynomials such that the product of any two different ones, mu

2、ltiplied by a function w(x) called a weight function and integrated over a certain interval, vanishes. Such a set is called a set of orthogonal polynomials. Among other things, this property makes it possible to expand an arbitrary function f(x) as a sum of the polynomials, each multiplied by a coef

3、ficient c(k), which is easily and uniquely determined by integration. A Fourier series is similar, but the orthogonal functions are not polynomials. These functions can also be used to specify basis states in quantum mechanics, which must be orthogonal.Legendre Polynomials DenifitionThe Legendre pol

4、ynomials Pn(x), n = 0, 1, 2 . are orthogonal on the interval from -1 to +1, which is expressed by the integral.The Kronecker delta is zero if n m, and unity if n = m. In most applications, x = cos , and varies from 0 to . In this case, dx = sin d, of course. The Legendre polynomials are a special ca

5、se of the more general Jacobi polynomials P(,)n(x) orthogonal on (-1,1). By a suitable change of variable, the range can be changed from (-1,1) to an arbitrary (a,b). The weight function w(x) of the Legendre polynomials is unity, and this is what distinguishes them from the others and determines the

6、m.ApplicationsThe Lengendre polynomials are very clearly motivated by a problem that often appears. For example, suppose we have an electric charge q at point Q in the figure at the left, one of a group whose positions are referred to an origin at O, and we desire the potential at some point P. The

7、distance PO is taken as unity for convenience; simply multiply all distances by the actual distance PO in any particular case. The potential due to this charge is q/R. We can find R as a function of r and by the Law of Cosines: R2 = 1 + r2 - 2r cos = 1 - 2rx + r2, where x = cos . Now we expand 1/R i

8、n powers of r, finding 1/R = Pn(x)rn. The function 1/R is called the generating function of the Legendre polynomials, and can be used to investigate their properties. Generating functions are available for most orthogonal polynomials, but only in the Legendre case does the generating function have a

9、 clear and simple meaning.If we let x = 1, we find that Pn(1) = 1, and Pn(-1) = (-1)n. By taking partial derivatives of 1/R with respect to x and r, and then considering the coefficients of individual powers of r, we can find a number of relations between the polynomials and their derivatives. These

10、 can be manipulated to find the recursion relation, (n + 1) Pn+1(x) = (2n + 1)x Pn(x) - n Pn-1(x), and the differential equation satisfied by the polynomials, (1 - x2) Pn(x) -2x Pn(x) + n(n + 1) Pn(x) = 0. The recurrence relation allows us to find all the polynomials, since it is easy to find that P

11、0(x) = 1, P1(x) = x directly from the generating function, and this starts us off. The differential equation allows us to apply the polynomials to problems arising in mathematics and physics, among which is the important problem of the solution of Laplaces equation and spherical harmonics.The recurr

12、ence relation shows that the coefficient An of the highest power of x satisfies the relation An+1 = (2k + 1)/(k + 1) An, and so from the known coefficients for n = 0, 1 we can find that the coefficient of the highest power of x in Pn is .(2n-1)/n!.The polynomials can also be found by solving the dif

13、ferential equation by determining the coefficients of a power series substituted in the equation. This method was often used in quantum mechanics texts (see Reference 3), since the students were not usually acquainted with the mathematics of orthogonal polynomials. This method does not allow one to

14、investigate the properties of the polynomials in any detail, however, yielding only the individual polynomials themselves.Consider the polynomials Gn(x) = dn/dxn (x2 - 1)n. The quantity to be differentiated is indeed a polynomial, of degree 2n, and consisting of only even powers. When differentiated

15、 n times, it becomes a polynomial of order n consisting of either all odd or all even powers of x, as n is odd or even. The coefficient of the highest power of x is 2n(2n-1)(2n-2).(n+1), and the first two polynomials are 1 and 2x. If G(x) is substituted in the recurrence relation for the Legendre po

16、lynomials, it is found to satisfy it. If we divide G(x) by the constant 2nn!, then the first two polynomials are 1 and x. Therefore, Pn(x) = (1/2nn!) dn/dxn (x2 - 1)n. This is called Rodriguess formula; similar formulas exist for other orthogonal polynomials.The great advantage of Rodrigues formula

17、is its form as an nth derivative. This means that in an integral, it can be used repeatedly in an integration by parts to evaluate the integral. The orthogonality of the Legendre polynomials follows very quickly when Rodrigues formula is used. There is a Rodrigues formula for many, but not all, orth

18、ogonal polynomials. It can be used to find the recurrence relation, the differential equation, and many other properties.For finding solutions to Laplaces equation in spherical coordinates, the Legendre polynomials are sufficient so long as the problem is axially symmetric, in which there is no -dep

19、endence. The more general problem requires the introduction of related functions called the associated Legendre functions that are actually built up from Jacobi polynomials, and can also be expressed in terms of derivatives of the Legendre polynomials. Physics texts generally approached the problem

20、from first principles, never mentioning Jacobi polynomials, and thereby losing valuable insight.The Jacobi polynomials P(,)n(x) are orthogonal on (-1,1) with weight function w(x) = (1 - x)(1 + x). Their Rodrigues formula is P(,)n(x) = (-1)n/2nn! (1 - x)-(1 + x)- dn/dxn (1 - x)+n(1 + x)+n. The ordina

21、ry Legendre polynomial Pn(x) = P(0,0)n(x). They satisfy the differential equation (1 - x2)P(,)n + - - ( + + 2)x P(,)n + n( + + n + 1) P(,)n = 0.In solving Laplaces equation by the method of separation of variables, one obtains for the dependence T(x), x = cos , the differential equationd/dx(1 - x2)d

22、T/dx = l(l+1) - m2/(1 - x2T = 0The substitution T(x) = (1 - x2)m/2y(x) now gives the equation(1-x2)y - 2(m + 1)xy + l(l+1) - m(m+1)y = 0,which we recognize as satisfied by the Jacobi polynomial P(m,m)l-m(x). Hence, T(x) = (1 - x2)m/2y(x) P(m,m)l-m(x). This is the associated Legendre function, often

23、denoted Pml(x) in physics texts (e.g., Reference 4), and defined there as (-1)m(1 - x2)m/2 dm/dxm Pl. The subscript is no longer the degree of the polynomial.All the above is for a positive m. Since the equation contains m2, the solution for negative m is essentially the same, except perhaps for a m

24、ultiplicative factor. This is of little consequence for the traditional applications of spherical harmonics, but is critical for quantum mechanics, where relative phases matter. The choice in physics is that P-ml(x) = (-1)m(l - m)!/(l + m)! Pml(x), where m is always positive on the right. If you wor

25、k the functions out explicitly, you will find that the functions for +m and -m are essentially the same, as might be expected, and differ at most by a factor of -1.For the same m, Pml(x) and Pml(x) are orthogonal, and the integral of the square of Pml(x) is the same as for Pl(x), multiplied by (l -

26、m)!/(l + m)!. The functions are not orthogonal for different values of m; orthogonality of the spherical harmonics in this case depends on the functions.ReferencesM. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Washington, D.C.: National Bureau of Standards, Applied Mathematics Seri

27、es 55, June 1964). Chapter 22.D. Jackson, Fourier Series and Orthogonal Functions (Mathematical Assoc. of America, Carus Mathematical Monographs No. 6, 1941). Chapter X.L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (New York: McGraw-Hill, 1935). Chapter V.J. D. Jackson, Classical El

28、ectrodynamics, 2nd . ed. (New York: McGraw-Hill, 1975), Chapter III.Gauss型積分Gauss型求積公式的構(gòu)造方法(1)求出區(qū)間a,b上權(quán)函數(shù)為W(x)的正交多項(xiàng)式pn(x) (2)求出pn(x)的n個(gè)零點(diǎn)x1 , x2 , xn 即為Gsuss點(diǎn). (3)計(jì)算積分系數(shù) 幾種Gauss型求積公式Gauss-Legendre求積公式區(qū)間-1,1上權(quán)函數(shù)W(x)=1的Gauss型求積公式，稱為Gauss-Legendre求積公式，其Gauss點(diǎn)為L(zhǎng)egendre多項(xiàng)式的零點(diǎn)。公式的Gauss點(diǎn)和求積系數(shù)可在數(shù)學(xué)用表中查到。由于因此

29、，a,b上權(quán)函數(shù)W(x)=1的Gauss型求積公式為Gauss 公式的余項(xiàng)Gauss-Laguerre求積公式區(qū)間0,)上權(quán)函數(shù)W(x)=e-x的Gauss型求積公式，稱為Gauss-Laguerre求積公式，其Gauss點(diǎn)為L(zhǎng)aguerre多項(xiàng)式的零點(diǎn)。公式的Gauss點(diǎn)和求積系數(shù)可在數(shù)學(xué)用表中查到。由，所以，對(duì)0, +)上權(quán)函數(shù)W(x)=1的積分，也可以構(gòu)造類似的Gauss-Laguerre求積公式：。Gauss-Hermite求積公式區(qū)間（-, )上權(quán)函數(shù)W(x)=e-x2的Gauss型求積公式，稱為Gauss-Hermite求積公式，其Gauss點(diǎn)為L(zhǎng)aguerre多項(xiàng)式的零點(diǎn)。公式的

30、Gauss點(diǎn)和求積系數(shù)可在數(shù)學(xué)用表中查到。Legendre多項(xiàng)式隱式表達(dá)式顯式表達(dá)式其中正交性質(zhì)：遞推關(guān)系： 高斯積分的計(jì)算過(guò)程（1）、將a,b區(qū)間積分變換到-1,1區(qū)間；（2）、求高斯點(diǎn)及權(quán)i(i=1,2,n)；（3）、代入高斯公式：，其中，最終，高斯偽譜法的軌跡優(yōu)化方法下面討論基于高斯偽譜法的質(zhì)點(diǎn)彈道優(yōu)化方法。質(zhì)點(diǎn)運(yùn)動(dòng)方程取，則考慮地球自轉(zhuǎn)，忽略地球扁率，不考慮側(cè)滑情況下的三自由度質(zhì)點(diǎn)運(yùn)動(dòng)模型可以描述為約束條件，具體介紹如下過(guò)程約束熱流、過(guò)載、動(dòng)壓以及擬平衡滑翔條件等。其中動(dòng)壓、熱流和過(guò)載必須嚴(yán)格滿足。熱流密度研究飛行器軌跡優(yōu)化時(shí)通常以駐點(diǎn)熱流作為約束條件，因?yàn)轳v點(diǎn)是飛行器加熱較為嚴(yán)重的區(qū)

31、域。其工程估算表達(dá)式為：對(duì)于高超聲速飛行器，一般取，。是飛行器頭部半徑相關(guān)的參數(shù)。按照常用的熱流密度表達(dá)式，熱流密度約束為：為飛行器頭部曲率半徑，與飛行器特性相關(guān)的參數(shù)。動(dòng)壓約束考慮到動(dòng)壓對(duì)飛行器控制系統(tǒng)的影響和側(cè)向穩(wěn)定性的要求，再入過(guò)程中動(dòng)壓滿足過(guò)載約束為了保證結(jié)構(gòu)安全，需考慮過(guò)載約束。彈箭類飛行器一般對(duì)法向過(guò)載進(jìn)行約束。而升力體飛行器其法向和軸向都可能產(chǎn)生大過(guò)載。因此對(duì)總過(guò)載進(jìn)行約束：擬平穩(wěn)滑翔條件相對(duì)于前面幾種約束，擬平穩(wěn)滑翔約束不太嚴(yán)格，是一種考慮飛行器控制能力的“軟約束”，即所謂的“no-skip”條件，保證飛行器不再跳躍出大氣層。無(wú)因次化后的表達(dá)式為：其中是考慮哥氏加速度和牽連加速

32、度的附加項(xiàng)，一般再入問(wèn)題中其取值較??；是平衡滑翔邊界對(duì)應(yīng)的傾側(cè)角?？紤]，則上述約束簡(jiǎn)化為控制變量約束終端約束條件到達(dá)指定點(diǎn)的約束，對(duì)于對(duì)敵打擊武器平臺(tái)，還附加終端航跡角和速度的約束，到達(dá)指定區(qū)域的約束定義飛行器距離目標(biāo)點(diǎn)的地面航程為，則約束條件為，定義航向角與相對(duì)目標(biāo)點(diǎn)的視線角的偏差為，為使末制導(dǎo)段能更準(zhǔn)確的到達(dá)目標(biāo)，要求到達(dá)指定區(qū)域時(shí)航向也滿足一定約束，即目標(biāo)函數(shù)可以是下面的一個(gè)或多個(gè)的加權(quán)和。吸熱最少?gòu)椀榔交陨涑套畲筌壽E優(yōu)化數(shù)學(xué)模型狀態(tài)變量為，。軌跡優(yōu)化問(wèn)題的轉(zhuǎn)化區(qū)間變換將原軌跡優(yōu)化問(wèn)題的時(shí)間區(qū)間變換到GPM求解需要的區(qū)間。因此令，優(yōu)化問(wèn)題化為狀態(tài)變量和控制變量的多項(xiàng)式近似設(shè)利用高斯偽譜法對(duì)優(yōu)化問(wèn)題進(jìn)行離散處理選取的插值點(diǎn)個(gè)數(shù)為，分別為，其中，。（）