B irkhoff系統(tǒng)的離散最優(yōu)控制及其在航天器交會(huì)對(duì)接中的應(yīng)用?

孔新雷 吳惠彬

1)(北方工業(yè)大學(xué)理學(xué)院,北京100144)2)(北京理工大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,北京100081)

B irkhoff系統(tǒng)的離散最優(yōu)控制及其在航天器交會(huì)對(duì)接中的應(yīng)用?

孔新雷1)?吳惠彬2)

1)(北方工業(yè)大學(xué)理學(xué)院,北京100144)2)(北京理工大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院,北京100081)

(2016年10月26日收到;2016年12月25日收到修改稿)

由于非線性,最優(yōu)控制問(wèn)題通常依賴于數(shù)值求解,即通過(guò)離散目標(biāo)泛函和受控運(yùn)動(dòng)方程轉(zhuǎn)化為一有限維的非線性最優(yōu)化問(wèn)題.最優(yōu)控制問(wèn)題中的受控運(yùn)動(dòng)方程在表示為受控Birkhoff方程的形式之后,可以利用受控Birkhoff方程的離散變分差分格式進(jìn)行離散.與按照傳統(tǒng)差分格式近似受控運(yùn)動(dòng)方程相比,此途徑可以誘導(dǎo)更加真實(shí)可靠的非線性最優(yōu)化問(wèn)題,進(jìn)而也會(huì)誘導(dǎo)更加精確有效的離散最優(yōu)控制.應(yīng)用于航天器交會(huì)對(duì)接問(wèn)題,該種數(shù)值求解最優(yōu)控制問(wèn)題的方法在較大時(shí)間步長(zhǎng)的情況下仍然求得了一個(gè)有效實(shí)現(xiàn)交會(huì)對(duì)接的離散最優(yōu)控制.模擬結(jié)果驗(yàn)證了該方法的有效性.

Birkhoff系統(tǒng),最優(yōu)控制,非線性規(guī)劃,交會(huì)對(duì)接

1 引言

Birkhoff系統(tǒng)是由如下形式的Birkhoff方程

所描述的一類動(dòng)力學(xué)系統(tǒng)[1],其中a(t)=(a1(t),a2(t),···,a2n(t))代表系統(tǒng)的位形空間變量,函數(shù)B(t,a)和Ri(t,a)分別稱為系統(tǒng)的Birkhoff函數(shù)和Birkhoff函數(shù)組.作為經(jīng)典HaMilton系統(tǒng)的自然推廣,Birkhoff系統(tǒng)不僅能夠涵蓋更多的實(shí)際力學(xué)系統(tǒng),而且由于其源于Pfaff-Birkhoff變分原理,因而具備優(yōu)良的變分特性.與Lagrange系統(tǒng)相比,Birkhoff系統(tǒng)在保持變分特性的同時(shí)還具備明顯的辛結(jié)構(gòu).因此,Birkhoff系統(tǒng)為變分特性和辛結(jié)構(gòu)提供了理想的共棲環(huán)境.這一共棲特性奠定了Birkhoff系統(tǒng)動(dòng)力學(xué)研究的基礎(chǔ).

基于Birkhoff系統(tǒng)的變分特性,相關(guān)研究成果涉及了動(dòng)力學(xué)系統(tǒng)的Birkhoff化[2,3]、Birkhoff系統(tǒng)的對(duì)稱性和積分方法[4?6]、Birkhoff系統(tǒng)的變分差分格式[7?9]以及進(jìn)一步向廣義Birkhoff系統(tǒng)推廣[10,11].基于Birkhoff系統(tǒng)所具有的時(shí)變辛形式,可以構(gòu)造系統(tǒng)的幾何表示和保辛算法[12?14],研究系統(tǒng)的對(duì)稱約化[15].這些研究成果有效充實(shí)了Birkhoff系統(tǒng)動(dòng)力學(xué)的理論體系,但大多只是集中于Birkhoff系統(tǒng)自身的分析、代數(shù)和幾何性質(zhì),而對(duì)Birkhoff系統(tǒng)的控制理論并沒(méi)有涉及.人們?cè)谘芯繉?shí)際物理運(yùn)動(dòng)時(shí)并不總是停留在觀測(cè)物體的現(xiàn)有運(yùn)動(dòng)上,更多時(shí)候是希望能夠影響或改變物體的運(yùn)動(dòng),這就涉及到力學(xué)系統(tǒng)的控制問(wèn)題.

本文將最優(yōu)控制問(wèn)題中的受控運(yùn)動(dòng)方程納入Birkhoff系統(tǒng)的框架下,得到Birkhoff系統(tǒng)的最優(yōu)控制問(wèn)題,并利用受控Birkhoff系統(tǒng)的離散變分差分格式離散運(yùn)動(dòng)方程,將最優(yōu)控制問(wèn)題轉(zhuǎn)化為非線性最優(yōu)化問(wèn)題,進(jìn)而通過(guò)非線性規(guī)劃方法實(shí)現(xiàn)對(duì)最優(yōu)控制問(wèn)題的有效數(shù)值求解.

2 Birkhoff系統(tǒng)的離散最優(yōu)控制

一般的最優(yōu)控制問(wèn)題可以描述為:對(duì)一個(gè)受控的動(dòng)力學(xué)系統(tǒng)或運(yùn)動(dòng)過(guò)程,從一類允許的控制方案中找出一個(gè)最優(yōu)控制方案,使系統(tǒng)的運(yùn)動(dòng)在由某個(gè)初始狀態(tài)轉(zhuǎn)移到指定目標(biāo)狀態(tài)的同時(shí),其性能指標(biāo)值為最優(yōu).借助數(shù)學(xué)語(yǔ)言,最優(yōu)控制問(wèn)題對(duì)應(yīng)如下條件極值問(wèn)題:

在約束條件

在約束條件

下,目標(biāo)泛函

關(guān)于F=(F1,···,Fj,···,F2n)取得極值,其中a0是原始系統(tǒng)經(jīng)Birkhoff化后對(duì)應(yīng)系統(tǒng)的初始狀態(tài),a1是系統(tǒng)所要達(dá)到的目標(biāo)狀態(tài),B(t,a)和Ri(t,a)分別是受控運(yùn)動(dòng)系統(tǒng)˙x=f(x,F)經(jīng)Birkhoff化后對(duì)應(yīng)的Birkhoff函數(shù)和Birkhoff函數(shù)組.

通常情況下,由于非線性,上述形式的最優(yōu)控制問(wèn)題往往依賴于數(shù)值求解而非解析求解.直接法作為數(shù)值求解方法之一,主要是通過(guò)利用傳統(tǒng)差分格式直接離散目標(biāo)泛函和受控運(yùn)動(dòng)方程,將最優(yōu)控制問(wèn)題轉(zhuǎn)化為一有限維的非線性規(guī)劃問(wèn)題進(jìn)行求解.然而,人為地采用傳統(tǒng)差分格式離散受控運(yùn)動(dòng)方程不可避免地會(huì)破壞系統(tǒng)原有的幾何結(jié)構(gòu),進(jìn)而降低所得到的非線性規(guī)劃問(wèn)題的真實(shí)性和可靠性,最終也會(huì)影響所求得的離散最優(yōu)控制的精確性和有效性.因此,為了更加精確有效地?cái)?shù)值求解最優(yōu)控制問(wèn)題,受控運(yùn)動(dòng)方程的離散方式需要加以改進(jìn).

對(duì)于受控Birkhoff系統(tǒng)而言,離散變分差分格式被驗(yàn)證是一種更加精確有效的數(shù)值算法.與傳統(tǒng)差分格式相比,受控Birkhoff系統(tǒng)的離散變分差分格式來(lái)源于離散后的Pfaff-Birkhoff-D’A lembert原理,在誘導(dǎo)過(guò)程中兼顧了系統(tǒng)原有的變分特性和幾何結(jié)構(gòu),因而在精度、穩(wěn)定性、長(zhǎng)時(shí)間能量跟蹤等方面呈現(xiàn)出明顯的計(jì)算優(yōu)越性[16].鑒于此,在數(shù)值求解Birkhoff系統(tǒng)的最優(yōu)控制問(wèn)題時(shí),受控Birkhoff系統(tǒng)的離散變分差分格式是離散受控運(yùn)動(dòng)方程更加理想的標(biāo)尺.依據(jù)上述思路,可按照如下方式將Birkhoff系統(tǒng)的最優(yōu)控制問(wèn)題直接離散.

首先,將所考慮的時(shí)間區(qū)間[0,T]等分為N份,令時(shí)間步長(zhǎng)τ=T/N,進(jìn)一步在離散時(shí)間節(jié)點(diǎn)處利用和分別近似和那么采用矩形格式或梯形格式等傳統(tǒng)積分逼近格式近似積分可得

對(duì)應(yīng)地,目標(biāo)泛函就可以離散為

其次,受控運(yùn)動(dòng)方程采用受控Birkhoff系統(tǒng)的離散變分差分格式進(jìn)行離散.在連續(xù)情形下,Pfaff-Birkhoff-D’A lembert原理,即

誘導(dǎo)了受控Birkhoff方程

對(duì)照連續(xù)情形,直接離散后的Pfaff-Birkhoff-D’A lembert原理,即

對(duì)應(yīng)地誘導(dǎo)了離散受控Birkhoff方程

最后,對(duì)照連續(xù)與離散情形下的Pfaff-Birkhoff-D’A lembert原理,令端點(diǎn)處的變分項(xiàng)對(duì)應(yīng)相等,就可以自然合理地離散最優(yōu)控制問(wèn)題中的邊值條件[17].

經(jīng)過(guò)上述離散過(guò)程之后,Birkhoff系統(tǒng)的最優(yōu)控制問(wèn)題就轉(zhuǎn)化為如下有限維的非線性最優(yōu)化問(wèn)題:

在約束條件

下,離散目標(biāo)泛函

對(duì)于上述離散所得的最優(yōu)化問(wèn)題,可以借助已有成熟的非線性規(guī)劃方法直接求解,諸如序列二次規(guī)劃[18]等.由于上述最優(yōu)控制問(wèn)題的離散過(guò)程考慮到了受控運(yùn)動(dòng)方程的內(nèi)在幾何結(jié)構(gòu),離散結(jié)果相對(duì)更加真實(shí),因而最終求得的離散最優(yōu)控制也相對(duì)更加精確.另外,在保證離散劃分結(jié)點(diǎn)充分細(xì)密的情況下,所求得的離散最優(yōu)控制可以實(shí)現(xiàn)預(yù)期控制目的[17].

3 航天器交會(huì)對(duì)接過(guò)程的離散最優(yōu)控制

真實(shí)的航天器交會(huì)對(duì)接過(guò)程分為四個(gè)階段:地面導(dǎo)引、自動(dòng)尋的、最后接近和停靠、對(duì)接合攏,每一個(gè)階段實(shí)際上都涉及到深刻的基礎(chǔ)理論和復(fù)雜的尖端技術(shù).本文僅考慮一個(gè)簡(jiǎn)化后的模型:假設(shè)目標(biāo)航天器在近圓軌道上做無(wú)動(dòng)力慣性飛行,而追蹤航天器在自身推力和地球引力的作用下,經(jīng)過(guò)變軌機(jī)動(dòng)達(dá)到與目標(biāo)航天器一致的運(yùn)動(dòng)狀態(tài).在這種假設(shè)下,兩個(gè)航天器的交會(huì)對(duì)接過(guò)程就轉(zhuǎn)化為追蹤航天器的變軌過(guò)程,如圖1所示.

圖1 航天器交會(huì)對(duì)接示意圖Fig.1.The sketch of the rendezvous and docking of spacecrafts.

根據(jù)牛頓第二定律和動(dòng)量矩定理,追蹤航天器在變軌過(guò)程中的運(yùn)動(dòng)方程為

其中(r,θ)是極坐標(biāo)系下追蹤航天器的位形空間坐標(biāo),u是追蹤航天器的自身推力,m是追蹤航天器的質(zhì)量,M代表地球質(zhì)量,G是萬(wàn)有引力常數(shù).對(duì)應(yīng)地,航天器交會(huì)對(duì)接過(guò)程的最優(yōu)控制問(wèn)題,即追蹤航天器變軌過(guò)程的最優(yōu)控制問(wèn)題為:

在約束條件

在受控運(yùn)動(dòng)方程中,令

則方程可以轉(zhuǎn)化為一階形式

進(jìn)一步,取Birkhoff函數(shù)組、Birkhoff函數(shù)和廣義力分別為

則一階常微分方程組等價(jià)于受控Birkhoff方程

在將受控運(yùn)動(dòng)方程重新表示為受控Birkhoff方程之后,就可以利用受控Birkhoff系統(tǒng)的離散變分差分格式進(jìn)行離散,從而實(shí)現(xiàn)更加精確有效的數(shù)值求解最優(yōu)控制問(wèn)題.離散后可得非線性最優(yōu)化問(wèn)題.

在約束條件

除地球質(zhì)量M=6×1024和萬(wàn)有引力常數(shù)G=6.67×10?11之外,參考神州十號(hào)飛船與天宮一號(hào)對(duì)接過(guò)程的有關(guān)數(shù)據(jù),人為地取定參數(shù):r0=(6371+330)×103,r1=(6371+350)×103,θ0=0,θ1=3π/2,T=2500,N=500,τ=5,m=8×103.求解上述非線性最優(yōu)化問(wèn)題可得離散最優(yōu)控制力如圖2所示.在所求得的離散最優(yōu)控制力的作用下,追蹤航天器的數(shù)值運(yùn)行軌跡如圖3所示.

受限于非線性規(guī)劃問(wèn)題的維數(shù),整個(gè)最優(yōu)控制過(guò)程被劃分為500段,因此,時(shí)間步長(zhǎng)τ=T/N=5.顯然,在這種時(shí)間步長(zhǎng)尺度下,經(jīng)離散所得到的非線性規(guī)劃問(wèn)題是對(duì)原始最優(yōu)控制問(wèn)題一種比較粗糙的近似,故而求得的離散最優(yōu)控制曲線并不平滑.然而,在該離散最優(yōu)控制力的作用下,追蹤航天器的數(shù)值運(yùn)動(dòng)軌跡卻近乎光滑,同時(shí)也符合預(yù)期.這更加說(shuō)明了本文所提出的Birkhoff系統(tǒng)的離散最優(yōu)控制方法在數(shù)值求解最優(yōu)控制問(wèn)題時(shí)的精確性和有效性.

圖2 離散最優(yōu)控制力{u}=0Fig.2.D iscrete op tiMal control{u}=0.

圖3 追蹤航天器的數(shù)值運(yùn)行軌跡Fig.3.The nuMerical tra jectory of the tracking spacecraft.

4 結(jié)論

在Birkhoff系統(tǒng)動(dòng)力學(xué)的框架下,通過(guò)將最優(yōu)控制問(wèn)題中的受控運(yùn)動(dòng)方程重新表示為受控Birkhoff方程,并進(jìn)一步利用受控Birkhoff系統(tǒng)的離散變分差分格式進(jìn)行離散,最優(yōu)控制問(wèn)題被轉(zhuǎn)化為一個(gè)有限維的非線性最優(yōu)化問(wèn)題.這一數(shù)值求解最優(yōu)控制問(wèn)題的方法在本文中被稱為Birkhoff系統(tǒng)的離散最優(yōu)控制方法.適用于航天器交會(huì)對(duì)接問(wèn)題,該方法在較大時(shí)間步長(zhǎng)的情況下依然求得了一個(gè)有效實(shí)現(xiàn)交會(huì)對(duì)接的離散最優(yōu)控制策略,其可靠性和有效性得到了驗(yàn)證.

限于最優(yōu)化問(wèn)題的維數(shù),航天器交會(huì)對(duì)接過(guò)程僅被劃分為500段.直觀上,較密離散劃分下得到的最優(yōu)化問(wèn)題要比相對(duì)較疏離散劃分下的最優(yōu)化問(wèn)題更能真實(shí)地近似原始最優(yōu)控制問(wèn)題,相應(yīng)地也就能夠誘導(dǎo)更加精確的離散最優(yōu)控制.因此,加細(xì)節(jié)點(diǎn)劃分是提高所求離散最優(yōu)控制精確性的一個(gè)途徑.然而,伴隨劃分節(jié)點(diǎn)的加細(xì)變密,最優(yōu)化問(wèn)題的維數(shù)也相應(yīng)增加,再加之非線性,求解最優(yōu)化問(wèn)題也變得愈加復(fù)雜.因此,如何在保證計(jì)算精度的前提下,有效求解高維非線性最優(yōu)化問(wèn)題就成為一個(gè)亟待解決的問(wèn)題.針對(duì)該問(wèn)題,可行的研究方案之一是利用Birkhoff系統(tǒng)的約化理論對(duì)最優(yōu)化問(wèn)題進(jìn)行約化降維[19,20].這種基于系統(tǒng)對(duì)稱性實(shí)現(xiàn)的約化,與通過(guò)直接減少離散節(jié)點(diǎn)數(shù)目實(shí)現(xiàn)的降維不同,理論上它不會(huì)降低整套方法的數(shù)值求解精度,伴隨先約化后重構(gòu)的過(guò)程實(shí)現(xiàn)最優(yōu)化問(wèn)題的有效求解.

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(Received 26 October 2016;revised Manuscrip t received 25 DeceMber 2016)

PACS:45.20.Jj,02.40.Yy,02.60.CbDOI:10.7498/aps.66.084501

*Pro ject supported by the National Natu ral Science Foundation of China(G rant Nos.11602002,11672032),the Ou tstand ing Talents PrograMof Beijing(Grant No.2015000020124G 025),and the Excellent Young Teachers PrograMof North China University of Technology(G rant No.XN 072-041).

?Corresponding author.E-Mail:kongxin lei@ncut.edu.cn

D iscrete op tiMal control for B irkhoffi an systeMs and its app lication to rendezvous and docking of spacecrafts?

Kong Xin-Lei1)?Wu Hui-Bin2)

1)(College of Science,North China University of Technology,Beijing 100144,China)2)(School ofMatheMatics and Statistics,Beijing Institu te of Technology,Beijing 100081,China)

In general,optimal control prob leMs rely on numerically rather than analytically solving methods,due to their nonlinearities.The direct Method,one of the nuMerically solving Methods,isMainly to transforMthe optimal control probleMinto a nonlinear optiMization probleMw ith finite diMensions,via discretizing the ob jective functional and the forced dynaMical equations directly.However,in the procedure of the directmethod,the classical discretizations of the forced equations w ill reduce or aff ect the accuracy of the resulting optiMization probleMas well as the discrete optiMal control.In view of this fact,More accurate and effi cient nuMerical algorithMs shou ld be eMp loyed to approxiMate the forced dynaMical equations.As verified,the discrete variational diff erence schemes for forced Birkhoffi an systeMsexhibit excellent numerical behaviors in terMs of high accuracy,long-tiMe stability and p recise energy p rediction.Thus,the forced dynaMical equations in optimal control prob leMs,after being represented as forced Birkhoffi an equations,can be discretized according to the discrete variational diff erence scheMes for forced Birkhoffi an systeMs.CoMpared w ith the Method of eMp loying traditional diff erence scheMes to discretize the forced dynaMical equations,thisway yields faithful nonlinear optiMization p robleMs and consequently gives accurate and effi cient discrete op timal control.Subsequently,in the paper we are to app ly the p roposed Method of nuMerically solving op tiMal control probleMs to the rendezvous and docking prob leMof spacecrafts.First,weMake a reasonable siMp lifi cation,i.e.,the rendezvous and docking process of two spacecrafts is reduced to the p robleMof op timally transferring the chaser spacecraft w ith a continuously acting force froMone circu lar orbit around the Earth to another one.During this transfer,the goal is toMiniMize the control eff ort.Second,the dynaMical equations of the chaser spacecraft are represented as the forMof the forced Birkhoffi an equation.Then in this case,the discrete variational diff erence scheme for forced Birkhoffi an systeMcan be eMp loyed to discretize the chaser spacecraft’s equations ofMotion.W ith further discretizing the controleff ort and the boundary conditions,the resu lting nonlinear optiMization probleMisobtained.Finally,theoptiMization prob leMis solved directly by thenon linear programMingMethod and then the discrete op tiMal control is achieved.The obtained optiMal control is effi cient enough to realize the rendezvous and docking p rocess,even though it is only an app roxiMation of the continuous one.Simu lation resu lts fu lly verify the effi ciency of the proposed method for numerically solving optimal control p robleMs,if the fact that the tiMe step is chosen to be very large to liMit the diMension of the optiMization prob leMis noted.

Birkhoffi an system,op timal control,nonlinear programMing,rendezvous and docking

10.7498/aps.66.084501

?國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):11602002,11672032)、北京市優(yōu)秀人才培養(yǎng)資助(青年骨干個(gè)人)(批準(zhǔn)號(hào):2015000020124G 025)和北方工業(yè)大學(xué)優(yōu)秀青年教師培養(yǎng)計(jì)劃(批準(zhǔn)號(hào):XN 072-041)資助的課題.

?通信作者.E-Mail:kongxin lei@ncut.edu.cn

?2017中國(guó)物理學(xué)會(huì)C h inese P hysica l Society

http://w u lixb.iphy.ac.cn