錢佳敏 陳林聰 陳虹霖

(華僑大學(xué)土木工程學(xué)院,廈門 361021)

多自由度擬不可積哈密頓系統(tǒng)的隨機(jī)分?jǐn)?shù)階最優(yōu)控制*

錢佳敏 陳林聰?陳虹霖

(華僑大學(xué)土木工程學(xué)院,廈門 361021)

推廣了適用于分?jǐn)?shù)階系統(tǒng)控制的隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略,提出了高斯白噪聲激勵(lì)下多自由度擬不可積哈密頓系統(tǒng)以響應(yīng)最小化為目標(biāo)的隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略.首先,應(yīng)用擬不可積哈密頓系統(tǒng)隨機(jī)平均法,將受控系統(tǒng)簡(jiǎn)化為關(guān)于能量的部分平均伊藤方程.然后,將控制性能指標(biāo)中關(guān)于控制力的部分表示為分?jǐn)?shù)階形式,結(jié)合隨機(jī)動(dòng)態(tài)規(guī)劃原理,建立并求解部分平均系統(tǒng)的無(wú)界遍歷控制的隨機(jī)動(dòng)態(tài)規(guī)劃方程,獲得了隨機(jī)分?jǐn)?shù)階最優(yōu)控制律.最后,采用一個(gè)算例驗(yàn)證了隨機(jī)分?jǐn)?shù)階控制策略的控制效果和控制效率.研究表明,隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略對(duì)傳統(tǒng)的整數(shù)階隨機(jī)動(dòng)力學(xué)系統(tǒng)同樣適用,能比傳統(tǒng)的整數(shù)階控制策略取得更好的控制效果.另外,隨著激勵(lì)強(qiáng)度增加,整數(shù)階控制策略的控制效率顯著降低;而分?jǐn)?shù)階控制策略的控制效率雖比整數(shù)階控制策略的控制效率略低,但隨著激勵(lì)強(qiáng)度的增加,分?jǐn)?shù)階控制策略的控制效率緩慢上升并趨于平穩(wěn), 可以有效地緩解控制效率與控制效果之間的矛盾.

隨機(jī)分?jǐn)?shù)階最優(yōu)控制, 隨機(jī)平均法, 隨機(jī)動(dòng)態(tài)規(guī)劃原理, 多自由度系統(tǒng), 擬不可積哈密頓系統(tǒng)

引言

隨機(jī)最優(yōu)控制具有廣泛的應(yīng)用前景,亦是相當(dāng)具有挑戰(zhàn)性的課題.經(jīng)過(guò)幾十年的發(fā)展,隨機(jī)最優(yōu)控制已取得了長(zhǎng)足的發(fā)展,出現(xiàn)了許多重要的研究成果,如文獻(xiàn)[1-4].但在工程中,應(yīng)用最廣泛的隨機(jī)最優(yōu)控制依然是基于線性系統(tǒng)模型的LGQ控制或最優(yōu)Bang-bang控制.因此,越來(lái)越多的學(xué)者開始研究非線性系統(tǒng)的隨機(jī)最優(yōu)控制.如,Beaman[5],Yohsida[6],以及Young和Chang[7]用統(tǒng)計(jì)線性化法將非線性隨機(jī)系統(tǒng)轉(zhuǎn)化為擬線性系統(tǒng), 再應(yīng)用LGQ控制策略;Crespo和Sun[8]用廣義胞映射法求解受控非線性隨機(jī)系統(tǒng)的動(dòng)態(tài)規(guī)劃方程;Crespo和Sun[9]還提出了另一種非線性控制方法,即無(wú)論系統(tǒng)初始條件如何,系統(tǒng)響應(yīng)最終都可達(dá)到一個(gè)預(yù)先設(shè)計(jì)好的概率密度函數(shù).Dimentberg等[10]提出線性隨機(jī)振動(dòng)系統(tǒng)最優(yōu)有界控制問(wèn)題的動(dòng)態(tài)規(guī)劃方程的混合解法;李杰等[11]將廣義概率密度演化方程用于隨機(jī)結(jié)構(gòu)系統(tǒng)的最優(yōu)控制;朱位秋及其合作者[12,13]提出了基于隨機(jī)平均法和隨機(jī)動(dòng)態(tài)規(guī)劃原理的擬哈密頓系統(tǒng)的非線性隨機(jī)最優(yōu)控制策略.然而,控制效率和控制效果這一對(duì)矛盾仍未得到很好解決.

近年來(lái),越來(lái)越多的學(xué)者開始從事分?jǐn)?shù)階最優(yōu)控制問(wèn)題的研究.如,Agrawal[14]給出了分?jǐn)?shù)階最優(yōu)控制問(wèn)題的一般求解方法.曾慶山和楊增芳[15]將求解整數(shù)階系統(tǒng)最優(yōu)控制的方法和步驟拓展到分?jǐn)?shù)階系統(tǒng)的最優(yōu)控制,獲得指標(biāo)函數(shù)終端自由和受函數(shù)約束的分?jǐn)?shù)階系統(tǒng)的最優(yōu)控制.紀(jì)增浩[16]采用分?jǐn)?shù)階變分法推導(dǎo)出分?jǐn)?shù)階最優(yōu)控制問(wèn)題的必要條件,然后采用多段線性插值法來(lái)求解,最終得到最優(yōu)控制律.分?jǐn)?shù)階最優(yōu)控制的意義是對(duì)整數(shù)階最優(yōu)控制理論的進(jìn)一步推廣,可獲得具有更佳的動(dòng)態(tài)性能和魯棒性的控制結(jié)果.然而,分?jǐn)?shù)階最優(yōu)控制問(wèn)題研究還處于起步階段,許多方面都需要進(jìn)一步完善,尤其是針對(duì)隨機(jī)激勵(lì)下非線性動(dòng)力學(xué)系統(tǒng).

最近,Hu及其合作者[17]提出了一種隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略,對(duì)隨機(jī)激勵(lì)下具有分?jǐn)?shù)階導(dǎo)數(shù)型阻尼的擬可積哈密頓系統(tǒng)進(jìn)行控制,取得了較好的控制效率和控制效果.本文將進(jìn)一步推廣隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略,對(duì)多自由度非線性擬不可積哈密頓系統(tǒng)的隨機(jī)響應(yīng)進(jìn)行控制.研究表明,分?jǐn)?shù)階隨機(jī)控制策略對(duì)傳統(tǒng)的整數(shù)階隨機(jī)動(dòng)力學(xué)系統(tǒng)仍然適用,能比整數(shù)階控制策略取得更佳的控制效果,同時(shí)還能有效地緩解控制效率與控制效果之間的矛盾.

1 隨機(jī)平均法

考慮受控?cái)M哈密頓系統(tǒng),其運(yùn)動(dòng)方程型為：

Qi(0)=Qi0,Pi(0)=Pi0

i,j=1,2,…n,k=1,2,…m

(1)

式中Qi,Pi分別是廣義位移與廣義動(dòng)量;H′=H′(Q,P)是相應(yīng)哈密頓系統(tǒng)的哈密頓函數(shù);c′=c′(Q,P)是擬線性阻尼系數(shù);ε1/2fik=ε1/2fik(Q,P)為激勵(lì)的幅值;Wk(t)是相關(guān)函數(shù)為2Dklδ(t)的高斯白噪聲;ui=ui(Q,P)為反饋控制力.經(jīng)Wong-Zakai修正項(xiàng)修正后,(1)式可以轉(zhuǎn)化為如下的It隨機(jī)微分方程：

(2)

式中H與cij分別為修正后的哈密頓函數(shù)與擬線性阻尼系數(shù);Bk(t)為Wiener過(guò)程.

設(shè)與系統(tǒng)(2)相應(yīng)的哈密頓系統(tǒng)為不可積,即H是與(2)相應(yīng)的哈密頓系統(tǒng)的唯一獨(dú)立對(duì)合的首次積分.應(yīng)用擬不可積哈密頓系統(tǒng)隨機(jī)平均法,可由(2)式導(dǎo)出如下關(guān)于H的部分平均It隨機(jī)微分方程：

(3)

·dq1dq2…dqndp2…dpn

·dq1dq2…dqndp2…dpn

p3,…pn)≤H}

(4)

2 隨機(jī)動(dòng)態(tài)規(guī)劃方程

考慮系統(tǒng)(3)在半無(wú)限長(zhǎng)時(shí)間區(qū)間上的控制.設(shè)性能指標(biāo)為:

(5)

引入值函數(shù):

(6)

應(yīng)用隨機(jī)動(dòng)態(tài)規(guī)劃原理,可以導(dǎo)出如下形式的動(dòng)態(tài)規(guī)劃方程:

(7)

其中,

(8)

表示最優(yōu)平均成本.式(7)右邊取極小值的必要條件為:

(9)

設(shè)

f(H,)=f1(H)+

(10)

其中R為正定對(duì)稱常數(shù)矩陣,則最優(yōu)控制律為:

(11)

(12)

求解上式,可獲得dV/dH.然后,再將dV/dH代入方程(11)可得最優(yōu)控制律的表達(dá)式.最后,將最優(yōu)控制律,代入部分平均It隨機(jī)微分方程(3)以取代ui,完成平均,得到最優(yōu)控制系統(tǒng)的平均It隨機(jī)微分方程:

(13)

其中,

(14)

求解與方程(13)相應(yīng)的FPK方程,可得最優(yōu)控制或未控系統(tǒng)的響應(yīng)統(tǒng)計(jì)量.為評(píng)價(jià)分?jǐn)?shù)階最優(yōu)控制策略的效果與效率,引入如下兩個(gè)準(zhǔn)則：

(15)

3 算例

考慮一個(gè)兩自由度受控的擬不可積哈密頓系統(tǒng),其運(yùn)動(dòng)方程為:

(16)

其中,w1,w2與a為正常數(shù);ξi(i=1,2)為強(qiáng)度2Di的高斯白噪聲;ui(i=1,2)為反饋控制力.

與系統(tǒng)(16)相應(yīng)的哈密頓函數(shù)為:

(17)

(18)

式中,

(19)

取成本函數(shù)形為式(10),其中u=[u1,u2]T,

R=diag(R1,R2)及

f1(H)=s0+s1H+s2H2+s3H3

(20)

運(yùn)用隨機(jī)動(dòng)態(tài)規(guī)劃原理,建立并求解形如式(7)的隨機(jī)動(dòng)態(tài)規(guī)劃方程,得到最優(yōu)控制律為：

(21)

(22)

pcon(H)=

(23)

其中C1為歸一化系數(shù).令u=0,求解與之相應(yīng)的平穩(wěn)FPK方程可得未控系統(tǒng)的平穩(wěn)概率密度:puncon(H)=

(24)

最優(yōu)控制和未控系統(tǒng)的廣義位移與廣義速度的聯(lián)合平穩(wěn)概率密度為:

(25)

(26)

(27)

以及最優(yōu)控制力的均方值

(28)最后,按方程(15)計(jì)算控制效果K1和控制效率K2.對(duì)如下系統(tǒng)參數(shù)：b1=b2=0.05, a=2.0, w1=1.0,w2=1.414, s1=0.05,s2=0.0, s3=1.5,dV(0)/dH=13.0,R1=R2=R=2.1,圖1與圖2給出了上述最優(yōu)控制下系統(tǒng)第一個(gè)自由度的若干個(gè)數(shù)值結(jié)果.

圖1 不同分?jǐn)?shù)階數(shù)時(shí)控制效果與激勵(lì)強(qiáng)度變化的關(guān)系Fig.1 Relationship between control effect and excitation intensity under different fractional orders

圖2 不同分?jǐn)?shù)階數(shù)時(shí)控制效率與激勵(lì)強(qiáng)度變化的關(guān)系Fig.2 Relationship between control efficiency and excitation intensity under different fractional orders

由圖1可知,本文提出的分?jǐn)?shù)階最優(yōu)控制策略在控制效果上比傳統(tǒng)的整數(shù)階要好,并且控制效果隨著分?jǐn)?shù)階階數(shù)的增加而提高.另外,隨著激勵(lì)強(qiáng)度增加,控制效果均變差.圖2給出了不同分?jǐn)?shù)階階數(shù)下隨激勵(lì)強(qiáng)度變化的最優(yōu)控制效率.由圖可知,當(dāng)a=1.0,即分?jǐn)?shù)階最優(yōu)控制策略退為整數(shù)階最優(yōu)控制策略,此時(shí)控制效率隨著激勵(lì)強(qiáng)度的增加而降低.然而,當(dāng)分?jǐn)?shù)階數(shù)a=1.5或2.5時(shí),隨著激勵(lì)強(qiáng)度的增加,分?jǐn)?shù)階控制策略的控制效率開始緩慢提高,并趨于平穩(wěn).因此,為了緩解控制效率與控制效果這一對(duì)矛盾,采用隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略取代傳統(tǒng)的整數(shù)階控制策略是一種可行的方案.

4 小結(jié)

本文進(jìn)一步推廣了適用于分?jǐn)?shù)階系統(tǒng)控制的隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略,提出了多自由度擬不可積哈密頓系統(tǒng)以響應(yīng)最小化為目標(biāo)的隨機(jī)分?jǐn)?shù)階最優(yōu)控制策略.數(shù)值算例表明,控制效果隨著分?jǐn)?shù)階階數(shù)的增加而提高;整數(shù)階控制策略的控制效果隨著激勵(lì)強(qiáng)度增加而顯著降低;分?jǐn)?shù)階控制效率比整數(shù)階控制效率略低,但是隨著激勵(lì)強(qiáng)度的增加而緩慢上升并趨于平緩,可為解決控制效果與控制效率是一對(duì)矛盾提供可行方案.

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*The project supported by the Natural Science Foundation of China through the Grants (11672111), by the Research Award Fund for Outstanding Young Researcher in Higher Education Institutions of Fujian Province and by the Research Fund for Excellent Young Scientific and Technological Project of Huaqiao University under the Grant (ZQN-YX307)

? Corresponding author E-mail:lincongchen@hqu.edu.cn

17 March 2017,revised 18 April 2017.

STOCHASCTICFRACTIONAL OPTIMAL CONTROL OF MDOF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS*

Qian Jiamin Chen Lincong?Chen Honglin

(CollegeofCivilEngineering,HuaqiaoUniversity,Xiamen361021,China)

In this paper, a fractional optimal control strategy suitable for the fractional order dynamical system is extended to the classical integral dynamical system. A stochastic fractional optimal control strategy is proposed to minimize the response of the multi-degree-of-freedom (MDOF) quasi-non-integrable Hamiltonian systems under Gaussian white noise excitations. First, the controlled system is reduced to a partially averaged Itequation for the energy process by applying the stochastic averaging method of quasi-non-integrable Hamiltonian systems. The control force part in the control performance index is then formulated as the form of fractional-order. By combining with the stochastic dynamics programming principle, the dynamical programming equation for the ergodic control of the partially averaged system is established and then solved to yield the fractional optimal control law. Finally, an example is given to illustrate the effectiveness and efficiency of the proposed control design procedure. The numerical results indicate that the stochastic fractional optimal control strategy can be suitable for the classical integral order stochastic dynamical systems. The proposed control strategy can obtain the better control effectiveness than that of the classical integral-order optimal control strategy. Furthermore, the efficiency of integral-order control strategy becomes worse remarkably as the excitation intensity increases, while the efficiency of fractional order control strategy is a bit lower than that of integral-order control strategy, but it slowly rises and then tends to stable. The proposed stochastic fractional order optimal control strategy effectively mitigates the contradiction between the control effectiveness and control efficiency.

stochastic fractional optimal control, stochastic averaging method, stochastic dynamics programming principle, multi-degree-of-freedom system, quasi non-integrable Hamiltonian systems

*國(guó)家自然科學(xué)基金資助項(xiàng)目(11672111)、福建省高校青年杰出項(xiàng)目、華僑大學(xué)優(yōu)秀青年科技創(chuàng)新人才(ZQN-YX307)

10.6052/1672-6553-2017-024

2017-03-17收到第1稿,2017-4-18收到修改稿.

? 通訊作者 E-mail: lincongchen@hqu.edu.cn