宋娜 洪小敏 周川

(南京理工大學(xué)自動(dòng)化學(xué)院, 南京 210094)

通信受限的網(wǎng)絡(luò)化多智能體系統(tǒng)編隊(duì)控制*

宋娜 洪小敏 周川?

(南京理工大學(xué)自動(dòng)化學(xué)院, 南京 210094)

主要研究了具有通信受限及不確定性的無(wú)線(xiàn)網(wǎng)絡(luò)下的二階多智能體系統(tǒng)的編隊(duì)控制問(wèn)題.針對(duì)無(wú)線(xiàn)網(wǎng)絡(luò)的介質(zhì)訪(fǎng)問(wèn)約束,采用二進(jìn)制序列設(shè)計(jì)智能體節(jié)點(diǎn)調(diào)度協(xié)議,使得滿(mǎn)足約束條件的智能體節(jié)點(diǎn)經(jīng)網(wǎng)絡(luò)傳輸其采樣信息；另一方面,為建立更符合實(shí)際的通信信道模型,區(qū)別于傳統(tǒng)的固定連接權(quán)重拓?fù)鋱D,考慮到無(wú)線(xiàn)網(wǎng)絡(luò)的不確定性及拓?fù)溥B接權(quán)重與智能體節(jié)點(diǎn)之間距離的關(guān)系,利用無(wú)線(xiàn)信道的度量指標(biāo)構(gòu)建了多智能體通信拓?fù)涞男履Ｐ?定義信息更新誤差及編隊(duì)誤差,將二階多智能體系統(tǒng)模型轉(zhuǎn)化為閉環(huán)延時(shí)系統(tǒng)模型.然后,構(gòu)造Lyapunov-Krasovskii函數(shù)并利用LMI方法設(shè)計(jì)了編隊(duì)控制協(xié)議.最后,通過(guò)仿真驗(yàn)證了所提方法的有效性.

多智能體系統(tǒng), 編隊(duì)控制, 無(wú)線(xiàn)網(wǎng)絡(luò), 通信受限, 不確定性

引言

編隊(duì)控制問(wèn)題已成為多智能體系統(tǒng)研究的一個(gè)熱點(diǎn).通常,多智能體系統(tǒng)編隊(duì)控制是指設(shè)計(jì)合適的控制協(xié)議,使得多個(gè)網(wǎng)絡(luò)化的智能體節(jié)點(diǎn)達(dá)到期望的速度及位置(或相對(duì)位置),并保持隊(duì)形運(yùn)動(dòng).

目前,已有很多文獻(xiàn)對(duì)多智能體系統(tǒng)的編隊(duì)控制進(jìn)行了研究[1-8,23].其中,文獻(xiàn)[2]針對(duì)一階多智能體系統(tǒng),通過(guò)相對(duì)位置的測(cè)量估計(jì)分布式位置設(shè)計(jì)了編隊(duì)控制策略,并將提出的策略應(yīng)用到輪式機(jī)器人編隊(duì)控制中.文獻(xiàn)[6]不僅針對(duì)具有變化有向通信拓?fù)涞囊苿?dòng)自主多智能體系統(tǒng)的分布式非線(xiàn)性控制問(wèn)題,提出了分布式非線(xiàn)性控制策略,使得二階多智能體系統(tǒng)的控制輸出漸近收斂到一致；且通過(guò)動(dòng)態(tài)反饋線(xiàn)性化將具有非完整約束的輪式機(jī)器人模型轉(zhuǎn)換為兩個(gè)二階積分模型,研究了輪式機(jī)器人的編隊(duì)控制問(wèn)題.可見(jiàn),以上文獻(xiàn)均假設(shè)網(wǎng)絡(luò)信道容量是理想無(wú)約束的.然而,多智能體系統(tǒng)作為一種特殊的網(wǎng)絡(luò)控制系統(tǒng),它包含多個(gè)智能體節(jié)點(diǎn),且通過(guò)網(wǎng)絡(luò)傳輸其采樣信息.在實(shí)際應(yīng)用中,通信網(wǎng)絡(luò)(特別是無(wú)線(xiàn)網(wǎng)絡(luò))往往存在介質(zhì)訪(fǎng)問(wèn)約束等問(wèn)題.介質(zhì)訪(fǎng)問(wèn)約束是指任一時(shí)刻僅有有限個(gè)智能體節(jié)點(diǎn)的信息能通過(guò)網(wǎng)絡(luò)傳輸.這將導(dǎo)致智能體節(jié)點(diǎn)的控制器不能獲得所有傳感器的采樣信息,從而影響系統(tǒng)的控制性能甚至使系統(tǒng)失穩(wěn).針對(duì)具有介質(zhì)訪(fǎng)問(wèn)約束的網(wǎng)絡(luò)控制系統(tǒng)的控制,主要是設(shè)計(jì)調(diào)度策略決定網(wǎng)絡(luò)節(jié)點(diǎn)信息的發(fā)送順序,如常見(jiàn)的RR(Round-Robin,輪詢(xún)調(diào)度)、TOD(Try Once Discard,嘗試一次丟棄)、EDF(Earliest Deadline First,最小截止期優(yōu)先)等.此外,針對(duì)由多個(gè)被控對(duì)象組成的具有介質(zhì)訪(fǎng)問(wèn)約束及位速率約束的網(wǎng)絡(luò)控制系統(tǒng),文獻(xiàn)[9]基于平均駐留時(shí)間方法調(diào)度系統(tǒng)節(jié)點(diǎn),并采用線(xiàn)性規(guī)劃優(yōu)化算法在線(xiàn)分配網(wǎng)絡(luò)帶寬.因此,介質(zhì)訪(fǎng)問(wèn)約束下的多智能體系統(tǒng)編隊(duì)控制的研究具有重要意義,而目前已有的研究結(jié)果中未涉及介質(zhì)訪(fǎng)問(wèn)約束下的多智能體編隊(duì)控制.

在多智能體系統(tǒng)編隊(duì)控制的研究中,通常采用拓?fù)鋱D表示智能體節(jié)點(diǎn)與其鄰接節(jié)點(diǎn)之間的信息傳輸通道.現(xiàn)有文獻(xiàn)中,大部分的研究均假設(shè)拓?fù)鋱D的交互權(quán)重固定不變(固定拓?fù)淝闆r下)或者根據(jù)某種規(guī)則在有限集里變化(切換拓?fù)淝闆r下)[10-13].文獻(xiàn)[10]針對(duì)具有時(shí)變延時(shí)的非線(xiàn)性多智能體系統(tǒng),采用人工勢(shì)能函數(shù)方法,研究了根據(jù)馬爾科夫隨機(jī)過(guò)程切換的通信拓?fù)湎碌姆植际骄庩?duì)問(wèn)題.文獻(xiàn)[12]采用多領(lǐng)導(dǎo)的方式研究了有向通信拓?fù)湎戮哂袝r(shí)變延時(shí)的二階多智能體系統(tǒng)的時(shí)變編隊(duì)以及時(shí)不變編隊(duì)控制問(wèn)題.然而,在實(shí)際的網(wǎng)絡(luò)傳輸中,隨著接收器和發(fā)送器之間的有效距離變化,通信拓?fù)涞慕换?quán)重往往是動(dòng)態(tài)變化的.如文獻(xiàn)[14]利用相應(yīng)的調(diào)度函數(shù)建立了不確定的通信拓?fù)淠Ｐ?研究了高階多智能體系統(tǒng)的跟蹤問(wèn)題.文獻(xiàn)[15]研究了具有動(dòng)態(tài)變化拓?fù)浼岸鄷r(shí)變延時(shí)的多智能體系統(tǒng)的連續(xù)時(shí)間平均一致性問(wèn)題.為建立更加符合實(shí)際的網(wǎng)絡(luò)通信模型,用于度量無(wú)線(xiàn)連接的性能指標(biāo)在移動(dòng)網(wǎng)絡(luò)中得到了研究,如中斷概率[16]、信噪比[17],及接收概率[18]等.因此,本文將針對(duì)具有介質(zhì)訪(fǎng)問(wèn)約束及信道不確定性的無(wú)線(xiàn)網(wǎng)絡(luò)通信下的二階多智能體系統(tǒng)的編隊(duì)控制問(wèn)題展開(kāi)研究,主要工作與貢獻(xiàn)如下：

(1)考慮到無(wú)線(xiàn)通信受到多徑衰竭、遮蔽及發(fā)送器和接收器之間距離變化等因素的影響,區(qū)別于傳統(tǒng)的固定連接權(quán)重的拓?fù)鋱D模型,引入接收概率建立更符合實(shí)際的交互權(quán)重通信拓?fù)湫履Ｐ?使其連接權(quán)重隨著智能體節(jié)點(diǎn)之間距離的變化而變化.這種方法避免了超過(guò)通信距離、遮蔽和多徑衰竭等不良情況下的傳輸請(qǐng)求,減少了不必要的數(shù)據(jù)傳輸,同時(shí)也便于實(shí)際應(yīng)用中實(shí)現(xiàn).

(2)針對(duì)無(wú)線(xiàn)通信網(wǎng)絡(luò)的介質(zhì)訪(fǎng)問(wèn)約束,首次采用二進(jìn)制序列設(shè)計(jì)智能體節(jié)點(diǎn)調(diào)度協(xié)議,并利用該調(diào)度協(xié)議在采樣時(shí)刻調(diào)度滿(mǎn)足條件的智能體節(jié)點(diǎn)接入網(wǎng)絡(luò).通過(guò)引入信息更新誤差及編隊(duì)誤差,并設(shè)計(jì)有效的編隊(duì)控制策略,將二階多智能體系統(tǒng)編隊(duì)控制問(wèn)題轉(zhuǎn)換為具有時(shí)變延時(shí)的閉環(huán)系統(tǒng)的穩(wěn)定性問(wèn)題求解.

1 問(wèn)題描述

考慮具有N個(gè)智能體節(jié)點(diǎn)(編號(hào)為1,2,……,N)和一個(gè)虛擬領(lǐng)導(dǎo)者(編號(hào)為0)的二階多智能體系統(tǒng).第i個(gè)智能體節(jié)點(diǎn)的模型為：

(1)

假設(shè)1N個(gè)智能體節(jié)點(diǎn)中最多允許C個(gè)智能體節(jié)點(diǎn)接入網(wǎng)絡(luò)(C

圖1 第i個(gè)智能體節(jié)點(diǎn)的結(jié)構(gòu)Fig. 1 Structure of the N agent

假設(shè)2 執(zhí)行器、控制器的工作方式均為事件驅(qū)動(dòng),傳感器的工作方式為時(shí)間驅(qū)動(dòng),其采樣周期為h,h>0.

如圖1所示,由于無(wú)線(xiàn)網(wǎng)絡(luò)通信通道的介質(zhì)訪(fǎng)問(wèn)約束,任一時(shí)刻最多允許C個(gè)智能體節(jié)點(diǎn)接入無(wú)線(xiàn)網(wǎng)絡(luò).為此,將通信時(shí)間劃分為多個(gè)具有相同時(shí)間間隔為h的時(shí)間槽.在每個(gè)采樣時(shí)刻(記為k,k=0,1,2,…),利用如(2)式所示[19]的二進(jìn)制節(jié)點(diǎn)調(diào)度策略調(diào)度智能體節(jié)點(diǎn)接入網(wǎng)絡(luò)：

si(k)=「(k+1)pi+θi?-「kpi+θi?

i=1,2,…,N

(2)

Wi≥Wi*

(3)

其中，Wi*為期望的平均傳輸速率.

(4)

進(jìn)一步,表示為：

(5)

考慮到實(shí)際無(wú)線(xiàn)通信受到多徑衰竭、遮蔽及發(fā)送器和接收器之間距離變化等因素的影響,理想的二進(jìn)制通道模型往往是不存在的,故本文采用文獻(xiàn)[20]中的接收概率以表征無(wú)線(xiàn)網(wǎng)絡(luò)信道的通信質(zhì)量,接收概率可定義為：

(6)

(7)

如圖2所示，智能體節(jié)點(diǎn)i的鄰接節(jié)點(diǎn)j為：

Ni={i∈υ|(i,j)∈ε且dij≤dT}

(8)

其中，dT={dij|P=PT}，PT為接收概率的閾值.若dij>dT，則視為通信質(zhì)量不可靠,舍棄相應(yīng)的信息.

圖2 智能體節(jié)點(diǎn)i的鄰接節(jié)點(diǎn)jFig. 2 Adjacent agent j of agent i

注2 在第k個(gè)采樣時(shí)刻，當(dāng)且僅當(dāng)(i,j)∈ε,dij≤dT,且si(k)=1時(shí)，智能體節(jié)點(diǎn)i才能與其遠(yuǎn)程控制器i及其鄰接節(jié)點(diǎn)j通信。

注3 從智能體節(jié)點(diǎn)的連接權(quán)重(7)及鄰接節(jié)點(diǎn)(8)的表示可以看出,隨著智能體節(jié)點(diǎn)的運(yùn)動(dòng),在不同的時(shí)刻其鄰接節(jié)點(diǎn)可能是不同的,多智能體系統(tǒng)的拓?fù)浣Y(jié)構(gòu)也可能是變化的.

設(shè)計(jì)如下編隊(duì)控制協(xié)議：

(9)

其中,t∈[kh,(k+1)h),k1,k2為待設(shè)計(jì)值,xdi∈Rn,xdj∈Rn分別為期望的智能體節(jié)點(diǎn)i及其鄰接智能體節(jié)點(diǎn)的位置信息.記C=diag{ci0},當(dāng)智能體節(jié)點(diǎn)i與虛擬領(lǐng)導(dǎo)者之間存在通信鏈路,且ri0≤dT時(shí),ci0≤ai0；否則ci0=0.

定義信息更新誤差為：

(10)

代入(4)式,得:

(11)

根據(jù)式(1)和編隊(duì)控制協(xié)議(9),智能體節(jié)點(diǎn)i可表示為：

(12)

其中,t∈[kh,(k+1)h).

定義智能體節(jié)點(diǎn)i的編隊(duì)誤差：

(13)

則(12)式可改寫(xiě)成：

其中,t∈[kh,(k+1)h).

記Mk=diag{1-si(k)},i=1,2,…,N,

則可得閉環(huán)多智能體系統(tǒng)：

t∈[kh,(k+1)h)

(15)

t∈[kh,(k+1)h)

(16)

簡(jiǎn)記為:

t∈[kh,(k+1)h)

(17)

綜上所述,本文設(shè)計(jì)目標(biāo)是針對(duì)無(wú)線(xiàn)網(wǎng)絡(luò)通信具有的信道約束及不確定性,設(shè)計(jì)節(jié)點(diǎn)調(diào)度協(xié)議(2)和編隊(duì)控制協(xié)議(9),使得閉環(huán)多智能體系統(tǒng)(17)漸近穩(wěn)定.

2 編隊(duì)控制協(xié)議的設(shè)計(jì)

定義1 若存在常數(shù)α>0和β>0使得狀態(tài)響應(yīng)滿(mǎn)足‖x(t)‖≤αe-β(t-t0)‖x(t0)‖,?t≥t0具有任意非負(fù)初始狀態(tài),則稱(chēng)系統(tǒng)指數(shù)穩(wěn)定.

(1)S<0

φT(t)(τ(t)HTR-1H-FTH-HTF)φ(t)

定理1 針對(duì)閉環(huán)多智能體系統(tǒng)(17),給定常數(shù)α>0,λ>0,對(duì)于第k個(gè)時(shí)間段及l(fā)=k-1個(gè)時(shí)間段,若存在一系列適維正定矩陣Pk,Qk,Yk,Hk,Rk,Pl,Ql,Yl,Hl,Rl,使得下列不等式成立：

(18)

Pk≤βPl,Qk≤βQl,Rk≤βRl,Yk≤βYl

(19)

(20)

其中,β≥1,γ=e-αh,G=Aε2+Bε4,J=GTRkG,

證明：令φ(t)=[zT(t),zT(t-τ(t)),zT(t-h),eT(t-τ(t))]T.

記εi(i=1,2,3,4)為塊矩陣,且εij=εi-εj.

(1)假設(shè)t∈(kh,(k+1)h)時(shí),系統(tǒng)處于第k個(gè)時(shí)間段.構(gòu)造Lyapunov-Krasovskii函數(shù)如下：

(h-τ(t))[z(t)-z(t-h)]T·

(21)

其中,V1(t)=zT(t)Pkz(t),

V4(t)= (h-τ(t))[z(t)-z(t-h)]T·

Hk[z(t)-z(t-h)],

e-αhzT(t-h)Qkz(t-h)-αV2(t)+

[z(t)-z(t-h)]THk[z(t)-z(t-h)]+

Hk[z(t)-z(t-h)]+[z(t)-z(t-h)]T·

(22)

由引理2,可得

(23)

由式(22)～(23)可知,

(24)

其中,

∑=Γ+hJ+(h-τ(t))(∏1+∏2)+τ(t)(∏3+∏4).可見(jiàn),∑是∏1+∏2和∏3在τ(t)∈[0,h]上的凸組合.因此,若(18)滿(mǎn)足,則有∑<0,即:

對(duì)上式在t∈[tk,tk+1)=[kh,(k+1)h)上積分可得:

Vk(t)≤e-α(t-tk)Vk(tk)

(25)

從而可知Vp在第P個(gè)時(shí)間段內(nèi)指數(shù)衰減.

(2)對(duì)任意采樣時(shí)刻tk=kh,根據(jù)不等式組(19),可知:

(26)

因此,當(dāng)t∈[tk,tk+1),由式(25)～(26)得:

≤e-α(t-tk)βe-α(tk-tk-1)Vk-1(tk-1)≤…

≤βδ(k)e-αδ(k)hV0(t0)

(27)

根據(jù)式(19)有:

則λN(tk-t0)≤αδ2(k)h<αδ(k)h,代入式(27)得:

Vk(t)<βδ(k)e-λN(tk-t0)V0(t0)

結(jié)合式(21),有:

κ1‖z(t)‖2≤V(t)≤βδ(k)e-λN(t-t0)κ2‖z(t0)‖2,進(jìn)一步可得:

(28)

(29)

(30)

且智能體節(jié)點(diǎn)調(diào)度協(xié)議(2)滿(mǎn)足式(3)及如下不等式:

(31)

其中,C是智能體節(jié)點(diǎn)接入網(wǎng)絡(luò)最多允許個(gè)數(shù), 0<σ<1表示了智能體節(jié)點(diǎn)μ1k接入網(wǎng)絡(luò)次數(shù)的漸近平均值,δ(k)為智能體系統(tǒng)接入網(wǎng)絡(luò)的總次數(shù),且:

則閉環(huán)多智能體系統(tǒng)(17)在具有介質(zhì)訪(fǎng)問(wèn)受限及不確定性的無(wú)線(xiàn)通信下能夠達(dá)到穩(wěn)定編隊(duì).

證明：式(18)等價(jià)于

根據(jù)Schur補(bǔ)引理,則有:

(32)

(33)

(34)

(35)

由于-PR-1P≤?2R-2?P,有:

(36)

將式(36)代入式(34)～(35),即得式(29)～(30).

3 算例仿真

圖3 多智能體系統(tǒng)通信拓?fù)鋱DFig. 3 Communication topology of multi-agent system

圖4 期望的正方形編隊(duì)隊(duì)形Fig. 4 Expected square formation

圖5 智能體節(jié)點(diǎn)的位置軌跡Fig. 5 Position trajectories of Agents

圖6 智能體節(jié)點(diǎn)的速度軌跡Fig. 6 Velocity trajectories of Agents

圖7 多智能體系統(tǒng)的位置響應(yīng)Fig. 7 Position response of multi-agent system

圖8 多智能體系統(tǒng)的速度響應(yīng)Fig. 8 Velocity response of multi-agent system

圖9 接入網(wǎng)絡(luò)的智能體節(jié)點(diǎn)的數(shù)量Fig. 9 Number of agents accessed to network

圖10 智能體節(jié)點(diǎn)接入網(wǎng)絡(luò)的時(shí)序Fig. 10 Time sequences of agents accessed to network

上述多智能體系統(tǒng)的編隊(duì)控制仿真結(jié)果如圖5～圖10所示.智能體節(jié)點(diǎn)的位置軌跡如圖5的(a)、(b)所示,可見(jiàn)兩個(gè)位置分量分別收斂并保持期望的距離.圖6的(a)、(b)給出了智能體節(jié)點(diǎn)的速度軌跡,可見(jiàn)兩個(gè)速度分量均保持一致.圖7和圖8分別為多智能體系統(tǒng)的位置和速度軌跡曲線(xiàn),表明了隨著時(shí)間推移,智能體節(jié)點(diǎn)的隊(duì)形達(dá)到期望的正方形編隊(duì)隊(duì)形,且速度達(dá)到一致,并保持隊(duì)形運(yùn)動(dòng).根據(jù)本文設(shè)計(jì)的節(jié)點(diǎn)調(diào)度協(xié)議,圖9表示接入網(wǎng)絡(luò)的智能體節(jié)點(diǎn)的數(shù)量,可以看出任意時(shí)刻最多有3個(gè)智能體節(jié)點(diǎn)接入網(wǎng)絡(luò)傳輸采樣信息.圖10給出了各個(gè)智能體節(jié)點(diǎn)在調(diào)度策略下的接入網(wǎng)絡(luò)的時(shí)序,可見(jiàn)智能體節(jié)點(diǎn)在調(diào)度策略下間斷性地傳輸信息,最終達(dá)到了穩(wěn)定的編隊(duì).以上結(jié)果表明,在具有介質(zhì)訪(fǎng)問(wèn)約束及不確定性的無(wú)線(xiàn)通信下,多智能體系統(tǒng)快速地形成期望的編隊(duì)隊(duì)形并保持該隊(duì)形運(yùn)動(dòng).為驗(yàn)證方法的有效性,本文僅采用了四個(gè)智能體節(jié)點(diǎn)進(jìn)行編隊(duì)分析,而對(duì)更多數(shù)量的多智能體系統(tǒng),此方法同樣有效.

4 結(jié)論

本文針對(duì)無(wú)線(xiàn)網(wǎng)絡(luò)環(huán)境下二階多智能體系統(tǒng)的編隊(duì)控制問(wèn)題,考慮到網(wǎng)絡(luò)的不確定性及介質(zhì)訪(fǎng)問(wèn)約束,采用二進(jìn)制序列設(shè)計(jì)智能體節(jié)點(diǎn)調(diào)度策略,并利用接收概率描述通信拓?fù)涞倪B接權(quán)重,使得任意時(shí)刻傳輸狀態(tài)信息的智能體節(jié)點(diǎn)數(shù)滿(mǎn)足約束且達(dá)到穩(wěn)定的編隊(duì)隊(duì)形.首先,通過(guò)定義信息更新誤差及編隊(duì)誤差,將二階多智能體系統(tǒng)模型轉(zhuǎn)化為閉環(huán)延時(shí)系統(tǒng)模型.其次,利用Lyapunov-Krasovskii函數(shù)及LMI技術(shù)給出了二進(jìn)制節(jié)點(diǎn)調(diào)度策略下編隊(duì)控制協(xié)議的設(shè)計(jì).最后,通過(guò)仿真驗(yàn)證了所提方法的有效性.本文后續(xù)工作是將進(jìn)一步研究基于事件觸發(fā)機(jī)制的無(wú)線(xiàn)網(wǎng)絡(luò)通信協(xié)議以節(jié)省無(wú)線(xiàn)網(wǎng)絡(luò)的帶寬和能量,以及將該方法推廣到高階非線(xiàn)性多智能體的編隊(duì)控制.

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*The project supported by the National Natural Science Foundation of China(61074023) and Undergraduate innovation training projects of Jiangsu province

? Corresponding author E-mail: njust_zc@126.com

20 April 2016,revised 23 June 2016.

FORMATION CONTROL OF NETWORKED MULTI-AGENT SYSTEM WITH COMMUNICATION CONSTRAINS*

Song Na Hong Xiaomin Zhou Chuan?

(SchoolofAutomation,NanjingUniversityofScienceandTechnology,Nanjing210094,China)

The formation control of second-order multi-agent system under wireless network with communication constrains and uncertainty is investigated in this paper. The binary sequence is used to design the agent scheduling protocol for the media access constrains in wireless network, where only the sampling information of the agents which satisfies the constraints is allowed to transmit through network. On the other hand, in order to build more realistic communication channel model, the wireless network metric is used to construct a new topology model rather than the traditional fixed interaction weights, which considers the uncertainties of wireless network and also the relationship between interaction weights and the distances among agents. Furthermore, the second-order multi-agent system model is transformed into close-loop system model with time-delay by defining information updating error and formation error. The formation control scheme is then designed by constructing Lyapunov- Krasovski function and taking advantage of LMI techniques. Finally, a simulation example illustrates the effectiveness of the proposed method.

multi-agent system, formation control, wireless network, communication constraints, uncertainties

*國(guó)家自然科學(xué)基金資助項(xiàng)目(61074023),江蘇省大學(xué)生科研創(chuàng)新訓(xùn)練項(xiàng)目資助

10.6052/1672-6553-2016-032

2016-04-20收到第1稿,2016-06-23收到修改稿.

? 通訊作者 E-mail: njust_zc@126.com