郭濤 梁燕軍

不確定非線性時(shí)滯關(guān)聯(lián)大系統(tǒng)自適應(yīng)分散容錯(cuò)控制

郭濤1梁燕軍1

針對(duì)一類不確定非線性時(shí)滯關(guān)聯(lián)大系統(tǒng),提出了一種基于時(shí)滯代換的自適應(yīng)分散容錯(cuò)控制方案.該方案采用模糊邏輯系統(tǒng)作為逼近器,提出了時(shí)滯代換的方法處理系統(tǒng)未知時(shí)滯關(guān)聯(lián)函數(shù),并結(jié)合自適應(yīng)技術(shù)處理代換誤差和逼近誤差.與現(xiàn)有方法相比,本文方法能在線補(bǔ)償所有四種類型的執(zhí)行器故障,系統(tǒng)控制器的設(shè)計(jì)也不再依賴于時(shí)滯假設(shè)條件,同時(shí)還可保證閉環(huán)系統(tǒng)所有信號(hào)全局一致最終有界.仿真結(jié)果進(jìn)一步驗(yàn)證了本文方法的有效性.

關(guān)聯(lián)大系統(tǒng),時(shí)滯系統(tǒng),時(shí)滯代換,全局穩(wěn)定性,模糊逼近,執(zhí)行器故障

非線性系統(tǒng)控制中,反推控制(Backstepping control, BC)是一種重要的設(shè)計(jì)方法,在不確定非線性系統(tǒng)、非線性時(shí)滯系統(tǒng)和容錯(cuò)控制等方面得到廣泛應(yīng)用.

對(duì)于不確定非線性系統(tǒng)的反推控制,常采用模糊邏輯系統(tǒng)或神經(jīng)網(wǎng)絡(luò)作為逼近器,并發(fā)展出基于逼近器的自適應(yīng)反推控制(Approximator-based adaptive backstepping control,ABABC).但由于這些逼近器的輸入僅當(dāng)保持在某一個(gè)緊集內(nèi)才是有效的,因此閉環(huán)系統(tǒng)只能保證為半全局穩(wěn)定[1?2].為了解決這個(gè)問題,最近發(fā)展出兩種可得全局穩(wěn)定結(jié)果的方法.一種方法是采用代換技術(shù)[3?6],將未知函數(shù)的輸入替換為有界的系統(tǒng)參考信號(hào),從而使得逼近器對(duì)未知函數(shù)的逼近始終成立.另一種方法是采用復(fù)合切換技術(shù)[7?10],通過在逼近器所成立的緊集外部設(shè)置額外的控制率保證閉環(huán)系統(tǒng)的全局穩(wěn)定性.基于代換的方法結(jié)構(gòu)簡(jiǎn)單,但主要適用于系統(tǒng)不確定項(xiàng)僅含系統(tǒng)輸出變量y的系統(tǒng);基于切換的方法可適用于系統(tǒng)不確定項(xiàng)含任意變量,但控制器結(jié)構(gòu)復(fù)雜.

對(duì)于非線性時(shí)滯系統(tǒng)的反推控制,主要是通過構(gòu)造Lyapunov-Krasovskii或Lyapunov-Razumikhin泛函,來消除時(shí)滯對(duì)閉環(huán)系統(tǒng)穩(wěn)定性的影響.Ge等針對(duì)一類含未知虛擬控制系數(shù)的不確定非線性時(shí)滯系統(tǒng),提出了神經(jīng)網(wǎng)絡(luò)自適應(yīng)反推控制方法[11],同時(shí)避免控制奇異性問題.同神經(jīng)網(wǎng)絡(luò)一樣,模糊邏輯系統(tǒng)(Fuzzy logic system,FLS)也被用來作為非線性逼近器,處理不確定非線性時(shí)滯系統(tǒng)中的未知非線性時(shí)滯函數(shù)[12].該領(lǐng)域最新的一些研究結(jié)果主要集中在嚴(yán)格反饋系統(tǒng)[5,13]、關(guān)聯(lián)大系統(tǒng)分散控制[14?15]、隨機(jī)系統(tǒng)[16]、多輸入多輸出非線性系統(tǒng)[17?18],以及實(shí)際應(yīng)用如二階化學(xué)反應(yīng)器[19]等.值得指出的是,時(shí)滯系統(tǒng)控制器的設(shè)計(jì)常依賴于對(duì)系統(tǒng)未知時(shí)滯所做的假設(shè)條件,如系統(tǒng)時(shí)滯為已知常數(shù)、未知常數(shù)、有界的未知時(shí)變時(shí)滯和時(shí)變時(shí)滯d(t)的導(dǎo)數(shù)滿足˙d(t)＜d?＜1等.如何構(gòu)造不依賴于時(shí)滯假設(shè)條件的控制器的研究結(jié)果并不多見.

針對(duì)日益復(fù)雜的控制系統(tǒng),反推控制也與容錯(cuò)控制(Fault-tolerant control,FTC)相結(jié)合,發(fā)展出基于FTC的反推控制方法,顯著提高了復(fù)雜系統(tǒng)的控制可靠性.文獻(xiàn)[20]將FTC與BC相結(jié)合,針對(duì)一類參數(shù)化嚴(yán)格反饋系統(tǒng),提出了一種自適應(yīng)容錯(cuò)補(bǔ)償控制方案,并應(yīng)用于“雙水獺”飛機(jī)的飛控模型中,取得了滿意的控制結(jié)果.沿著這個(gè)思路,文獻(xiàn)[21]討論了不確定非線性系統(tǒng)的容錯(cuò)控制問題,模糊邏輯系統(tǒng)或神經(jīng)網(wǎng)絡(luò)用來在線逼近系統(tǒng)未知函數(shù),文獻(xiàn)[22]討論了具有隨機(jī)執(zhí)行器故障的非線性系統(tǒng)容錯(cuò)控制問題,文獻(xiàn)[23]則討論了具有時(shí)變執(zhí)行器的非線性系統(tǒng)的容錯(cuò)控制問題.容錯(cuò)控制在多輸入多輸出非線性系統(tǒng)、關(guān)聯(lián)大系統(tǒng)方面的推廣見文獻(xiàn)[24–25].值得注意的是,針對(duì)時(shí)滯非線性系統(tǒng)容錯(cuò)控制的研究結(jié)果較少,僅有如文獻(xiàn)[25]給出了基于BC和FTC的控制結(jié)果,但控制器的設(shè)計(jì)需依賴于時(shí)滯假設(shè)條件

本文針對(duì)一類含有未知時(shí)滯關(guān)聯(lián)項(xiàng)的關(guān)聯(lián)大系統(tǒng),提出了一種基于時(shí)滯代換的自適應(yīng)分散容錯(cuò)控制方案.采用模糊邏輯系統(tǒng)逼近系統(tǒng)未知時(shí)滯關(guān)聯(lián)項(xiàng),采用時(shí)滯代換技術(shù)處理逼近器輸入中的時(shí)滯信號(hào),并基于FTC的理論構(gòu)建了全局穩(wěn)定的自適應(yīng)容錯(cuò)控制器.與現(xiàn)有研究結(jié)果相比,本文的主要貢獻(xiàn)可歸納如下.首先,本文用參考信號(hào)yr代換未知時(shí)滯關(guān)聯(lián)函數(shù)中的未知時(shí)滯輸出yd,使得自適應(yīng)模糊控制器的構(gòu)建不再依賴于對(duì)系統(tǒng)時(shí)滯所做的假設(shè)條件,大大增加了控制器設(shè)計(jì)的便易性.其次,借鑒文獻(xiàn)[3]的思想,本文也用代換的方法保證系統(tǒng)的全局穩(wěn)定性.不同的是,由于代換的是系統(tǒng)輸出時(shí)滯信號(hào)yd,本文精心設(shè)計(jì)了特殊方法來消除含有時(shí)滯的代換誤差.第三,本文考慮了容錯(cuò)控制中的所有四種執(zhí)行器故障模型,這些故障模型均可由所構(gòu)建的自適應(yīng)控制器有效補(bǔ)償.

1 問題描述

考慮如下關(guān)聯(lián)大系統(tǒng),其第i個(gè)子系統(tǒng)為

其中為系統(tǒng)狀態(tài),yi為系統(tǒng)輸出;為系統(tǒng)輸入,同時(shí)也是執(zhí)行器的輸出,下標(biāo)mi表示系統(tǒng)輸入的個(gè)數(shù),注意本文考慮了可能發(fā)生的執(zhí)行器失效情況;關(guān)聯(lián)項(xiàng)為未知光滑函數(shù),為未知時(shí)滯; ωTi=[ωi,1,ωi,2,···,ωi,mi]∈Rmi為常數(shù)向量.

控制目標(biāo):所考慮的執(zhí)行器失效均可得到有效的補(bǔ)償,閉環(huán)系統(tǒng)為全局一致最終有界(Global uniformly ultimately bounded,GUUB),跟蹤誤差可以收斂到原點(diǎn)附近的一個(gè)小鄰域內(nèi).

假設(shè)1.在區(qū)間[0,+∞)上,參考信號(hào)yi,r(t)及其前ni階導(dǎo)數(shù)已知,分段連續(xù)且有界.

2 預(yù)備知識(shí)

2.1 執(zhí)行器失效模型

本文考慮的四種執(zhí)行器失效模型在實(shí)際控制系統(tǒng)中常常會(huì)發(fā)生.這四種模型為損傷(Loss of e ff ectiveness,LOE)、卡死(Lock in place,LIP)、飛車或飽和(Hard over fault, HOF)、松浮(Float),可將其表述如下

假設(shè)2.對(duì)于關(guān)聯(lián)大系統(tǒng)(1)的任何一個(gè)子系統(tǒng)來說,對(duì)于其mi個(gè)執(zhí)行器輸出,若其中有任意不大于mi?1個(gè)執(zhí)行器發(fā)生LIP、HOF或Float,剩余的執(zhí)行器仍可驅(qū)使閉環(huán)系統(tǒng)達(dá)到上述控制目標(biāo).這也是研究容錯(cuò)控制問題的基本假設(shè).

注 1. 文獻(xiàn) [26]首次提出了式 (2)所示的飛控系統(tǒng)執(zhí)行器四種失效模型,但現(xiàn)有一些研究結(jié)果如文獻(xiàn)[22,24?25,27?29]等僅考慮了LOE和LIP模型.由于松浮(操縱面脫離控制)、飛車或飽和(執(zhí)行器處于極限位置)這兩種故障類型也可導(dǎo)致嚴(yán)重的故障(如1985年日本的Flight123,2002年阿拉斯加的Flight85),故在設(shè)計(jì)容錯(cuò)控制器時(shí)考慮到這兩種執(zhí)行器故障的影響,具有重要的現(xiàn)實(shí)意義.本文同時(shí)考慮了這四種故障模型,這是本文的第一個(gè)主要特點(diǎn).

2.2 模糊逼近

本文采用FLS作為系統(tǒng)(1)中未知關(guān)聯(lián)函數(shù)的逼近器.若模糊逼近器采用單點(diǎn)模糊化、乘積運(yùn)算的模糊蘊(yùn)含規(guī)則、重心法解模糊和高斯函數(shù)的隸屬度函數(shù)時(shí),可以表示為[30]

其中,x=[x1,···,xn]T為模糊逼近器的輸入;f(x|θ)為模糊逼近器的輸出;為未知參數(shù)向量,為模糊基函數(shù)向量,M 為模糊規(guī)則集合中的規(guī)則數(shù)目.根據(jù)模糊邏輯系統(tǒng)的逼近定理[30],對(duì)于緊集?Fuzzy∈Rn中的連續(xù)非線性函數(shù)存在式(5)所示的模糊邏輯系統(tǒng),使得

2.3 時(shí)滯代換

根據(jù)式(8),有

這里的li,j,1和li,j,2為未知的Lipschitz常數(shù).

注2.用時(shí)滯代換的方法來處理系統(tǒng)中的時(shí)滯項(xiàng),可以使得控制器的設(shè)計(jì)過程不再依賴于時(shí)滯假設(shè)條件,進(jìn)而增加系統(tǒng)控制器設(shè)計(jì)的便利性.這是本文的第二個(gè)主要特點(diǎn).

3 控制器設(shè)計(jì)

3.1 系統(tǒng)結(jié)構(gòu)變形

根據(jù)式(6)可知式(10)中hi,j(yr)可寫為

其中,ψi,j,2＞0為一未知常數(shù).令ei,j=qi,j+εi,j,則

項(xiàng)ei,j中包含了逼近誤差和代換誤差的一部分,其邊界ψi將在下文中采用自適應(yīng)的方法進(jìn)行處理.

將式(10)、(13)、(18)和(15)中的ei,j代入式(1),有

3.2 控制器設(shè)計(jì)

定義誤差坐標(biāo)變換

步驟1.根據(jù)式(20),有

穩(wěn)定化函數(shù)αi,1設(shè)計(jì)為

從步驟1可以看出,穩(wěn)定化函數(shù)αi,1的設(shè)計(jì)與現(xiàn)有的反推控制方法類似,為了簡(jiǎn)化,本文省略了下面步驟的詳細(xì)設(shè)計(jì)過程.有關(guān)反推法的詳細(xì)設(shè)計(jì)請(qǐng)參考文獻(xiàn)[31].

步驟2～ni.穩(wěn)定化函數(shù)αi,j和控制器ui0設(shè)計(jì)為

4 穩(wěn)定性分析

本節(jié)討論由系統(tǒng)(1)、控制器(25)和參數(shù)自適應(yīng)率(26)構(gòu)成的閉環(huán)系統(tǒng)的穩(wěn)定性,這需要用到如下三個(gè)不等式.

對(duì)于pi,jzi,j,由前文yd和yrd的定義及式(11),有

對(duì)于未知參數(shù)a及其估計(jì)值?a,有估計(jì)誤差?a=a??a,則

下面引入本文的穩(wěn)定性定理.

定理1.考慮由時(shí)滯關(guān)聯(lián)大系統(tǒng)(1)、控制器(25)和參數(shù)自適應(yīng)率(26)構(gòu)成的閉環(huán)系統(tǒng).當(dāng)滿足假設(shè)1和2的條件時(shí),通過正確選擇設(shè)計(jì)參數(shù)ci,j,ri,Γi,j,γi,1和γi,2,所提出的基于時(shí)滯代換的分散模糊控制可以在線補(bǔ)償式(2)所示的執(zhí)行器故障,保證閉環(huán)系統(tǒng)GUUB,同時(shí)可使跟蹤誤差收斂到原點(diǎn)(zi,1=0)附近的小鄰域內(nèi).

證明.取李雅普諾夫函數(shù)為

其中,Vi定義為

對(duì)Vi求導(dǎo),并將式(26)、(28)～(30)和(22)定義的ρi(ρi=代入,有

再考慮到式(30),有

其中a0為一正數(shù)且滿足

定義集合

當(dāng)(zi,j, θi,j,ψi,ρi)處于集合?外部時(shí),也就是說,當(dāng)有因此,閉環(huán)系統(tǒng)(27)的所有解最終都將一致收斂于緊集? 內(nèi),即閉環(huán)系統(tǒng)的信號(hào)zi,j, θi,j,ψi,ρi均有界.根據(jù)式(22),式(24)和(25),可知αi,j,ui0有界.再由式(20)知,xi,j均有界.因此,閉環(huán)系統(tǒng)(27)為GUUB.

由式(35)直接可得

即跟蹤誤差zi,1最終將一致收斂到包含原點(diǎn)在內(nèi)的小鄰域內(nèi).□

由式(36)可知,增大a0,減小b0,會(huì)使得這一鄰域變小,即系統(tǒng)的跟蹤性能得到增強(qiáng).也就是說,通過增大參數(shù)的值,減小參數(shù)ri的值,可以改善閉環(huán)系統(tǒng)的跟蹤性能.

5 仿真實(shí)例

考慮如下互聯(lián)雙倒立擺模型[25]

其中,xi,1=θi為雙擺相對(duì)于垂直線的偏轉(zhuǎn)角度,i=1,2, xi,2= ˙θi.雙擺均由執(zhí)行器驅(qū)動(dòng),即執(zhí)行器的輸入為系統(tǒng)控制器的輸出ui0.為搭建系統(tǒng)模型,兩倒立擺之間的未知關(guān)聯(lián)函數(shù)假設(shè)為受彈性滯后影響的彈簧,則

模型中其他參數(shù)取值如下.擺端質(zhì)量m1=2kg,m2= 2.5kg,慣性矩J1=0.5kg,J2=0.625kg,彈簧彈性常數(shù)k=100 N/m,擺高r=0.5 m,彈簧自然長(zhǎng)度l=0.5m,擺軸間距b=0.5m,重力加速度g=9.81m/s2.參考信號(hào)選擇為y1r=sin(u)和y2r=cos(0.8u),系統(tǒng)時(shí)滯選為d1(t)=1.6(1+sin(t))和d2(t)=1.6(1?cos(t)).

模糊隸屬度函數(shù)選擇為

模糊基函數(shù)選擇為

根據(jù)式(6),可構(gòu)造模糊逼近器?h1,2(y2r)和

分散控制器的設(shè)計(jì)參數(shù)選擇如下.對(duì)于子系統(tǒng)1,有c1,1=25,c1,2=25,γ1,1=10,γ1,2=10,Γ1,2=10I, r1=5,δ1,2=0.1;對(duì)于子系統(tǒng)2,有c2,1=25,c2,2=25, γ2,1=10,γ2,2=10,Γ2,2=10I,r2=5,δ2,2=0.1.式(17)中的比例函數(shù)選為b1,k(x)=0.5,b2,k=0.8,k=1,2,3.

在仿真過程中,執(zhí)行器故障模型選擇如下.對(duì)于子系統(tǒng)1來說,當(dāng)t＞10時(shí)發(fā)生LOE,u1,1=0.8uc1,1;當(dāng)t＞8時(shí)發(fā)生HOF,u1,2=?100;當(dāng)t＞5時(shí)發(fā)生LIP,u1,3=13.對(duì)于子系統(tǒng)2來說,當(dāng)t＞6時(shí)發(fā)生LOE,u2,1=0.6uc2,1;當(dāng)t＞12時(shí)發(fā)生LIP,u2,2=5;當(dāng)t＞9時(shí)發(fā)生Float,上述由式(17)確定.系統(tǒng)初始狀態(tài)選擇為

圖1 子系統(tǒng)1仿真結(jié)果Fig.1 Simulation results of Subsystem 1

仿真圖1和2中,子圖(a)所示為子系統(tǒng)輸出跟蹤參考信號(hào)仿真曲線,子圖(b)為執(zhí)行器輸出曲線,子圖(c)為系統(tǒng)未知參數(shù)的估計(jì)曲線.從這些曲線圖上可以看出,本文方法可以保證閉環(huán)系統(tǒng)的所有信號(hào)有界,同時(shí)跟蹤誤差可以收斂到原點(diǎn)附近的小鄰域內(nèi).圖3所示為不同系統(tǒng)參數(shù)對(duì)控制性能造成的影響.由于在式(36)中,參數(shù)ri的增大或減小會(huì)導(dǎo)致參數(shù)的值相應(yīng)增大或減小,所以我們?cè)诒緦?shí)例中采用固定ri而改變?chǔ),j,γi,1和γi,2的方式來考證這些參數(shù)對(duì)控制效果的影響.仿真過程中,保持ri=5和δi,2=0.1不變,第一組參數(shù)選擇為ci,j=5, γi,j=3,Γi,2=3I,i=1,2,j=1,2,所得仿真結(jié)果如y1(1)和y2(1)所示;第二組參數(shù)選擇為ci,j=15,γi,j=6, Γi,2=6I,所得仿真結(jié)果如y1(2)和y2(2)所示;第三組參數(shù)選擇為ci,j=25,γi,j=10,Γi,2=10I,所得仿真結(jié)果如y1(3)和y2(3)所示.從圖中可以看出,增大設(shè)計(jì)參數(shù)ci,j,γi,j和Γi,2的值,即增大式(36)中a0的值,可以顯著改善閉環(huán)系統(tǒng)的跟蹤性能.

圖2 子系統(tǒng)2仿真結(jié)果Fig.2 Simulation results of Subsystem 2

圖3 不同設(shè)計(jì)參數(shù)下仿真結(jié)果Fig.3 Simulation results under di ff erent design parameters

6 結(jié)論

本文討論了一類關(guān)聯(lián)時(shí)滯大系統(tǒng)的自適應(yīng)模糊容錯(cuò)控制問題.所提出的控制方案能夠在線補(bǔ)償所有常見的四種執(zhí)行器故障.通過代換的方法,使得模糊逼近器的輸入信號(hào)為有界的參考信號(hào),從而保證了閉環(huán)系統(tǒng)所有信號(hào)全局一致最終有界.這種代換還使得控制器的設(shè)計(jì)不再依賴于對(duì)時(shí)滯信號(hào)所做的假設(shè),大大增加了控制器設(shè)計(jì)的便易性.本文方法可以直接推廣到系統(tǒng)輸出含有時(shí)滯的其他不確定系統(tǒng)中,如輸出反饋系統(tǒng)、控制增益不為1的系統(tǒng)等.對(duì)含有時(shí)變或隨機(jī)執(zhí)行器故障的非線性時(shí)滯系統(tǒng)的容錯(cuò)控制則是我們下一步研究的方向.

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郭 濤 安陽(yáng)師范學(xué)院計(jì)算機(jī)與信息工程學(xué)院副教授.2009年獲得西安電子科技大學(xué)博士學(xué)位.主要研究方向?yàn)榉赐瓶刂?自適應(yīng)模糊控制和非線性控制.本文通信作者.E-mail:gtmailbox@126.com

(GUO Tao Associate professor at the School of Computer and Information Engineering,Anyang Normal University.He received his Ph.D.degree from Xidian University in 2009.His research interest covers backstepping control,adaptive fuzzy control,and nonlinear control.Corresponding author of this paper.)

梁燕軍 安陽(yáng)師范學(xué)院計(jì)算機(jī)與信息工程學(xué)院副教授.2010年獲得中國(guó)海洋大學(xué)博士學(xué)位.主要研究方向?yàn)檎駝?dòng)控制和自適應(yīng)控制.

E-mail:myluck0404@126.com

(LIANG Yan-Jun Associate professor at the School of Computer and Information Engineering,Anyang Normal University. He received his Ph.D.degree from Ocean University of China in 2010.His research interest covers vibration control and adaptive control.)

Adaptive Decentralized Fault-tolerant Control for Uncertain Nonlinear Time-delay Large Scale Systems

GUO Tao1LIANG Yan-Jun1

In this paper,a delay replacement based adaptive decentralized fault-tolerant control method is proposed for a class of uncertain nonlinear time-delay large-scale systems.With fuzzy logic systems as approximators,a novel delay replacement technique is proposed to deal with the unknown delayed nonlinear interconnection functions,and adaptive technique is combined to deal with the errors of replacement and approximation. Compared with the existing results,all of the four types of actuator faults can be compensated for online,the controller design is no longer depended on the assumptions of the time-delays,and the globally uniformly ultimately boundedness of the closed-loop system is guaranteed.Simulation results are provided to show the e ff ectiveness of the control approach.

Interconnected large-scale system,time-delay system,delay replacement,global stability,fuzzy approximation, actuator fault

郭濤,梁燕軍.不確定非線性時(shí)滯關(guān)聯(lián)大系統(tǒng)自適應(yīng)分散容錯(cuò)控制.自動(dòng)化學(xué)報(bào),2017,43(3):486?492

Guo Tao,Liang Yan-Jun.Adaptive decentralized fault-tolerant control for uncertain nonlinear time-delay large scale systems.Acta Automatica Sinica,2017,43(3):486?492

2015-12-16 錄用日期2016-04-18

Manuscript received December 16,2015;accepted April 18,2016

河南省創(chuàng)新型科技團(tuán)隊(duì)項(xiàng)目 (C20150028),河南省高校創(chuàng)新人才項(xiàng)目(15HASTIT021),河南省基礎(chǔ)與前沿技術(shù)研究計(jì)劃(142300410114),河南省教育廳自然科學(xué)基金項(xiàng)目(13A520017)資助

Supported by the Innovation Scientists and Technicians Troop Construction Projects of Henan Province(C20150028),the Program for Science&Technology Innovation Talents in Universities of Henan Province(15HASTIT021),the Science and Technology Project of Henan Province(142300410114),and the Project of Natural Science Foundation of Henan Provincial Department of Education(13A520017)

本文責(zé)任編委 劉允剛

Recommended by Associate Editor LIU Yun-Gang

1.安陽(yáng)師范學(xué)院計(jì)算機(jī)與信息工程學(xué)院安陽(yáng) 455000

1.School of Computer and Information Engineering,Anyang Normal University,Anyang 455000

DOI10.16383/j.aas.2017.c150827