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      具有輸入時(shí)滯和預(yù)設(shè)性能的非線性系統(tǒng)有限時(shí)間動(dòng)態(tài)面控制

      2024-05-31 13:47:29夏曉南尹治林李春張鑫磊吳嵩
      關(guān)鍵詞:時(shí)滯預(yù)設(shè)動(dòng)態(tài)

      夏曉南 尹治林 李春 張鑫磊 吳嵩

      DOI: 10.3969/j.issn.1671-7775.2024.03.010

      開放科學(xué)(資源服務(wù))標(biāo)識(shí)碼(OSID):

      摘要:針對一類具有輸入時(shí)滯和動(dòng)態(tài)不確定性的非嚴(yán)格反饋非線性系統(tǒng)的跟蹤問題,提出一種新的基于預(yù)設(shè)性能的有限時(shí)間自適應(yīng)跟蹤控制方案.利用Pade逼近和輔助中間變量將有時(shí)滯系統(tǒng)轉(zhuǎn)化為無時(shí)滯系統(tǒng),采用由一階輔助系統(tǒng)生成的動(dòng)態(tài)信號(hào)處理未建模動(dòng)態(tài),引入雙曲正切函數(shù)實(shí)現(xiàn)預(yù)設(shè)性能跟蹤控制,并給出基于動(dòng)態(tài)面控制方法的穩(wěn)定性分析.在MATLAB環(huán)境中,以具有未建模動(dòng)態(tài)和輸入時(shí)滯的二階非線性系統(tǒng)為例,對所提出的控制策略進(jìn)行數(shù)值仿真.結(jié)果表明:所提出的控制方案能夠避免現(xiàn)有有限時(shí)間控制所出現(xiàn)的虛擬控制求導(dǎo)奇異性問題,所有信號(hào)有限時(shí)間有界, 跟蹤誤差收斂到預(yù)設(shè)的時(shí)變區(qū)間,可見控制算法切實(shí)有效.

      關(guān)鍵詞:? 非線性系統(tǒng); 非嚴(yán)格反饋系統(tǒng); 未建模動(dòng)態(tài); 輸入時(shí)滯; 預(yù)設(shè)性能; 自適應(yīng)控制; 動(dòng)態(tài)面控制; 有限時(shí)間穩(wěn)定

      中圖分類號(hào): TP273? 文獻(xiàn)標(biāo)志碼:? A? 文章編號(hào):?? 1671-7775(2024)03-0316-07

      引文格式:? 夏曉南,尹治林,李? 春,等. 具有輸入時(shí)滯和預(yù)設(shè)性能的非線性系統(tǒng)有限時(shí)間動(dòng)態(tài)面控制[J].江蘇大學(xué)學(xué)報(bào)(自然科學(xué)版),2024,45(3):316-322.

      收稿日期:?? 2023-02-15

      基金項(xiàng)目:? 江蘇省產(chǎn)學(xué)研合作項(xiàng)目(BY2021489); 揚(yáng)州市科技計(jì)劃項(xiàng)目產(chǎn)業(yè)前瞻技術(shù)研發(fā)項(xiàng)目(YZ2021022); 揚(yáng)州大學(xué)2022年創(chuàng)新創(chuàng)業(yè)“揚(yáng)帆計(jì)劃”項(xiàng)目(YZYF202220)

      作者簡介:?? 夏曉南(1970—),女,江蘇鎮(zhèn)江人,副教授(xnxia@yzu.edu.cn),主要從事自適應(yīng)控制與智能控制研究.

      尹治林(2002—),男,重慶墊江人,本科生(共同一作,2973479175@qq.com),揚(yáng)帆計(jì)劃項(xiàng)目主持人,主要從事系統(tǒng)建模與仿真研究.

      Finite-time dynamic surface control for nonlinear systems with

      input delay and prescribed performance

      XIA Xiaonan, YIN Zhilin, LI Chun, ZHANG Xinlei, WU Song

      (College of Information Engineering, Yangzhou University, Yangzhou, Jiangsu 225127, China)

      Abstract: A new finite-time adaptive tracking control scheme based on prescribed performance was developed to solve the control problem of non-strict feedback systems with input delays and dynamic uncertainties. The time-delay systems were transformed into delay-free systems by Pade approximation and auxiliary intermediate variable, and the unmodeled dynamics was handled by the dynamic signal generated by the first-order auxiliary system. The prescribed performance adaptive tracking control was implemented by the hyperbolic tangent function, and the stability analysis was presented based on dynamic surface control method. Taking the second-order nonlinear system with unmodeled dynamics and input delay as example, the numerical simulation of the proposed control strategy was conducted in MATLAB environment . The results show that the proposed control scheme can avoid the singularity in the derivation of virtual control, and all signals in the closed-loop system are bounded in finite time. The tracking error can converge to prescribed time-varying region, and the control algorithm is effective.

      Key words:? nonlinear system; non-strict feedback system; unmodeled dynamics; input delay; prescribed performance; adaptive control; dynamic surface control; finite-time stability

      非線性系統(tǒng)的自適應(yīng)控制一直是控制理論研究的熱點(diǎn)問題.文獻(xiàn)[1-2]研究了復(fù)雜分?jǐn)?shù)階非線性系統(tǒng)的自適應(yīng)控制問題.近年來,非線性系統(tǒng)的有限時(shí)間控制問題逐漸得到控制領(lǐng)域?qū)W者的普遍關(guān)注.有限時(shí)間控制是系統(tǒng)的狀態(tài)在有限時(shí)間內(nèi)趨于穩(wěn)定,然后保持在穩(wěn)定狀態(tài).對于非線性系統(tǒng),文獻(xiàn)[3]利用李雅普諾夫-克拉索夫斯基函數(shù)求解了時(shí)變時(shí)滯系統(tǒng)的有限時(shí)間穩(wěn)定性.文獻(xiàn)[4]綜述了非線性系統(tǒng)的有限時(shí)間控制方法.文獻(xiàn)[5]研究了一類嚴(yán)格反饋非線性系統(tǒng)的有限時(shí)間問題.文獻(xiàn)[6-7]研究了基于后推設(shè)計(jì)的不同系統(tǒng)的有限時(shí)間控制.然而,后推技術(shù)可能會(huì)導(dǎo)致計(jì)算復(fù)雜度問題.文獻(xiàn)[8]采用動(dòng)態(tài)面控制,降低了控制器復(fù)雜度,但文獻(xiàn)[8]中仍然存在奇異性問題.文獻(xiàn)[9]研究了具有全狀態(tài)約束的非線性系統(tǒng)有限時(shí)間控制問題,預(yù)設(shè)性能控制和輸出約束都是要求系統(tǒng)滿足穩(wěn)定性能的同時(shí)保證瞬時(shí)性能.文獻(xiàn)[10-11]針對具有輸入故障的非線性系統(tǒng)給出了基于命令濾波器的有限時(shí)間控制方案.文獻(xiàn)[12]基于后推技術(shù)構(gòu)造了自適應(yīng)預(yù)設(shè)性能跟蹤控制器,利用雙曲正切函數(shù)的有界性進(jìn)行了誤差變換.上述文獻(xiàn)都未考慮輸入時(shí)滯對控制系統(tǒng)的影響.

      輸入時(shí)滯是核反應(yīng)堆、電網(wǎng)和生物系統(tǒng)等許多實(shí)際工程中經(jīng)常遇到的普遍現(xiàn)象.輸入時(shí)滯可能導(dǎo)致閉環(huán)系統(tǒng)性能下降甚至不穩(wěn)定.文獻(xiàn)[13-14]基于Pade近似信息處理輸入時(shí)滯,對于較小的已知輸入時(shí)滯是一種有效的處理方法.與現(xiàn)有文獻(xiàn)比較,文獻(xiàn)[11-12]沒有考慮輸入時(shí)滯問題,文獻(xiàn)[5-8]只考慮了有限時(shí)間控制,沒有考慮預(yù)設(shè)性能控制和系統(tǒng)存在輸入時(shí)滯問題,文獻(xiàn)[13-14]沒有考慮有限時(shí)間控制和預(yù)設(shè)性能控制.

      文中受文獻(xiàn)[13-14]的啟發(fā),在利用Pade逼近的方法并引入中間變量對含輸入時(shí)滯的非嚴(yán)格反饋系統(tǒng)進(jìn)行變換基礎(chǔ)上,考慮系統(tǒng)具有動(dòng)態(tài)不確定性,基于預(yù)設(shè)性能和動(dòng)態(tài)面控制方法,提出有限時(shí)間自適應(yīng)跟蹤控制方案.文中擬利用Pade逼近、加項(xiàng)減項(xiàng)、對數(shù)變換方法分別處理系統(tǒng)中的輸入時(shí)滯、非嚴(yán)格反饋及預(yù)設(shè)性能問題,并給出基于動(dòng)態(tài)面控制方法的有限時(shí)間控制器設(shè)計(jì)策略和穩(wěn)定性證明.

      1? 問題描述與基本假設(shè)

      考慮如下含未建模動(dòng)態(tài)的純反饋非線性系統(tǒng):

      h·=Q(h,x,t),

      x·i=fi(xi,xi+1)+di(h,x,t),i=1,2,…,n-1,

      x·n=fn(xn)+u(t-τ)+dn(h,x,t),

      y=x1,(1)

      式中:狀態(tài)向量xi=[x1x2…xi]T∈Ri, i=1,2,…,n,x=xn;輸出信號(hào)y∈R;u(t-τ)是時(shí)滯為τ的輸入信號(hào);fi(xi,xi+1)、fn(xn)是光滑未知函數(shù);h∈Rn0是未建模動(dòng)態(tài);Q(h,x,t)是由滿足Lipschitz條件的未知連續(xù)函數(shù)組成的向量;di(h,x,t)是未知擾動(dòng),i=1,2,…,n,n≥2.

      控制目標(biāo)是對系統(tǒng)(1)設(shè)計(jì)自適應(yīng)控制u(t), 使得輸出y跟蹤給定期望軌跡yd, 閉環(huán)系統(tǒng)所有信號(hào)在有限時(shí)間半全局一致終結(jié)有界, 且滿足預(yù)設(shè)性能.

      假設(shè)1[15]? 參考輸入Xd=[yd? y·d? y··d]T∈Ωd光滑可測,且Ωd={xd:y2d+y·2d+y··2d≤B0},B0為已知正常數(shù).

      假設(shè)2[15]? 未建模動(dòng)態(tài)h為指數(shù)輸入狀態(tài)實(shí)用穩(wěn)定,即對系統(tǒng)h·=Q(h,x,t),存在K∞函數(shù)α1、α2和Lyapunov函數(shù)V(h)使得

      α1(‖h‖)≤V(h)≤α2(‖h‖),? (2)

      V(h)hQ(t,h,x)≤-cV0(h)+γ(|x1|)+dr(3)

      成立,其中c>0,dr≥0是已知常數(shù),γ(·)是已知K∞類函數(shù).

      假設(shè)3[15]? 存在未知非負(fù)連續(xù)函數(shù)Δi1(·)和非減連續(xù)函數(shù)Δi2(·)使得|di(h,x,t)|≤Δi1(‖xi‖)+Δi2(‖h‖),其中Δi2(0)=0,i=1,2,…,n.

      引理1[15]? 對于系統(tǒng)h·=Q(h,x,t),如果V是指數(shù)輸入狀態(tài)實(shí)用穩(wěn)定函數(shù),即式(2)、(3)成立,則對任一常數(shù)c∈(0,c),任一初始條件h0=h(t0),r0>0,任一連續(xù)函數(shù)γ(|y|)≥γ(|y|),存在有限時(shí)間T0=max{0,ln[V(h0)/r0]/(c-c)}≥0和非負(fù)函數(shù)D(t0,t),設(shè)計(jì)信號(hào)r·=-cr+γ(‖xi‖)+dr,r(t0)=r0,使得當(dāng)D(t0,t)=0,有V(h)≤r(t)+D(t0,t),D(t0,t)=max{0,e-c(t-t0)V(h0)-e-c(t-t0)r0}.不失一般性,假設(shè)γ(‖xi‖)=γ(‖xi‖).

      引理2[10]? 對于非線性系統(tǒng)x·=f(x),如果存在正定C1函數(shù)V(x):Rn→R,緊集ΣRn,K∞類函數(shù)ζ1,ζ2,以及常量α>0,β>0,0

      Tr=t0+1α(1-q)lnαV1-m(x0)+νβνβ.

      引理3[12]? 對于η1,η2,…,ηN∈R+以及0

      在緊集ΠZi上利用徑向基函數(shù)神經(jīng)網(wǎng)絡(luò)逼近未知函數(shù)Φi(Zi),即Φi(Zi)=θ*iTφi(Zi)+εi(Zi),其中基函數(shù)向量φi(Zi)=[φi1(Zi)φi2(Zi)…φili(Zi)]T∈Rli,φij(Zi)可選為以φij和μij為寬度和中心的高斯函數(shù)φij(Zi)=exp-(Zi-μij)T(Zi-μij)φ2ij,i=1,2,…,n;j=1,2,…,li,li是節(jié)點(diǎn)數(shù)量,理想權(quán)值θ*i=argminθi∈Rli(supZi∈ΩZi|θTiφi(Zi)-Φi(Zi)|).

      2? 預(yù)設(shè)性能變換與Pade逼近

      引入預(yù)設(shè)性能函數(shù)(t)=(0-∞)e-k0t+∞,其中0、∞、k0是正的設(shè)計(jì)常數(shù),且0>∞.跟蹤誤差e1=y-yd需要滿足-kc(t)0是正的函數(shù),引入如下坐標(biāo)變換S=12×logkc(t)+e1kc(t)-e1,其逆變換可表示為e1=kc(t)tanh S.對變換求導(dǎo)可得

      S·=1kc(t)(1-tanh2S)e·1-

      e1·2kc(t)(1-tanh2S)-k·c(t)tanh Skc(t)(1-tanh2S).(4)

      為方便表示,定義

      m(t)=1kc(t)(1-tanh2S)>0,

      n(t)=-e1·2kc(t)(1-tanh2S)-k·c(t)tanh Skc(t)(1-tanh2S),(5)

      因此,S·=m(t)e·1+n(t).

      利用Pade逼近處理較小的輸入時(shí)滯,用(·)表示拉普拉斯變換,即

      {u(t-τ)}=e-τq(u(t))=

      e-τq/2eτq/2(u(t))≈1-τq/21+τq/2(u(t)),(6)

      式中:q是拉氏變換因子.引入中間變量xn+1,則有

      (xn+1(t))-(u(t))=1-τq/21+τq/2(u(t)),(7)

      2(u(t))=(xn+1(t))+τq(xn+1(t))/2.(8)

      令J=2τ,則有

      x·n+1=2Ju-Jxn+1.(9)

      3? 自適應(yīng)有限時(shí)間控制

      系統(tǒng)(1)轉(zhuǎn)換成如下形式:

      h·=Q(h,x,t),

      x·i=Fi(xi+1)+xi+1+di(h,x,t),

      x·n=Fn(xn)-u+xn+1+dn(h,x,t),

      xn+1=2Ju-Jxn+1,

      y=x1,(10)

      式中: Fi(xi+1)=fi(xi,xi+1)-xi+1,i=1,2,…,n-1;Fn(xn)=fn(xn).

      定義誤差系統(tǒng)為

      z1=S,

      zi=xi-ωi,i=2,3,…,n-1,

      zn=xn-ωn+xn+1/J,(11)

      式中: ωi是濾波器τω·i+ωi=αi-1的輸出,αi-1為該濾波器輸入,i=2,3,…,n.

      定義如下記號(hào): λi=‖θ*i‖2,λ~i=λ^i-λi,zi=[z1z2…zi]T,yj=[y2y3…yj]T,λ^i=[λ^1? λ^2… λ^i]T,1≤i≤n,2≤j≤n,其中λ^i為λi的估計(jì),yj=ωj-αj-1.定義Lyapunov函數(shù)Vi=12z2i+12λ~2i.

      動(dòng)態(tài)面控制器設(shè)計(jì)步驟分為n步.其中第1步的設(shè)計(jì)過程如下:

      z·1=m(t)(F1(x2)+y2+z2+α1+

      d1(t,h,x)-y·d)+n(t).(12)

      V1的導(dǎo)數(shù)為

      V·1=z1[m(t)(F1(x2)+y2+z2+α1+

      d1(t,h,x)-y·d)+n(t)]+λ~1λ^·1.(13)

      利用Young不等式、假設(shè)3和引理1,可得

      z1m(t)d1(t,h,x)≤

      z1m(t)[Δ11(|x1|)+Δ12(‖h1‖)]≤

      z21m2[Δ11(|x1|)+Δ12α1(r+D0)]2+14,(14)

      z1z2m(t)≤14z21m2+z22,(15)

      z1y2m(t)≤14z21m2+y22.(16)

      令Φ1(Z1)=mF1(x2)-my·d+z1m2[Δ11(|x1|)+Δ12α-11(r+D0)]2+n(t),其中Z1=[xT2z1yd? y·d·kc(t)? k·c(t)r]T.利用Young不等式,可得z1θ*1Tφ1(Z1)≤‖φ1(Z1)‖2z21λ12a21+a212,其中a1是正常數(shù).

      構(gòu)造虛擬控制律和自適應(yīng)律如下:

      α1=-1mk1+1+12m2z1+k1zq1+‖φ1(Z1)‖2z1λ^12a21,(17)

      λ^·1=‖φ1(Z1)‖2z212a21-σ1λ^1,(18)

      式中: σ1>0、k1>0是設(shè)計(jì)常數(shù).

      非負(fù)連續(xù)函數(shù)δ1(z2,λ^1,yd,y·d,y2,r,kc(t),k·c(t))滿足

      ε1(Z1)≤δ1(z2,λ^1,yd,y·d,y2,r,kc(t),k·c(t)).(19)

      由Young不等式,可得

      z1ε1≤z21+14δ21,(20)

      -σ1λ~1λ^1≤-σ12λ~21+σ12λ21.(21)

      因此,可得

      V·1≤-k1z21-k1z1q+1+y22+z22-σ12λ~21+σ12λ21+a212+14+14δ21.(22)

      第i步的設(shè)計(jì)過程如下:

      z·i=Fi(xi+1)+yi+1+zi+1+αi-ω·i+di(t,h,x),(23)

      V·i=zi[Fi(xi+1)+yi+1+zi+1+αi-ω·i+

      di(t,h,x)]+λ~iλ^·i.(24)

      令Φi(Zi)=Fi(xi+1)-ω·i+zi[Δi1(‖xi‖)+Δi2α-11(r+D0)]2,其中Zi=[xTi+1ziyir]T.構(gòu)造虛擬控制律和自適應(yīng)律如下:

      αi=-kizi-kizqi+‖φi(Zi)‖2ziλ^i2a2i,(25)

      λ^·i=‖φi(Zi)‖2z2i2a2i-σiλ^i,(26)

      式中: σi>0、ki>0是設(shè)計(jì)常數(shù).將式(25)和(26)代入式(24),可得

      V·i≤-ki-32z2i-kiziq+1+14δ2i+z2i+1+

      y2i+1+12a2i-σi2λ~2i+σi2λ2i+14.(27)

      由控制誤差定義,可知y·i=ω·i-α·i-1=-yiτi-α·i-1,存在非負(fù)函數(shù)ηi(zi+1,yi+1,λ^i,yd,y·d,y··d,r,kc(t),k·c(t),k··c(t),,·,··)滿足

      |y·i+yiτi|≤ηi(zi+1,yi+1,λ^i,yd,y·d,y··d,r,kc(t),??? k·c(t),k··c(t),,·,··).(28)

      因而,可以得到

      yiy·i≤-1τiy2i+y2i+14η2i.(29)

      第n步的設(shè)計(jì)過程如下: 由zn=xn-ωn+xn+1/J,可得z·n=Fn(xn)+u+dn(h,x,t)-ω·n.類似地,設(shè)計(jì)

      u=-knzn-knzqn+‖φn(Zn)‖2znλ^n2a2n,(30)

      λ^·n=‖φn(Zn)‖2z2n2a2n-σnλ^n,(31)

      式中:σn>0、kn>0是設(shè)計(jì)常數(shù).由式(30)、(31),可得

      V·n≤-(kn-12)z2n-knznq+1+14δ2n+12a2n-

      σn2λ~2n+σn2λ2n+14.(32)

      由y·n=-ynτn-α·n-1可知存在非負(fù)函數(shù)ηn(zn,ynλ^n,yd,y·d,y··d,r,kc(t),k·c(t),k··c(t),,·,··)滿足|y·n+ynτn|≤ηn(zn,yn,λ^n,yd,y·d,y··d,r,kc(t),k·c(t),k··c(t),,·,··).因而,可以得到

      yny·n≤-1τny2n+y2n+14η2n.(33)

      4? 主要結(jié)果

      構(gòu)造總的李雅普諾夫函數(shù)如下:

      V(X)=∑ni=1Vi+∑ni=212y2i.(34)

      定義緊集:

      Ω={[zTn? yTn? λ~Tn]T:V≤P}Rpn,(35)

      Ωkc={[kc? k·c? k··c]T:k2c+k·2c+k··2c≤Pkc},(36)

      式中:X=zTn? yTn? λTnT;P是任意給定常數(shù);pn=3n-1.連續(xù)函數(shù)ηi(·)和δi(·)在緊集Ω×Ωkc上的最大值分別為Mi和Ni.

      定理1? 考慮由系統(tǒng)(1)、假設(shè)1和3、控制律(30),虛擬控制律(17)和(25),以及自適應(yīng)律(18)、(26)和(31)構(gòu)成的閉環(huán)系統(tǒng).如果初始條件滿足V(0)≤P,則存在設(shè)計(jì)常數(shù)σi、τi和ki使得閉環(huán)系統(tǒng)所有信號(hào)有界,且跟蹤誤差e(t)能收斂到預(yù)設(shè)緊集(-kc(t),kc(t)),其中設(shè)計(jì)常數(shù)σi、τi和ki滿足下面的不等式:

      ki>52+a2,1≤i≤n,

      1τi>94+a2,2≤i≤n,

      a=min12σi,1≤i≤n,

      b=min(2mkmin,2m-2σi,2m-2).(37)

      證明? 引入常數(shù)m∈(0,1],m=μ+12,對V求導(dǎo)并做如下變換:

      V·≤-∑ni=1ki-52z2i-kmin∑ni=1z2mi+14∑ni=1δ2i-

      ∑ni=21τi-2y2i-∑ni=1σi2λ~2i+∑ni=1σi2λ2i-

      ∑ni=114σi(λ~2i)m+∑ni=114σi(λ~2i)m-∑ni=214y2im+

      ∑ni=214(y2i)m+14∑ni=2η2i+12∑ni=1a2i+n4,(38)

      式中: kmin=min{ki}>0,i=1,2,…,n.

      14σiλ~2im≤14σiλ~2i+14σi1-mmm1-m,

      14y2im≤14y2i+141-mmm1-m.(39)

      將式(39)代入式(38),得到

      V·≤-∑ni=1ki-52z2i-kmin∑ni=1z2mi+14∑ni=1δ2i-

      ∑ni=21τi-2y2i-∑ni=1σi2λ~2i+∑ni=1σi2λ2i-

      ∑ni=114σi(λ~2i)m+14σiλ~2i-∑ni=214(y2i)m+

      14y2i+14(1+σi)(1-m)mm1-m+

      14∑ni=2η2i+12∑ni=1a2i+n4.(40)

      如果V(X)≤P,zn、yn、λ^n是有界的.由式(25)和(30),得到αi和u是有界的,由yi=ωi-hi-1,得到ωi是有界的.根據(jù)xi=zi+yi+αi-1,得到xi有界.因此ηi(·)和δi(·)在緊集Ω×Ωkc上分別有最大值Mi和Ni.因此,可以得到

      V·≤-aV-bVm+μ,(41)

      式中:μ=12∑ni=1a2i+∑ni=1σi2λ2i+∑ni=114N2i+∑ni=214M2i+4(1+σi)(1-m)mm1-m+n4.

      如果V(X)=P及滿足b>μPm,由于a>0,則V·(X)≤0.因而對于t≥0,如果V(X(0))≤P,則存在V(X)≤P.因此,如果V(X(0))≤P初始條件滿足,類似于上述的討論,由引理2,可以得到閉環(huán)系統(tǒng)所有信號(hào)在有限時(shí)間內(nèi)半全局一致終結(jié)有界,有限時(shí)間為Tr=1a(1-m)lnaV1-m(0)+νbνb.

      5? 仿真結(jié)果

      本節(jié)針對動(dòng)態(tài)模型

      z·=-z+0.5x21,

      x·1=x31+x2+z2sin x1,

      x·2=x1x22+u(t-τ)+x1z,

      y=x1,(42)

      給出仿真實(shí)例,驗(yàn)證所提出方案的有效性.

      期望軌跡選為yd=0.5sin t.輸入時(shí)滯τ=0.1 s,(t)=0.7e-3t+0.1;時(shí)變約束函數(shù)kc(t)=0.7+0.2cos t,動(dòng)態(tài)信號(hào)設(shè)計(jì)為r·=-r+1.5x31+0.8.

      初始條件設(shè)置如下:x1(0)=0.2,x2(0)=0.1,x3(0)=0.15,λ^1(0)=0.1,λ^2(0)=0.6,λ^3(0)=0.3,ω2(0)=0.1,r(0)=0.3,z(0)=0.1,l1=l2=10.

      設(shè)計(jì)參數(shù)設(shè)置如下:k1=2,k2=50,a1=a2=1,σ1=0.1,σ2=0.1,τ2=0.3,m=0.5.仿真結(jié)果如圖1-4所示.

      6? 結(jié)? 論

      1) 文中針對具有輸入時(shí)滯、未建模動(dòng)態(tài)和預(yù)設(shè)性能的非嚴(yán)格反饋非線性系統(tǒng),提出了一種新的有限時(shí)間自適應(yīng)控制方案.

      2) 利用Pade逼近和輔助中間變量處理輸入時(shí)滯,結(jié)合相應(yīng)的Lyapunov函數(shù),解決了具有輸入時(shí)滯系統(tǒng)有限時(shí)間控制設(shè)計(jì)和穩(wěn)定性分析問題.利用雙曲正切函數(shù)的特性構(gòu)造了一個(gè)可逆非線性映射處理預(yù)設(shè)性能.

      3) 理論分析表明,控制系統(tǒng)所有信號(hào)在有限時(shí)間內(nèi)都有界,系統(tǒng)的跟蹤誤差可以收斂到規(guī)定的時(shí)變區(qū)域.

      [WT5HZ]參考文獻(xiàn)(References)[WT5”BZ]

      [1]? 付海燕,雷騰飛,賀金滿,等.具有非線性時(shí)滯項(xiàng)的分?jǐn)?shù)階混沌系統(tǒng)ADM求解與動(dòng)力學(xué)分析[J].吉林大學(xué)學(xué)報(bào)(理學(xué)版),2022,60(2):432-438.

      FU H Y, LEI T F, HE J M, et al. ADM solution and dynamic analysis of fractional-order chaotic systems with nonlinear delay[J]. Journal of Jilin University(Science Edition),2022,60(2): 432-438. (in Chinese)

      [2]? 王春彥,邸金紅,毛北行.不確定大氣分?jǐn)?shù)階混沌系統(tǒng)的自適應(yīng)滑模同步[J].吉林大學(xué)學(xué)報(bào)(理學(xué)版),2022,60(2):439-444.

      WANG C Y, DI J H, MAO B X. Adaptive sliding mode synchronization of uncertain fractional-order atmospheric chaotic system[J]. Journal of Jilin University(Science Edition), 2022,60(2):439-444. (in Chinese)

      [3]? YANG R M, SUN L Y. Finite-time robust control of a class of nonlinear time-delay systems via Lyapunov functional method[J]. Journal of Franklin Institute, 2019,356(3):1155-1176.

      [4]? 劉洋,井元偉,劉曉平,等. 非線性系統(tǒng)有限時(shí)間控制研究綜述[J]. 控制理論與應(yīng)用,2020,37(1):1-12.

      LIU Y, JING Y W, LIU X P, et al. Survey on finite-time control for nonlinear systems[J]. Control Theory and Applications, 2020,37(1):1-12.(in Chinese)

      [5]? LI H Y, ZHAO S Y, HE W, et al. Adaptive finite-time tracking control of full state constrained nonlinear systems with dead-zone[J]. Automatica,2019,100:99-107.

      [6]? ZHANG Y, WANG F, ZHANG J. Adaptive finite-time tracking control for output-constrained nonlinear systems with non-strict-feedback structure[J]. International Journal of Adaptive Control and Signal Processing, 2020,34(4):560-574.

      [7]? JIN X. Adaptive fixed-time control for MIMO nonlinear systems with asymmetric output constraints using universal barrier functions[J]. IEEE Transactions on Automa-tic Control, 2019,64(7):3046-3053.

      [8]? LI Y M, LI K W, TONG S C. Finite-time adaptive fuzzy output feedback dynamic surface control for MIMO nonstrict feedback systems[J]. IEEE Transactions on Fuzzy Systems,2019,27(1):96-110.

      [9]? LIU L, GAO T T, LIU Y J, et al. Time-varying asymmetrical BLFs based adaptive finite-time neural control of nonlinear systems with full state constraints[J]. IEEE/CAA Journal of Automatica Sinica, 2020,7(5):1335-1343.

      [10]? LI Y X. Finite time command filtered adaptive fault to-lerant control for a class of uncertain nonlinear systems[J].Automatica, 2019,106:117-123.

      [11]? YU J P, ZHAO L, YU H S, et al. Fuzzy finite-time command filtered control of nonlinear systems with input saturation[J]. IEEE Transactions on Cybernetics, 2018,48(8):2378-2387.

      [12]? LIU C G, WANG H Q, LIU X P, et al. Adaptive prescribed performance tracking control for strict-feedback nonlinear systems with zero dynamics[J]. International Journal of Robust and Nonlinear Control,2019,29(18):6507-6521.

      [13]? LI D P, LIU Y J, TONG S C, et al. Neural networks-based adaptive control for nonlinear state constrained sys-tems with input delay[J]. IEEE Transactions on Cybernetics, 2019,49(4):1249-1258.

      [14]? WANG W, LIANG H J, ZHANG Y H, et al. Adaptive cooperative control for a class of nonlinear multi-agent systems with dead zone and input delay[J]. Nonlinear Dynamics,2019,96(4):2707-2719.

      [15]? ZHANG T P, XIA M Z, ZHU J M. Adaptive backstepping neural control of state-delayed nonlinear systems with full-state constraints and unmodeled dynamics[J]. International Journal of Adaptive Control and Signal Processing, 2017,31(11):1704-1722.

      (責(zé)任編輯? 梁家峰)

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