隨機(jī)系統(tǒng)的多模型直接自適應(yīng)解耦控制器

1、第36卷第9期自動(dòng)化學(xué)報(bào)Vol.36,No.9 2010年9月ACTA AUTOMATICA SINICA September,2010隨機(jī)系統(tǒng)的多模型直接自適應(yīng)解耦控制器鄭益慧1王昕1李少遠(yuǎn)2姜建國(guó)3摘要針對(duì)多變量離散時(shí)間隨機(jī)系統(tǒng),提出了一種采用廣義最小方差性能指標(biāo)的多模型直接自適應(yīng)解耦控制器.該多模型控制器由多個(gè)固定控制器和兩個(gè)自適應(yīng)控制器構(gòu)成.固定控制器用以覆蓋系統(tǒng)參數(shù)的可能變化范圍,自適應(yīng)控制器用以保證系統(tǒng)的穩(wěn)定性和提高暫態(tài)性能.該多模型控制器利用矩陣的偽交換性和擬Diophantine方程性質(zhì),基于廣義最小方差性能指標(biāo),將隨機(jī)系統(tǒng)辨識(shí)算法和最優(yōu)控制器設(shè)計(jì)相結(jié)合,直接辨識(shí)出控制器的參數(shù)

2、,通過(guò)廣義最小方差性能指標(biāo)中加權(quán)多項(xiàng)式的選取,不但實(shí)現(xiàn)了多變量系統(tǒng)的動(dòng)態(tài)解耦控制,而且消除了穩(wěn)態(tài)誤差、配置了閉環(huán)極點(diǎn).文末給出了全局收斂性分析.仿真結(jié)果表明該方法明顯優(yōu)于常規(guī)自適應(yīng)控制器.關(guān)鍵詞多模型,隨機(jī)系統(tǒng),直接自適應(yīng)控制,動(dòng)態(tài)解耦DOI10.3724/SP.J.1004.2010.01295Multiple Models Direct Adaptive Decoupling Controller for a Stochastic System ZHENG Yi-Hui1WANG Xin1LI Shao-Yuan2JIANG Jian-Guo3Abstract For a multivar

3、iable discrete-time stochastic system,a multiple models direct adaptive decoupling controller based on generalized minimum variance performance index is presented.It is composed of multiplexed models and two adaptive models.Thexed models are used to cover the region where the system parameters jump,

4、while the adaptive models are used to guarantee the stability and improve the transient response.For thexed models,it utilizes the matrix pseudo-commutativity and quasi-Diophantine equation to design the generalized minimum variance controller.For the adaptive models,it adopts the stochastic system

5、identication algorithm with optimal controller design method to identify the controller parameter directly.Then,through the choice of the weighting polynomial matrices,it not only realizes the dynamic decoupling control but also eliminates the steady state error and places the closed-loop poles arbi

6、trarily.Finally, the global convergence is given.The simulation proves the eectives of the controller proposed.Key words Multiple models,stochastic system,direct adaptive control,dynamic decoupling多模型控制器的設(shè)計(jì)可以追溯到70年代.從1971年Lainiotis提出的基于后驗(yàn)概率加權(quán)的多模型控制器13,到目前基于切換指標(biāo)的多模型自適應(yīng)控制器46,已經(jīng)經(jīng)歷了近40年的發(fā)展.1984年, Badr等提

7、出了加權(quán)多模型控制器,每個(gè)控制器采用二次型最優(yōu)指標(biāo)進(jìn)行設(shè)計(jì),控制器輸出采用加權(quán)和形式7.該方法本質(zhì)上相當(dāng)于一種軟切換8,因而切換過(guò)程比較平滑,對(duì)系統(tǒng)的執(zhí)行機(jī)構(gòu)損害小,易于收稿日期2009-02-23錄用日期2010-04-16Manuscript received February23,2009;accepted April16,2010國(guó)家高技術(shù)研究發(fā)展計(jì)劃(863計(jì)劃(2008AA04Z129,國(guó)家自然科學(xué)基金(60504010,60864004,60774015,上海市教委支出預(yù)算項(xiàng)目(2008093,上海市教委科研創(chuàng)新項(xiàng)目(09YZ241資助Supported by National

8、High Technology Research and Devel-opment Program of China(863Program(2008AA04Z129,Na-tional Natural Science Foundation of China(60504010,60864 004,60774015,Disbursal Budget Program of Shanghai Munic-ipal Education Commission of China(2008093,and Innovation Program of Shanghai Municipal Education Co

9、mmission of China (09YZ2411.上海交通大學(xué)電工與電子技術(shù)中心上海2002402.上海交通大學(xué)自動(dòng)化系上海2002403.上海交通大學(xué)電氣系上海2002401.Center of Electrical and Electronic Technology,Shanghai Jiao Tong University,Shanghai2002402.Department of Automation,Shanghai Jiao Tong University,Shanghai2002403.Department of Electrical Engineering,Shangha

10、i Jiao Tong University,Shanghai200240在實(shí)際工業(yè)過(guò)程中使用9,但也正因?yàn)椴捎每刂破鬏敵龅募訖?quán)和形式,因而難以得到穩(wěn)定性、收斂性等理論證明.1986年,Fu等針對(duì)狀態(tài)空間模型,構(gòu)造多個(gè)不同的狀態(tài)反饋建立多模型集,按編號(hào)順序從小到大進(jìn)行控制器切換,成功地保證了閉環(huán)系統(tǒng)是指數(shù)穩(wěn)定的10.但由于采用了順序切換方式,導(dǎo)致系統(tǒng)的過(guò)渡過(guò)程很差11,難以在實(shí)際中得到應(yīng)用. 1994年,Narendra等為了提高系統(tǒng)的暫態(tài)性能,采用多個(gè)初值不同的自適應(yīng)模型構(gòu)成多模型集,然后根據(jù)切換指標(biāo)選出最優(yōu)模型,進(jìn)而選出與之相對(duì)應(yīng)的控制器實(shí)行切換控制12.該控制器在系統(tǒng)首次發(fā)生跳變時(shí)可以提

11、高系統(tǒng)的暫態(tài)性能,但當(dāng)多個(gè)初值不同的自適應(yīng)模型收斂到同一鄰域時(shí),會(huì)退化成常規(guī)自適應(yīng)控制器而喪失了多模型的優(yōu)點(diǎn).為了解決上述問(wèn)題,文獻(xiàn)13采用多個(gè)固定模型構(gòu)成多模型集覆蓋系統(tǒng)參數(shù)變化的可能范圍,但由于固定模型不具有自適應(yīng)能力而無(wú)法消除系統(tǒng)的穩(wěn)態(tài)誤差.文獻(xiàn)14在上述多固定模型的基礎(chǔ)上加入一個(gè)自適應(yīng)模型解決了上述問(wèn)題,同時(shí)給出了全局收斂性證明.為了進(jìn)一步加快系統(tǒng)的辨識(shí)速度,提高系統(tǒng)的暫態(tài)性能,文獻(xiàn)15在上述模型集的基礎(chǔ)上,再加入一個(gè)可重新賦值的自適應(yīng)模型.該自適應(yīng)模型的參數(shù)值1296自動(dòng)化學(xué)報(bào)36卷可被重新賦值為最優(yōu)模型的參數(shù)值以加快系統(tǒng)的辨識(shí)速度,同時(shí)給出全局收斂性證明.但以上方法都是針對(duì)連續(xù)時(shí)間

12、系統(tǒng).1998年,Narendra等將上述方法推廣到離散時(shí)間系統(tǒng)16.文獻(xiàn)1719采用直接算法、分層遞階和逐維定位策略優(yōu)化多模型集的數(shù)目,減少計(jì)算量,但上式方法都是針對(duì)確定性系統(tǒng),沒(méi)有考慮到實(shí)際過(guò)程中噪聲的影響,因而難以在工業(yè)生產(chǎn)中得到應(yīng)用.為此,文獻(xiàn)20考慮有色噪聲的影響,將以上結(jié)果推廣到隨機(jī)系統(tǒng).但該方法僅限于單輸入單輸出系統(tǒng),采用辨識(shí)系統(tǒng)參數(shù)的間接自適應(yīng)算法,不但加大了計(jì)算量,而且容易造成矩陣方程求解的病態(tài)問(wèn)題21.另外,該方法的性能指標(biāo)采用最小方差形式,沒(méi)有對(duì)控制輸入加以限制,往往造成性能指標(biāo)最優(yōu)時(shí),控制輸入要求過(guò)大,在實(shí)際工業(yè)過(guò)程中會(huì)造成閥門開(kāi)度超過(guò)最大值而無(wú)法實(shí)現(xiàn)22.本文針對(duì)多變

13、量離散時(shí)間隨機(jī)系統(tǒng),提出了一種采用廣義最小方差性能指標(biāo)的多模型直接自適應(yīng)解耦控制器.該多模型控制器由多個(gè)固定控制器和兩個(gè)自適應(yīng)控制器構(gòu)成.固定控制器利用矩陣的偽交換性和擬Diophantine方程性質(zhì),基于廣義最小方差性能指標(biāo)進(jìn)行設(shè)計(jì),用以覆蓋系統(tǒng)參數(shù)的可能變化范圍.自適應(yīng)控制器是將隨機(jī)系統(tǒng)辨識(shí)算法和最優(yōu)控制器設(shè)計(jì)相結(jié)合,直接辨識(shí)出控制器參數(shù)的直接算法,用以保證系統(tǒng)的穩(wěn)定性和提高暫態(tài)性能.最后通過(guò)廣義最小方差性能指標(biāo)中加權(quán)多項(xiàng)式的選取,不但實(shí)現(xiàn)了多變量系統(tǒng)的動(dòng)態(tài)解耦控制,而且消除了穩(wěn)態(tài)誤差、配置了閉環(huán)極點(diǎn).文末給出了全局收斂性分析.仿真結(jié)果表明與常規(guī)自適應(yīng)控制器相比,無(wú)論模型參數(shù)發(fā)生跳變,還是

14、動(dòng)態(tài)解耦能力都得到明顯改善.1被控系統(tǒng)描述設(shè)多輸入多輸出線性離散時(shí)間隨機(jī)系統(tǒng)用下述ARMAX模型描述A(t,z1y(t+k=B(t,z1u(t+C(t,z1(t+k(1式中,u(t,y(t分別為n維輸入、輸出向量,(t為n維獨(dú)立同分布白噪聲向量,滿足:E(t|F t1=0,a.s.(2E (t |F t1=2,a.s.(3lim Nsup1NNt=1(t 2<,a.s.(4式中,a.s.是almost surely的縮略語(yǔ),F t表示非降子代數(shù)族,A(t,z1,B(t,z1,C(t,z1是時(shí)間t 和單位后移算子z1的矩陣多項(xiàng)式,具有如下形式A(t,z1=I+A1(tz1+·&#

15、183;·+A na(tzn a(5B(t,z1=B0(t+B1(tz1+···+B nb(tzn b(6C(t,z1=I+C1(tz1+···+C na(tzn a(7且t,B0(t非奇異,k為系統(tǒng)的傳輸時(shí)延.系統(tǒng)滿足如下假設(shè):假設(shè)1.系統(tǒng)為時(shí)不變系統(tǒng)或含跳變參數(shù)的時(shí)變系統(tǒng),同時(shí)假設(shè)相鄰跳變時(shí)間間隔足夠長(zhǎng),系統(tǒng)在此期間內(nèi)參數(shù)保持不變;假設(shè)2.t變化時(shí),A(t,z1,B(t,z1,C(t, z1構(gòu)成的參數(shù)矩陣在一緊集中變化;假設(shè)3.A(t,z1,B(t,z1,C(t,z1的階次上限n a,n b,n a和時(shí)延k已知;假設(shè)4.

16、C(t,z1是穩(wěn)定的,即detC(t,z1 =0,|z|1,t;假設(shè)5.系統(tǒng)是最小相位系統(tǒng),即detB(t,z1 =0,|z|1,t.由假設(shè)1可知,A i(t,B j(t和C k(t為常值矩陣(時(shí)不變系統(tǒng)或分段常值矩陣(含跳變參數(shù)的時(shí)變系統(tǒng).因此,在相鄰跳變時(shí)間間隔內(nèi),系統(tǒng)可寫成時(shí)不變形式而不失一般性:A(z1y(t+k=B(z1u(t+C(z1(t+k(8下文將針對(duì)系統(tǒng)(8進(jìn)行研究.2多模型自適應(yīng)控制器針對(duì)被控系統(tǒng)模型的各個(gè)參數(shù)在一定范圍內(nèi)變化的特點(diǎn),首先確定系統(tǒng)模型參數(shù)的變化區(qū)域,然后將該變化區(qū)域劃分為m個(gè)小的子區(qū)域,在每個(gè)子區(qū)域中建立一個(gè)固定模型,得到m個(gè)固定模型.接著對(duì)每個(gè)固定模型設(shè)計(jì)

17、最優(yōu)控制器,得到m個(gè)參數(shù)不變的固定控制器.在此基礎(chǔ)上,添加一個(gè)常規(guī)自適應(yīng)控制器保證系統(tǒng)的穩(wěn)定性,再添加一個(gè)可重新賦值的自適應(yīng)控制器用以提高系統(tǒng)的暫態(tài)性能.這樣得到的m個(gè)固定控制器與2個(gè)自適應(yīng)控制器一起構(gòu)成多控制器集,基于切換指標(biāo)選取最優(yōu)控制器作為當(dāng)前控制器進(jìn)行控制.2.1m個(gè)固定控制器2.1.1m個(gè)系統(tǒng)參數(shù)模型定義 1.由矩陣多項(xiàng)式A(z1,B(z1和C(z1的各系數(shù)矩陣構(gòu)成的矩陣稱為系統(tǒng)參數(shù)模型.(t所有取值構(gòu)成的集合稱為系統(tǒng)參數(shù)模型集.將系統(tǒng)參數(shù)模型集分為m個(gè)模型子集s,并且s滿足1s,s非空,s=1,···,m;9期鄭益慧等:隨機(jī)系統(tǒng)的多模型直接自適應(yīng)解耦控

18、制器12972ms=1s;3s,ss,0r s<,滿足 s r s,稱s為模型子集s的中心,r s為半徑,其存在性由假設(shè)2保證.這樣建立的m個(gè)模型s(s=1,···,m就構(gòu)成m個(gè)固定模型.2.1.2廣義最小方差控制器的設(shè)計(jì)設(shè)性能指標(biāo)為J p=E P(z1y(t+kR(z1y(t+Q(z1u(t 2(9式中,y(t為n維期望輸出向量,P(z1,Q(z1, R(z1為加權(quán)多項(xiàng)式矩陣,且滿足P(z1穩(wěn)定.針對(duì)式(8中的A(z1和式(9中的P(z1,引入¯A(z1,¯P(z1滿足矩陣的偽交換性23¯A(z1P(z1=¯P(z1

19、A(z1(10式中,¯A(z1,¯P(z1滿足det¯A(z1=detA(z1(11det¯P(z1=detP(z1(12令¯C(z1=¯P(z1C(z1(13對(duì)于式(10中的¯A(z1,引入Diophantine方程24,得¯C(z1=¯A(z1F(z1+zk G(z1(14同理,針對(duì)式(14中F(z1,G(z1,引入¯F(z1,¯G(z1滿足矩陣的偽交換性23,有¯F(z1G(z1=¯G(z1F(z1(15且滿足det¯F(z1=detF(z1(16d

20、et¯G(z1=detG(z1(17利用上述矩陣多項(xiàng)式構(gòu)造C(z1,滿足C(z1=¯F(z1¯A(z1+zk¯G(z1(18下面用三個(gè)定理給出C(z1的性質(zhì).定理1.擬Diophantine方程性質(zhì).C(z1P(z1=F(z1A(z1+zkG(z1(19式中F(z1=¯F(z1¯P(z1(20G(z1=¯G(z1¯P(z1(21證明.用P(z1右乘式(18,考慮式(10得C(z1P(z1=¯F(z1¯A(z1P(z1+zk¯G(z1P(z1=¯F(z1¯P(z1A(

21、z1+zk¯G(z1P(z1=F(z1A(z1+zkG(z1定理2.偽交換性.¯C(z1F(z1=¯F(z1C(z1(22證明.用F(z1右乘式(18,利用式(13(15和式(20,可得C(z1F(z1=¯F(z1¯A(z1F(z1+zk¯G(z1F(z1=¯F(z1¯A(z1F(z1+zk¯F(z1G(z1=¯F(z1¯A(z1F(z1+zk G(z1=¯F(z1¯P(z1C(z1=F(z1C(z1定理3.穩(wěn)定性.C(z1是穩(wěn)定的,即detC(z1=0,|z|1(

22、23證明.由式(20和式(22可知C(z1F(z1=F(z1C(z1=¯F(z1¯P(z1C(z1考慮式(12,(16及假設(shè)條件P(z1、C(z1穩(wěn)定,可得detC(z1=detP(z1detC(z1=0,|z|1為了求得最優(yōu)控制律和系統(tǒng)閉環(huán)方程,用F(z1左乘式(8,得F(z1A(z1y(t+k=F(z1B(z1u(t+F(z1C(z1(t+k(24利用定理1和定理2,可得C(z1P(z1y(t+k=G(z1y(t+F(z1B(z1u(t+F(z1C(z1(t+k=1298自動(dòng)化學(xué)報(bào)36卷G(z1y(t+F(z1B(z1u(t+C(z1F(z1(t+k即C(z1P(z1y

23、(t+kF(z1(t+k=G(z1y(t+F(z1B(z1u(t(25定義y0(t+k=P(z1y(t+kF(z1(t+k(26代入性能指標(biāo)(9中,并考慮式(2,得J P=E P(z1y(t+kR(z1y(t+Q(z1u(t 2=E y0(t+k+F(z1(t+kR(z1y(t+Q(z1u(t 2=y0(t+kR(z1y(t+Q(z1u(t 2+E F(z1(t+k 2為使性能指標(biāo)最小,可得y0(t+kR(z1y(t+Q(z1u(t=0(27此時(shí),性能指標(biāo)最小值為J pmin =k1i=0f i2(28將式(27代入式(25和式(26中,可得最優(yōu)控制律為C(z1R(z1y(tQ(z1u(t=G

24、(z1y(t+F(z1B(z1u(t即F(z1B(z1+C(z1Q(z1u(t+G(z1y(t=C(z1R(z1y(t(29聯(lián)立式(28和式(29,利用定理1和定理2,得G(z1y(t+F(z1A(z1y(t+k+C(z1Q(z1B1(z1A(z1y(t+kF(z1C(z1(t+kC(z1Q(z1B1(z1C(z1(t+k=C(z1R(z1y(tC(z1P(z1+Q(z1B1(z1A(z1y(t+kC(z1F(z1+Q(z1B1(z1C(z1(t+k=C(z1R(z1y(t由定理3,可得系統(tǒng)的閉環(huán)方程為P(z1+Q(z1B1(z1A(z1y(t+kF(z1+Q(z1B1(z1C(z1(t+k=

25、 R(z1y(t(30由假設(shè)5知B(z1穩(wěn)定,因此可以令Q(z1=R1B1(z1(31P(z1+R1A1(z1=T(z1(32R(z1=T(z1(33式中,R1為定常矩陣,T(z1=T0+T1(z1+···+T nt(zn t為穩(wěn)定的對(duì)角形多項(xiàng)式矩陣,其零點(diǎn)為期望的閉環(huán)系統(tǒng)極點(diǎn),滿足n p=n a,n tn a,則系統(tǒng)閉環(huán)方程為y(t+k=y(t+T1(z1F(z1+R1C(z1(t+k(34因此通過(guò)上述加權(quán)多項(xiàng)式矩陣的選取,系統(tǒng)不但配置了閉環(huán)極點(diǎn),而且實(shí)現(xiàn)了動(dòng)態(tài)解耦.式(29得到的即為固定控制器的最優(yōu)控制律,但為了和自適應(yīng)控制器采用相同的結(jié)構(gòu)形式,下面給出該控制

26、器的另一種形式.首先用C(z1R1左乘式(8,得C(z1R1A(z1y(t+k=C(z1R1B(z1u(t+C(z1R1C(z1(t+k(35與式(25相加,得C(z1P(z1+R1A(z1y(t+kF(z1+R1C(z1(t+k=G(z1y(t+F(z1B(z1+C(z1R1B(z1u(t(36考慮式(31和式(32,得C(z1T(z1y(t+kF(z1+R1C(z1(t+k=G(z1y(t+F(z1B(z1+C(z1Q(z1u(t(37定義廣義輸出預(yù)測(cè)y0(t+k= y(t+k (t+k(38y0(t+k=T(z1y(t+k(39(t+k=F(z1+R1C(z1(t+k(409期鄭益慧等:

27、隨機(jī)系統(tǒng)的多模型直接自適應(yīng)解耦控制器1299代入式(37,得C(z1 y0(t+k=G(z1y(t+F(z1B(z1+C(z1Q(z1u(t(41即y0(t=H1(z1y(tk+H2(z1u(tk+H3(z1 y0(tk(42式中,H1(z1=G(z1(43H2(z1=F(z1B(z1+C(z1Q(z1(44H3(z1=C1+···+C nc(znc+1=zC(z1I(45為了求取最優(yōu)控制律,令y0(t= y(t(46y(t=T(z1y(t(47則最優(yōu)控制律為H1(z1y(t+H2(z1u(tH3(z1 y(t1= y(t(482.1.3m個(gè)固定控制器定義2.系統(tǒng)

28、參數(shù)模型經(jīng)過(guò)式(24,(25,(43(48變換后得到的矩陣H1(z1,H2(z1,H3(z1構(gòu)成的矩陣稱為控制器模型.對(duì)應(yīng)(t的所有(t的集合稱為控制器模型集.對(duì)應(yīng)(ts的(t的集合s稱為控制器模型子集.對(duì)應(yīng)ss的s稱為該模型子集的中心.因此,上述變換得到的m個(gè)控制器s,s=1,···,m就構(gòu)成了多模型集中的m個(gè)固定控制器.2.2常規(guī)自適應(yīng)控制器定義數(shù)據(jù)向量X0(tk和控制器矩陣0為X0(tk=y(tkT,···,y(tkn h1T,u(tkT,···,u(tkn h2T, y0(t1T,·&#

29、183;·, y0(tn h3TT(490=01,···,0n=H10,H11,···,H1nh1,H20,H21,···,H2nh2,H30,H31,···,H3nh3T(500i=h10i1,···,h10in,h11i1,···,h11in,···,h20i1,···,h20in,h21 i1,···,h21in

30、,···,h30i1,···,h30in,h31 i1,···,h31in,···T,i=1,···,n(51則方程(42可以寫為y0(t=TX0(tk(52由式(52可以定義自適應(yīng)控制器的辨識(shí)方程為 y(t=(tkT X(tk(53式中,(tk=1(tk,···,n(tk=H10(tk,H11(tk,···,H1nh1(tk,H20(tk,H21(tk,··

31、3;,H2nh2(tk,H30(tk,H31(tk,···,H3nh3(tkT(54i(tk=h10i1(tk,···,h20i1(tk,···,h30i1(tk,···T,i=1,2,···,n(55X(tk=y(tkT,···,y(tkn h1T,u(tkT,···,u(tkn h2T,¯ y(t1T,···,¯ y(tkn h3TT

32、(56式中,¯ y(t=(tT X(tk(57表示廣義輸出 y(t+k=T(z1y(t+k的后驗(yàn)預(yù)報(bào).采用如下辨識(shí)算法i(t=i(t1+X(tkr(tk1+X(tkT X(tk× yi(t yi(t(58 yi(t=X(tkTi(t1(59r(tk=r(tk1+X(tkT X(tk,r(1,r(2,···,r(d>0(60為了求取最優(yōu)控制律,令 y(t= y(t(61則最優(yōu)控制律為H1(z1y(t+H2(z1u(tH1(z1 y(t1= y(t(622.3可重新賦值自適應(yīng)控制器1當(dāng)可重新賦值自適應(yīng)控制器是當(dāng)前選定的最優(yōu)控制器時(shí),采用式(5

33、8(60進(jìn)行辨識(shí),以獲得控制器參數(shù),最優(yōu)控制律可由式(62給出.2當(dāng)可重新賦值自適應(yīng)控制器不是當(dāng)前選定的最優(yōu)控制器時(shí),則將可重新賦值自適應(yīng)控制器的參數(shù)初值賦值為當(dāng)前選定的最優(yōu)控制器的參數(shù)值.定義3.多模型控制器由m 個(gè)參數(shù)已知的控制器s ,s =1,···,m 及常規(guī)自適應(yīng)控制器m +1和可重新賦值的自適應(yīng)控制器m +2構(gòu)成.2.4最優(yōu)控制器對(duì)于上述m +2個(gè)控制器,采用如下切換指標(biāo):J s =1N Nt =1e s (t 2,s =1,···,m +2(63e s (t =y s (t y (t (64式中,y s (t =T (z

34、 1y s (t 為第s 個(gè)模型的廣義輸出, y (t =T (z 1y (t 為系統(tǒng)期望的廣義輸出,則 e s (t = y s (t y (t 為第s 個(gè)模型的廣義輸出誤差.為了選取最優(yōu)控制器,首先,類似于式(53自適應(yīng)控制器的表達(dá)形式,將所有m +2個(gè)控制器寫為y s (t =T s X (t k ,s =1,···,m +2(65式中,s 為第s 個(gè)模型的控制器參數(shù)矩陣.然后,任一時(shí)刻,根據(jù)切換指標(biāo),選取廣義輸出誤差最小的控制器作為當(dāng)前控制器實(shí)行控制.即,若J j =min(J s ,s =1,···,m +2(66記作j =

35、arg min(J s ,s =1,···,m +2(67則選取j 為當(dāng)前控制器實(shí)現(xiàn)控制.3全局收斂性分析為了分析上述多模型控制器的全局收斂性,現(xiàn)增加如下假設(shè):假設(shè)6. 1C (z 112是嚴(yán)格正實(shí)的.下面給出全局收斂性的證明.引理1.對(duì)于隨機(jī)系統(tǒng)(8,有1lim N 1NN t =1u (t 2lim N K 1NNt =1y (t 2+K 2(682lim N 1NN t =1y (t 2lim N K 3NNt =1y (t 2+K 4(693lim N 1NNt =1y (t 2lim N K 5NNt =1e (t (t 2+K 6(70式中,e (t

36、=y (t y (t =T (z 1y (t T (z 1y (t (71(t +k =F (z 1+R 1C (z 1(t +k (724limN 1NNt =1¯y (t 2lim N K 7NNt =1e (t (t 2+K 8(7351N r (N 1K 9N Nt =1e (t (t 2+K 10(74證明.1在隨機(jī)系統(tǒng)(8中,由假設(shè)4和假設(shè)5可知B (z 1,C (z 1穩(wěn)定,又由式(3可知白噪聲均方有界,故由文獻(xiàn)25可證.2由式(39中y (t +k =T (z 1y (t +k ,及T (z 1穩(wěn)定可證.3由式(71知y (t =e (t + y (t = e (t

37、(t + y (t +(t (75y (t 2=e (t (t + y (t +(t 23| e(t (t 2+3 y(t 2+3 (t 2(76針對(duì)式(72及(t的性質(zhì)式(2(4,有E (t|F tna=0,a.s.(77 E (t |F t1=2,a.s.(78lim Nsup1NNt=1(t 2<,a.s.(79由式(79及期望廣義輸出 y(t有界,所以得證.4將式(58左乘X(tkT,有X(tkTi(t=X(tkTi(t1+X(tkT X(tkr(tk1+X(tkT X(tk× yi (t yi(t(80即¯ y i (t= yi(t+X(tkT X(tkr(

38、tk1+X(tkT X(tk× e i(t(81用 y i(t減式(81,寫成多變量形式,有(t= e(tX(tkT X(tkr(tk1+X(tkT X(tk× e(t(82(t= y(t¯ y(t=T(z1y(tT(z1¯y(t(83由式(60,得(t=r(tk1r(tke(t(84代入式(83,有¯ y(t 2= y(t (t 2=y(tr(tk1r(tke(t 2=y(tr(tk1r(tk e (t+ (t 23 y(t 2+3 r(tk1r(tk e (t 2+3 r(tk1r(tk(t 23 y(t 2+3 e (t 2+3 (t 2

39、(85由式(73可證.5由式(60可知,r(N1=r(N2+X(N1T X(N1(86則r(N1可求解得到r(N1=r(1+N1t=0X(tT X(t(87由X(t是由u(t、 y(t、¯ y(t構(gòu)成及式(68(73可證.引理2.對(duì)于常規(guī)自適應(yīng)模型,具有如下性質(zhì):1limN1NNt=1e m+1(t (t 2=0(882limN1NNt=1e m+1(t (t (t =0(89證明.1對(duì)于自適應(yīng)辨識(shí)算法(58(60,有25limN1NNt=1e m+1(t (tr(t1<,a.s.(90由式(74及隨機(jī)的基本引理25可證.2對(duì)于自適應(yīng)辨識(shí)算法(58(60,由式(74及隨機(jī)的基本

40、引理25,可以得到limNsup1Nr(N1<,a.s.(91由式(78,(90及(91,有Nt=11t2E e m+1(t (t 2· (t 2|F tna<(92由文獻(xiàn)25中引理D.5.1得證.定理4.在上述假設(shè)16下,多模型直接自適應(yīng)控制算法作用于隨機(jī)系統(tǒng)(8時(shí),有1limN1NNt=1e(t(t 2=0(93式中,e(t=y(ty(t(94(t=T1(z1 (t(95 2lim Nsup1NNt=1y(t 2<,a.s.(96lim Nsup1NNt=1u(t 2<,a.s.(97證明.1對(duì)于隨機(jī)系統(tǒng),類似于切換指標(biāo)(63,定義J c=1NNt=1e(

41、t 2(98e(t= y(t y(t=T(z1y(tT(z1y(t(99式中, y(t=T(z1y(t為被控系統(tǒng)實(shí)際的廣義輸出, y(t=T(z1y(t為被控系統(tǒng)期望的廣義輸出, e(t= y(t y(t為被控系統(tǒng)的廣義輸出誤差.展開(kāi)式(99,知J c=1NNt=1e(t (t 2+1NNt=1(t 2+2NNt=1e(t (t · (t (100由切換指標(biāo)(63知:任何時(shí)刻,J c=J jJ m+1,即1 NNt=1e(t (t 2+1NNt=1(t 2+2NNt=1e(t (t · (t 1NNt=1e m+1(t (t 2+1NNt=1(t 2+2NNt=1e m+1

42、(t (t · (t (101考慮上述不等式兩邊第2項(xiàng)相等,則1 NNt=1e(t (t 2+2NNt=1e(t (t · (t 1NNt=1e m+1(t (t 2+2NNt=1e m+1(t (t · (t (102由引理2中式(88和式(89,可知0limN1NNt=1e(t (t 2+2NNt=1e(t (t · (t limN1NNt=1e m+1(t (t 2+limN2NNt=1e m+1(t (t · (t =0(103由此可得limN2NNt=1e(t (t · (t =0(104limN1NNt=1e(t (t 2

43、=0(105將式(71,(94和(95代入,得limN1NNt=1T(z1e(t(t 2=0(106考慮到T(z1是穩(wěn)定的,可得limN1NNt=1e(t(t =02將式(105代入引理1中式(70,可知limNsup1NNt=1y(t 2=0(107由引理1中式(68和式(69,定理得證.4仿真實(shí)驗(yàn)多變量系統(tǒng)模型如下所示(I+A1z1+A2z2y(t=(B0+B1z1u(t2+(I+C1z1(t(108式中,A 1= 0.2240.0140.0090.179,A 2= 0.0170.0040.0020.008 ,B 0= 0.8962.1471.98510.041 ,B 1=0.4594.0

44、870.4232.209 ,C 1= 0.1 .參考輸入y = 105.圖1表示采用常規(guī)自適應(yīng)控制器時(shí)系統(tǒng)的響應(yīng),圖2表示采用多模型自適應(yīng)控制器(100個(gè)固定模型和2個(gè)自適應(yīng)模型時(shí)系統(tǒng)的響應(yīng),其中常規(guī)自適應(yīng)模型和多模型中的兩個(gè)自適應(yīng)模型的初值取值相同,并且距離參數(shù)真值很近.為了檢測(cè)系統(tǒng)的解耦效果,在t =40步時(shí),y 1單獨(dú)由10變?yōu)?;在t =80步時(shí),y 2單獨(dú)由5變?yōu)?;在t =120步時(shí),y 1,y 2同時(shí)發(fā)生變化.從圖1和圖2中可以看出,多模型自適應(yīng)控制器的解耦效果優(yōu)于常規(guī)自適應(yīng)控制器.同樣,為了檢測(cè)系統(tǒng)在參數(shù)發(fā)生跳變時(shí)的響應(yīng)情況,在t =160步時(shí),系統(tǒng)參數(shù)發(fā)生變

45、化;在t =200步時(shí),系統(tǒng)參數(shù)和參考輸入y 同時(shí)發(fā)生變化.從圖中可以明顯看出,特別當(dāng)系統(tǒng)參數(shù)發(fā)生變化時(shí),多模型自適應(yīng)控制器由于采用多個(gè)模型覆蓋系統(tǒng)參數(shù)的變化范圍,其效果(圖2遠(yuǎn)遠(yuǎn)優(yōu)于常規(guī)自適應(yīng)控制器(圖1.5結(jié)論 本文針對(duì)多變量離散時(shí)間隨機(jī)系統(tǒng)提出了一種多模型自適應(yīng)直接算法.該算法首先利用矩陣的偽交換性和擬Diophantine 方程性質(zhì),采用廣義最小方差性能指標(biāo)進(jìn)行設(shè)計(jì);然后將隨機(jī)系統(tǒng)辨識(shí)算法和最優(yōu)控制器設(shè)計(jì)相結(jié)合,直接辨識(shí)出多模型自適應(yīng)控制器的參數(shù);最后通過(guò)加權(quán)多項(xiàng)式的選取,實(shí)現(xiàn)了多變量系統(tǒng)的動(dòng)態(tài)解耦控制.文末給出了全局收斂性分析.仿真結(jié)果證明了該方法的有效性.(a系統(tǒng)輸出y 1(t (

46、aSystem output y 1(t (b系統(tǒng)輸出y 2(t (bSystem output y 2(t 圖1常規(guī)自適應(yīng)控制器仿真結(jié)果Fig.1The simulation of the conventionaladaptive controller(a系統(tǒng)輸出y 1(t (aSystem output y 1(t (b系統(tǒng)輸出y 2(t (bSystem output y 2(t 圖2102個(gè)模型的多模型自適應(yīng)控制器仿真結(jié)果Fig.2The simulation of the multiple models adaptivecontroller with 102modelsReferen

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