随机系统的多模型直接自适应解耦控制器

1、第36卷第9期自动化学报Vol.36,No.9 2010年9月ACTA AUTOMATICA SINICA September,2010随机系统的多模型直接自适应解耦控制器郑益慧1王昕1李少远2姜建国3摘要针对多变量离散时间随机系统,提出了一种采用广义最小方差性能指标的多模型直接自适应解耦控制器.该多模型控制器由多个固定控制器和两个自适应控制器构成.固定控制器用以覆盖系统参数的可能变化范围,自适应控制器用以保证系统的稳定性和提高暂态性能.该多模型控制器利用矩阵的伪交换性和拟Diophantine方程性质,基于广义最小方差性能指标,将随机系统辨识算法和最优控制器设计相结合,直接辨识出控制器的参数

2、,通过广义最小方差性能指标中加权多项式的选取,不但实现了多变量系统的动态解耦控制,而且消除了稳态误差、配置了闭环极点.文末给出了全局收敛性分析.仿真结果表明该方法明显优于常规自适应控制器.关键词多模型,随机系统,直接自适应控制,动态解耦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多模型控制器的设计可以追溯到70年代.从1971年Lainiotis提出的基于后验概率加权的多模型控制器13,到目前基于切换指标的多模型自适应控制器46,已经经历了近40年的发展.1984年, Badr等提

7、出了加权多模型控制器,每个控制器采用二次型最优指标进行设计,控制器输出采用加权和形式7.该方法本质上相当于一种软切换8,因而切换过程比较平滑,对系统的执行机构损害小,易于收稿日期2009-02-23录用日期2010-04-16Manuscript received February23,2009;accepted April16,2010国家高技术研究发展计划(863计划(2008AA04Z129,国家自然科学基金(60504010,60864004,60774015,上海市教委支出预算项目(2008093,上海市教委科研创新项目(09YZ241资助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.上海交通大学电工与电子技术中心上海2002402.上海交通大学自动化系上海2002403.上海交通大学电气系上海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在实际工业过程中使用9,但也正因为采用控制器输出的加权和形式,因而难以得到稳定性、收敛性等理论证明.1986年,Fu等针对状态空间模型,构造多个不同的状态反馈建立多模型集,按编号顺序从小到大进行控制器切换,成功地保证了闭环系统是指数稳定的10.但由于采用了顺序切换方式,导致系统的过渡过程很差11,难以在实际中得到应用. 1994年,Narendra等为了提高系统的暂态性能,采用多个初值不同的自适应模型构成多模型集,然后根据切换指标选出最优模型,进而选出与之相对应的控制器实行切换控制12.该控制器在系统首次发生跳变时可以提

11、高系统的暂态性能,但当多个初值不同的自适应模型收敛到同一邻域时,会退化成常规自适应控制器而丧失了多模型的优点.为了解决上述问题,文献13采用多个固定模型构成多模型集覆盖系统参数变化的可能范围,但由于固定模型不具有自适应能力而无法消除系统的稳态误差.文献14在上述多固定模型的基础上加入一个自适应模型解决了上述问题,同时给出了全局收敛性证明.为了进一步加快系统的辨识速度,提高系统的暂态性能,文献15在上述模型集的基础上,再加入一个可重新赋值的自适应模型.该自适应模型的参数值1296自动化学报36卷可被重新赋值为最优模型的参数值以加快系统的辨识速度,同时给出全局收敛性证明.但以上方法都是针对连续时间

12、系统.1998年,Narendra等将上述方法推广到离散时间系统16.文献1719采用直接算法、分层递阶和逐维定位策略优化多模型集的数目,减少计算量,但上式方法都是针对确定性系统,没有考虑到实际过程中噪声的影响,因而难以在工业生产中得到应用.为此,文献20考虑有色噪声的影响,将以上结果推广到随机系统.但该方法仅限于单输入单输出系统,采用辨识系统参数的间接自适应算法,不但加大了计算量,而且容易造成矩阵方程求解的病态问题21.另外,该方法的性能指标采用最小方差形式,没有对控制输入加以限制,往往造成性能指标最优时,控制输入要求过大,在实际工业过程中会造成阀门开度超过最大值而无法实现22.本文针对多变

13、量离散时间随机系统,提出了一种采用广义最小方差性能指标的多模型直接自适应解耦控制器.该多模型控制器由多个固定控制器和两个自适应控制器构成.固定控制器利用矩阵的伪交换性和拟Diophantine方程性质,基于广义最小方差性能指标进行设计,用以覆盖系统参数的可能变化范围.自适应控制器是将随机系统辨识算法和最优控制器设计相结合,直接辨识出控制器参数的直接算法,用以保证系统的稳定性和提高暂态性能.最后通过广义最小方差性能指标中加权多项式的选取,不但实现了多变量系统的动态解耦控制,而且消除了稳态误差、配置了闭环极点.文末给出了全局收敛性分析.仿真结果表明与常规自适应控制器相比,无论模型参数发生跳变,还是

14、动态解耦能力都得到明显改善.1被控系统描述设多输入多输出线性离散时间随机系统用下述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分别为n维输入、输出向量,(t为n维独立同分布白噪声向量,满足: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的缩略语,F t表示非降子代数族,A(t,z1,B(t,z1,C(t,z1是时间t 和单位后移算子z1的矩阵多项式,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非奇异,k为系统的传输时延.系统满足如下假设:假设1.系统为时不变系统或含跳变参数的时变系统,同时假设相邻跳变时间间隔足够长,系统在此期间内参数保持不变;假设2.t变化时,A(t,z1,B(t,z1,C(t, z1构成的参数矩阵在一紧集中变化;假设3.A(t,z1,B(t,z1,C(t,z1的阶次上限n a,n b,n a和时延k已知;假设4.

16、C(t,z1是稳定的,即detC(t,z1 =0,|z|1,t;假设5.系统是最小相位系统,即detB(t,z1 =0,|z|1,t.由假设1可知,A i(t,B j(t和C k(t为常值矩阵(时不变系统或分段常值矩阵(含跳变参数的时变系统.因此,在相邻跳变时间间隔内,系统可写成时不变形式而不失一般性:A(z1y(t+k=B(z1u(t+C(z1(t+k(8下文将针对系统(8进行研究.2多模型自适应控制器针对被控系统模型的各个参数在一定范围内变化的特点,首先确定系统模型参数的变化区域,然后将该变化区域划分为m个小的子区域,在每个子区域中建立一个固定模型,得到m个固定模型.接着对每个固定模型设计

17、最优控制器,得到m个参数不变的固定控制器.在此基础上,添加一个常规自适应控制器保证系统的稳定性,再添加一个可重新赋值的自适应控制器用以提高系统的暂态性能.这样得到的m个固定控制器与2个自适应控制器一起构成多控制器集,基于切换指标选取最优控制器作为当前控制器进行控制.2.1m个固定控制器2.1.1m个系统参数模型定义 1.由矩阵多项式A(z1,B(z1和C(z1的各系数矩阵构成的矩阵称为系统参数模型.(t所有取值构成的集合称为系统参数模型集.将系统参数模型集分为m个模型子集s,并且s满足1s,s非空,s=1,···,m;9期郑益慧等:随机系统的多模型直接自适应解耦控

18、制器12972ms=1s;3s,ss,0r s<,满足 s r s,称s为模型子集s的中心,r s为半径,其存在性由假设2保证.这样建立的m个模型s(s=1,···,m就构成m个固定模型.2.1.2广义最小方差控制器的设计设性能指标为J p=E P(z1y(t+kR(z1y(t+Q(z1u(t 2(9式中,y(t为n维期望输出向量,P(z1,Q(z1, R(z1为加权多项式矩阵,且满足P(z1稳定.针对式(8中的A(z1和式(9中的P(z1,引入¯A(z1,¯P(z1满足矩阵的伪交换性23¯A(z1P(z1=¯P(z1

19、A(z1(10式中,¯A(z1,¯P(z1满足det¯A(z1=detA(z1(11det¯P(z1=detP(z1(12令¯C(z1=¯P(z1C(z1(13对于式(10中的¯A(z1,引入Diophantine方程24,得¯C(z1=¯A(z1F(z1+zk G(z1(14同理,针对式(14中F(z1,G(z1,引入¯F(z1,¯G(z1满足矩阵的伪交换性23,有¯F(z1G(z1=¯G(z1F(z1(15且满足det¯F(z1=detF(z1(16d

20、et¯G(z1=detG(z1(17利用上述矩阵多项式构造C(z1,满足C(z1=¯F(z1¯A(z1+zk¯G(z1(18下面用三个定理给出C(z1的性质.定理1.拟Diophantine方程性质.C(z1P(z1=F(z1A(z1+zkG(z1(19式中F(z1=¯F(z1¯P(z1(20G(z1=¯G(z1¯P(z1(21证明.用P(z1右乘式(18,考虑式(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.伪交换性.¯C(z1F(z1=¯F(z1C(z1(22证明.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.稳定性.C(z1是稳定的,即detC(z1=0,|z|1(

22、23证明.由式(20和式(22可知C(z1F(z1=F(z1C(z1=¯F(z1¯P(z1C(z1考虑式(12,(16及假设条件P(z1、C(z1稳定,可得detC(z1=detP(z1detC(z1=0,|z|1为了求得最优控制律和系统闭环方程,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自动化学报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定义y0(t+k=P(z1y(t+kF(z1(t+k(26代入性能指标(9中,并考虑式(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为使性能指标最小,可得y0(t+kR(z1y(t+Q(z1u(t=0(27此时,性能指标最小值为J pmin =k1i=0f i2(28将式(27代入式(25和式(26中,可得最优控制律为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联立式(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,可得系统的闭环方程为P(z1+Q(z1B1(z1A(z1y(t+kF(z1+Q(z1B1(z1C(z1(t+k=

25、 R(z1y(t(30由假设5知B(z1稳定,因此可以令Q(z1=R1B1(z1(31P(z1+R1A1(z1=T(z1(32R(z1=T(z1(33式中,R1为定常矩阵,T(z1=T0+T1(z1+···+T nt(zn t为稳定的对角形多项式矩阵,其零点为期望的闭环系统极点,满足n p=n a,n tn a,则系统闭环方程为y(t+k=y(t+T1(z1F(z1+R1C(z1(t+k(34因此通过上述加权多项式矩阵的选取,系统不但配置了闭环极点,而且实现了动态解耦.式(29得到的即为固定控制器的最优控制律,但为了和自适应控制器采用相同的结构形式,下面给出该控制

26、器的另一种形式.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与式(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考虑式(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定义广义输出预测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期郑益慧等:

27、随机系统的多模型直接自适应解耦控制器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为了求取最优控制律,令y0(t= y(t(46y(t=T(z1y(t(47则最优控制律为H1(z1y(t+H2(z1u(tH3(z1 y(t1= y(t(482.1.3m个固定控制器定义2.系统

28、参数模型经过式(24,(25,(43(48变换后得到的矩阵H1(z1,H2(z1,H3(z1构成的矩阵称为控制器模型.对应(t的所有(t的集合称为控制器模型集.对应(ts的(t的集合s称为控制器模型子集.对应ss的s称为该模型子集的中心.因此,上述变换得到的m个控制器s,s=1,···,m就构成了多模型集中的m个固定控制器.2.2常规自适应控制器定义数据向量X0(tk和控制器矩阵0为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则方程(42可以写为y0(t=TX0(tk(52由式(52可以定义自适应控制器的辨识方程为 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表示广义输出 y(t+k=T(z1y(t+k的后验预报.采用如下辨识算法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为了求取最优控制律,令 y(t= y(t(61则最优控制律为H1(z1y(t+H2(z1u(tH1(z1 y(t1= y(t(622.3可重新赋值自适应控制器1当可重新赋值自适应控制器是当前选定的最优控制器时,采用式(5

33、8(60进行辨识,以获得控制器参数,最优控制律可由式(62给出.2当可重新赋值自适应控制器不是当前选定的最优控制器时,则将可重新赋值自适应控制器的参数初值赋值为当前选定的最优控制器的参数值.定义3.多模型控制器由m 个参数已知的控制器s ,s =1,···,m 及常规自适应控制器m +1和可重新赋值的自适应控制器m +2构成.2.4最优控制器对于上述m +2个控制器,采用如下切换指标: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 为第s 个模型的广义输出, y (t =T (z 1y (t 为系统期望的广义输出,则 e s (t = y s (t y (t 为第s 个模型的广义输出误差.为了选取最优控制器,首先,类似于式(53自适应控制器的表达形式,将所有m +2个控制器写为y s (t =T s X (t k ,s =1,···,m +2(65式中,s 为第s 个模型的控制器参数矩阵.然后,任一时刻,根据切换指标,选取广义输出误差最小的控制器作为当前控制器实行控制.即,若J j =min(J s ,s =1,···,m +2(66记作j =

35、arg min(J s ,s =1,···,m +2(67则选取j 为当前控制器实现控制.3全局收敛性分析为了分析上述多模型控制器的全局收敛性,现增加如下假设:假设6. 1C (z 112是严格正实的.下面给出全局收敛性的证明.引理1.对于随机系统(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证明.1在随机系统(8中,由假设4和假设5可知B (z 1,C (z 1稳定,又由式(3可知白噪声均方有界,故由文献25可证.2由式(39中y (t +k =T (z 1y (t +k ,及T (z 1稳定可证.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针对式(72及(t的性质式(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及期望广义输出 y(t有界,所以得证.4将式(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减式(81,写成多变量形式,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可证.5由式(60可知,r(N1=r(N2+X(N1T X(N1(86则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构成及式(68(73可证.引理2.对于常规自适应模型,具有如下性质:1limN1NNt=1e m+1(t (t 2=0(882limN1NNt=1e m+1(t (t (t =0(89证明.1对于自适应辨识算法(58(60,有25limN1NNt=1e m+1(t (tr(t1<,a.s.(90由式(74及随机的基本引理25可证.2对于自适应辨识算法(58(60,由式(74及随机的基本

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由文献25中引理D.5.1得证.定理4.在上述假设16下,多模型直接自适应控制算法作用于随机系统(8时,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证明.1对于随机系统,类似于切换指标(63,定义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为被控系统实际的广义输出, y(t=T(z1y(t为被控系统期望的广义输出, e(t= y(t y(t为被控系统的广义输出误差.展开式(99,知J c=1NNt=1e(t (t 2+1NNt=1(t 2+2NNt=1e(t (t · (t (100由切换指标(63知:任何时刻,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考虑上述不等式两边第2项相等,则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将式(71,(94和(95代入,得limN1NNt=1T(z1e(t(t 2=0(106考虑到T(z1是稳定的,可得limN1NNt=1e(t(t =02将式(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,定理得证.4仿真实验多变量系统模型如下所示(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 .参考输入y = 105.图1表示采用常规自适应控制器时系统的响应,图2表示采用多模型自适应控制器(100个固定模型和2个自适应模型时系统的响应,其中常规自适应模型和多模型中的两个自适应模型的初值取值相同,并且距离参数真值很近.为了检测系统的解耦效果,在t =40步时,y 1单独由10变为0;在t =80步时,y 2单独由5变为5;在t =120步时,y 1,y 2同时发生变化.从图1和图2中可以看出,多模型自适应控制器的解耦效果优于常规自适应控制器.同样,为了检测系统在参数发生跳变时的响应情况,在t =160步时,系统参数发生变

45、化;在t =200步时,系统参数和参考输入y 同时发生变化.从图中可以明显看出,特别当系统参数发生变化时,多模型自适应控制器由于采用多个模型覆盖系统参数的变化范围,其效果(图2远远优于常规自适应控制器(图1.5结论 本文针对多变量离散时间随机系统提出了一种多模型自适应直接算法.该算法首先利用矩阵的伪交换性和拟Diophantine 方程性质,采用广义最小方差性能指标进行设计;然后将随机系统辨识算法和最优控制器设计相结合,直接辨识出多模型自适应控制器的参数;最后通过加权多项式的选取,实现了多变量系统的动态解耦控制.文末给出了全局收敛性分析.仿真结果证明了该方法的有效性.(a系统输出y 1(t (

46、aSystem output y 1(t (b系统输出y 2(t (bSystem output y 2(t 图1常规自适应控制器仿真结果Fig.1The simulation of the conventionaladaptive controller(a系统输出y 1(t (aSystem output y 1(t (b系统输出y 2(t (bSystem output y 2(t 图2102个模型的多模型自适应控制器仿真结果Fig.2The simulation of the multiple models adaptivecontroller with 102modelsReferen

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