1、陈 晶(1981,男,江苏泰兴人,博士,讲师,研究方向:混沌控制与同步,非线性系统控制。具有输出噪声干扰的线性系统的辨识是系统辨识领域的一个热点,许多国内外学者对此进行了较为深入的研究,并随之出现了许多辨识方法16,但是在实际应用中,许多学者发现,在辨识过程中系统的输入项也不可避免地受到噪声的污染,即实际辨识过程中所用到的输入项实际上是被污染过的输入项,利用受污染的输入项代替实际输入项辨识出的参数非系统真实的参数。针对具有输入输出噪声干扰的系统,文献7提出利用极大似然函数来实现对系统参数的辨识,文献8利用谱密度分解来实现对参数的估计。在控制领域来实现系统参数辨识的目的,就是为了对系统设计更加合
2、理的控制器,从而使系统能跟踪到目标系统,所以理想的辨识算法是设计合理的控制器的基础。针对具有输入输出误差的OE 模型,先将OE 模型转化成FI R 模型,接着通过递阶辨识算法辨识出FI R 模型的参数,再通过矩阵转换法,求出OE 模型的参数,最后针对参数已知的OE 模型,设计出合理的控制器使OE 模型的输出能渐近跟踪到目标系统的输出。1 算法设计考虑OE 模型描述的系统y (t=B (z A (zu 0(t+$y 0(t(1式(1中y (t=y 0(t+$y 0(t,u (t=u 0(t+$u 0(t,y 0(t,u 0(t分别为系统的实际输入和输出序列,是不可测的,$y 0(t,$u 0(t
3、是未知的输出输入噪声干扰,y (t,u (t是系统的可测输出与输入,而实际的系统可用下式表示y 0(t=A (z u 0(t(2令G (z =B (z A (z ,对G (z 采用长除法,可将G (z 转化为G (z =g 0+g 1z-1+g 2z-2+,+g i z -i+,。于是系统式(1变为y (t=(g 0+g 1z+,+g i z-i+,u 0(t+$y 0(t。假设系统式(1是稳定的,则当i y 时,g i y 0,因此上述系统可用一个有限脉冲响应模型近似为y (t=(g 0+g 1z -1+,+g p -1z-p +1u 0(t+$y 0(t,(3G (p,z :=g 0+g
4、1z。这里G (p,z 称为F I R 模型的传递函数,p 为FI R 模型参数数目,或者称为FI R模型阶次,只要阶次p足够大,式(3可以任意精度逼近式(1,式(3逼近式(1的精度取决于p的大小,p的选择又依赖于系统的快速性,一种可行的方法是通过一个准则函数来选择一个合适的p值。定义F I R模型参数向量;和输入数据向量5如下;:=g0,g1,g2,g p-1T I R p;5(t:=u0(t,u0(t-1,u(t-p+1I R p。于是式(3可以写成最小二乘格式的辨识模型y(t=5T(t;+$y0(t(4当5(t是可测的,递推最小二乘算法可以获得FI R 模型参数向量;的无偏估计,算法如下
5、;(t=;(t-1+L(ty(t-5T(t;(t-1;L(t=P(t5(t=P(t-15(t1+5(tP(t-15(tP(t=P(t-1-P(t-15(t5T(tP(t-1 1+5(tP(t-15(t=I-L(t5T(tP(t-1,P(0=p0I(5 5(t=u0(t,u0(t-1,u0(t-p+1T;(t=g0(t,g1(t,gp-1(tT。由于上面算法中假设5T(t是可测的,而在实际中5T(t是不可测的,所以对于(4式可以利用递阶辨识法5(t=;(t;T(t-1;(ty(t(6(t+1=5(t5T(t-15(ty(t(7根据式(6和式(7可以辨识出FI R模型的参数,进一步辨识原系统的参数
6、。2由FI R模型参数确定系统参数根据式(3可得b0+b1z-1+b2z-2+,+b n z-n1+a1z-1+a2z-2+,+a n z-n=g0+g1z-1+g2z-2+,+gp-1z-p+1。即b0+b1z-1+b2z-2+,+b n z-n=(g0+g1z-1+g2z-2+,+gp-1z-p+(1+a1z-1+a2z+,+a n z-n。设p2n+1,比较上式两边的z-i的系数,可建立2n+1个方程z0:b0=g0z-1:b1=g0a1+g1z-2:b2=g0a2+g1a1+g2sz-n:b n=g0a n+g1a n-1+,+gn-1a1+gnz-n-j:0=gj a n+gj+1a
7、 n-1+,+gj+n-1a1+gj+n,j=1,2,n。令a:a1a2a nI R n,b:b0b1b nI R n+1,ga:g0g1gnI R n+1,gb:gn+1gn+2g2nQ1:000,0g000,0g1g00,0g2g1g0,0s s w s sgn-1gn-2gn-3,g0I R(n+1n,Q2:gn gn-1,g1gn+1gn,g2s s w sg2n-1g2n-2,gnI R nn。将上式2n+1个方程的前n+1个方程和后n个方程分别联立起来,可得b=ga+Q1ag b=-Q2a。于是可得(8871735期闾立新,等:具有输入输出误差的OE模型参数辨识及控制3 控制器的设
8、计重新考虑系统式(1y (t=A (zu 0(t+$y 0(t(9式(9中A (z是系统的参数,当系统的参数未知时,很难设计控制器来使系统的输出来跟踪已知目标系统的输出,但是通过前两节的推导可知系统的参数是已知的。假设系统的输出y (t可测且可导,$y 0(t是系统的未知输出噪声干扰,且|$y 0(t|M ,M 是未知的正常数,y r (t是目标系统的输出,且y r (t也是可导的,设计的目标是对系统(9设计合适的控制器,使得li m t y (y (t-y r (t=li m t y e(t=0(10式(9两边同时减去目标输出得e(t=y (t-y r (t=B (zu 0(t+$y 0(t
9、-y r (t(11定理1 对于系统式(11,设计如下控制器和自适应率u 0(t=A (z 2y (t-y r (t+y (t-y r (t+M sgn (e(t(12M #(t=+e(t+(13则系统式(9输出y (t能渐近跟踪到目标y r (t。证明 设计李亚普诺夫函数V(t=12e 2(t+12(M -M 2。两边求导,并利用式(12。V(t=e(te (t+(M -M M #=e(t(-e(t-M sgn (e(t-$y 0(t+(M -M M #=-e 2(t-M +e(t+-e(t$y 0(t+(M -M M #-e 2(t-M +e(t+M +e(t+(M -M M #=-e 2
10、(t+(M -M +e (t+(M -M M #(t+(M -M (M #-e(t。将式(13代入上式得V(t-e 2(t0。由上式e(tI L 2,而e(tI L ,e (tI L ,根据baba lat 引理得e(t=0。 所以系统式(9输出y (t能渐近跟踪到目标。4 结论借助于递阶辩识理论,解决了具有输入输出误差OE 模型参数辩识问题,同时利用辩识出的参数对OE 模型设计控制器,使OE 模型的输出能渐近跟踪到目标输出,理论结果证明了本文多设计的辩识方法和控制器的设计是有效的。参 考 文 献1 D i ng F ,Chen T.B i as co m pensati on based r
11、ecursi ve l east squaresi den ti fi cati on al gorit hm forM ISO s yste m s.I EEE Tran sacti ons on C ir -cu its and Syste m s -II :Exp ress B ri efs ,2006;53(5:3493532 Di ng F ,Ch en T .Param eter esti m ation f or dua-l rate s yste m s w it h f-in item eas u re m en t dat a .Dyn a m ics of con ti
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15、ifica -ti on of nois y i nput -ou t pu t m odels .Auto m atica ,2007;43(2:4644728 Tors t en Um bert o S overi n iK aush i k M ahata .Pers pecti ves on errors i nvariab -les esti m ati on for dyna m i c syste m s .S i gnal Processi ng ,2002;82(4;11391154(下转第8724页A C l ass ofM I S O Sy mm etric Nonli
16、near Syste m s and Self-tuni ng ControllersHOU X iao-qiu,SONG E-na,C H EN Yu-jie1(S chool of E lectron ics and In f or m ati on,Li b rary1,H e il ong ji ang Insti tute of S ci en ce and Technol ogy,H arbin150027,P.R.Ch i naAbstractBased on physica lm ean i n g of t h e practica l ob jecti v e,the pr
17、ob le m s of M I SO Random Generalized H a mm erste i n M ode l(M I SO RGHMbe i n g no t applied to the sy mm e tric non li n ear syste m s w ere analyzed,and a m odified M ISO RGHM w as developed by add i n g a sy m bo lic functi o n w ith contro l i n put into the m ode.l A hyper-quadratic ob ject
18、 functi o n w as put up by addi n g h i g hest or der contr o l input ter m w ith a sy m bolic function i n to the ob-ject functi o n,and an algorith m for the constrained sel-f tun i n g contro ller be i n g suitab l e for the non-m i n i m um phase syste m s w ith open-l o op unstable c haracteriz
19、ation w as estab lished by forcing the control input w ith saturated li m ita-ti o n.The a l g orit h m w ith one contro l po licy can guarantee the si m ulati v e resu lts w ithout steady state dev iation and t h e contr o l i n pu t being converged to a vary i n g reg i o n centered in the zero-po
20、 i n.t Opti m izati o n results of the ob ject f u nction usi n g the projection grad i e nt a l g orit h m for the constrained opti m ization and t h e variab le m etric a l g orith m for t h e un-constra i n ed opti m ization are ind i c ated that any po i n t i n t h e feasi b le reg i o n can be
21、 e m p l o yed as the i n iti a l fea-si b le po i n t for the projection grad ient al g orith m and the feasi b le reg ion consisted of any c l o se l y-constra i n ts co m bina-ti o n is a regular do m a i n.Va li d ation o f the contro l algorith m is de m onstrated by t h e si m u l a ted result
22、s.Key wordsnon li n ear syste m se l-f tuning contro l ob j e ct f u ncti o n op ti m ization(上接第8718页Errors-in-variabl es E sti m ation and Contro l for OE Dyna m ic Syste m sL B L-i x i n g1,C HEN Jing1,2*(W ux iProf essi onalC ollege of Sci en ce and T echnol ogy,W ux i214028,P.R.Ch i n a;School
23、of Co mmun ication and Contro lEng i neeri ng2,Ji angnan Un i v.,W ux i214122,P.R.C hinaAbstractThe para m eter esti m ati o n prob le m o f OE m odels w ith errors-in-variables is stud i e d.Firs,t the OE m ode ls can turn into a fi n ite i m pulse response(FI Rm ode,l The basic i d ea i s to esti
24、m ate the FI R m ode l para m e-ters.Then w ith the FI R m odel para m eters,the para m eters of the OE m ode l can be ca lculated.F i n all y,w ith t h e kno w n para m eters of the OE m ode,l a contr o ller can be presented to track the tar ge.tKey wordsOE m ode l FI R m odel hierarch ica l identification a l g orit h m contro ller
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