锥补线性化问题和应用线性矩阵不等式解决控制问题

1、3线性矩阵不等式应用实例采用MATLAB中LM I工具街求解矩阵不等式的问题对了对称矩阵不等式可以采用徘补线性化方法，ZuX *k解AnA22 * A i.】)A 2<jr-l) A 2n (rr- Dn< 0(1)*尸*-1*即在满足A 12小J*A 224 2/f-D4 2 < 0(2)* 一MA ( n 1) n*KX、*、Np/0,Q/VO1.1N(3)的条件下使tra(r(PA/4- 0V)取得最小值锥补线性化的LMI算法为1)首先找到满足和中3个知阵不等式的所有未知矩阵变最的一个可行射(丸.Qs必“ No),令迭代次数Jt = O,2)对印阵变量(A 0.MN)求

2、解如下最小化问题:M inimi?p| traref PM + MtP + 0N+ NQ)|Subject to 2 X 3)令求出的最优解为；?上一，。叶1 A/h 1, Ni).一.3)3证所求出的最优解是否满足1)"；游足,则得解，若果不满足，检立k是否达到规定的迭代 次数,如果达到则系统无解：否则.令k-k+ 1.转到步骤2继续执行程序.以卜对一具体问题所做的仿肉所需要描述的LM I为：S-P 000A + BK、FC*-5000FC*共R-Q0BKFC- A< 0* *-R00* *px0* *，Lp>aa /?> a s> a 扭> a ao

3、 a f. k 是适维矩阵按照上面的谁补线性化方法算法，所描述的LM1如下：A=l 0.25;0 1;B=0.0063;0.0500;C=l 0;D=0;U=0 0;0 0;setlmis()P=lmivar(l,2 1);Q=lmivar(l,2 1);R=lmivar(l,2 1);S=lmivar(l,2 1);M=lmivar(l,2 1);N=lmivar(l,2 1);K=lmivar(2,l 2);F=lmivar(2,2 1);lmiterm( 1 1 1 S,l,l);lmiterm( 1 1 1lmiterm( 1 5 1 0,A);lmiterm( 1 5 1 K,B)；

4、 lmiterm( 1 6 1 F,-1,C);lmiterm( 1 2 2lmiterm( 1 6 2 F,1,C);lmiterm( 1 3 3 R, 1,1);lmiterm( 1 3 3 lmiterm( 1 5 3 lmiterm( 1 6 3 F,1,C);lmiterm( 1 6 3 0,-A);lmiterm( 1 4 4 lmiterm( 1 5 5 lmiterm( 1 6 6 lmiterm( -2 1 1 P,l,l);lmiterm( -2 2 1 0,l);lmiterm( -2 2 2 M, 1,1);lmiterm( -3 1 1lmiterm( -3 2 1

5、0,l);lmiterm( -3 2 2 N, 1,1);lmiterm( -4 1 1lmiterm( -5 1 1 Q,l,l);lmiterm( -6 1 1 R,l,l);lmiterm( -7 1 1 S,l,l);lmiterm( -8 1 1lmiterm( -9 1 1 N,l,l); linisys =getlmis ;tmin,xfeasp =feasp(lmisys)PP =dec2mat(lmisys,xfeasp,P)QQ =dec2mat(lmisys,xfeasp,Q)RR =dec2mat(lmisys,xfeasp,R) SS =dec2mat(lmisys,

6、xfeasp,S) MM =dec2mat(lmisys,xfeasp,M) NN =dec2mat(lmisys,xfeasp,N) KK =dec2mat(lmisys,xfeasp,K)FF =dec2mat(lmisys,xfeasp,F) fori =1:100 n =decnbr(lmisys);c =zeros(n ,1);forj=l:nPj,Qj,Rj,Sj,Mj,Nj,Kj,Fj =defcx(lmisys,j,P,Q,R,S,M,N,K,F); c(j)=trace(PP*Mj+MM*Pj +QQ*Nj+NN*Qj);endoptions = le-4,0,0,0,0；c

7、opt, xopt=mincx(lmisys,c,options)PPP =dec2mat(lmisys,xopt,P);QQQ =dec2mat(lmisys,xopt,Q);RRR =dec2mat(lmisys,xopt,R);SSS =dec2mat(lmisys,xopt,S);MMM =dec2mat(lmisys,xopt,M);NNN =dec2mat(lmisys,xopt,N);KKK =dec2mat(lmisys,xopt,K);FFF =dec2mat(lmisys,xopt,F);Z = SSS-PPP,U,U,U,(A+B*KKK)',-(FFF*C)&#

8、39;U,-SSS,U,U,U,(FFF*C),;U,U,RRR-QQQ,U,-(B*KKK)：-(A-FFF*Cy； U,U,U,-RRR,U,U;A+B*KKK,U,-B*KKK,U,-inv(PPP),U;-FFF*C,FFF*C,-(A -FFF*C),U,U,-inv(QQQ);Y =eig(Z);i2 =0;for il =1 :length(Y),if(Y(il , l)<0),12 =i2 +1 ;endendif(i2 =length(Y), break ;endPP =PPPQQ=QQQRR =RRRSS =SSSMM=MMMNN =NNNKK=KKKFF =FFFe

9、ndif(i=10),disp(' There is no result'); endSolver for linear objective mmiimzation under LMI constrauitsIterations :Best objective value so far149.439545236.063874323.102877414.567845514.567845613.442476new lower bound:0.296420713.442476*new lower bound:1.679189812.669593*new lower bound:3.5

10、15934912.669593*new lower bound:4.2857311011.916238*new lower bound:4.3956051111.916238*new lower bound:5.063925129.631247*new lower bound:5.242941139.178411*new lower bound:5.879022148.769358*new lower bound:6390317158.502939*new lower bound:6.790746168.326950*new lower bound:7.100688178.238591*new

11、 lower bound:7338510188.176522*new lower bound:7.520225198.161872*new lower bound:7.658027208.161872*new lower bound:7.790606218.147783*new lower bound:8.066778228.114291238.106330*new lower bound:8.074037248.104751*new lower bound:8.080386258.104153*new lower bound:8.085205268.103217*new lower boun

12、d:8.089119278.102513*new lower bound:8.091864288.101985*new lower bound:8.093950298.101589*new lower bound:8.095533308.101291*new lower bound:8.096731318.101177*new lower bound:8.097637328.100996*new lower bound:8.098334338.100859*new lower bound:8.098850348.100806*new lower bound:8.099240358.100721

13、*new lower bound:8.099951Result: feasible solution of required accuracybest objective value:8.100721guaranteed relative accuracy: 9.50e-05f-radiiis saniration: 0.000% of R= 1.00e+09copt =8.1007xopt =0.16390.00301.35730.36631.027718.70110.0038-0.01730.08120.0001-0.00070.00616.1002-0.01350.73683.2584-0.17090.0646-0.5553-2.78230.0001-0.0000PP =0.16390.0