💥1 概述
传统的系统状态估计方法只用到连续信号，而离散测量信号所包含的信息没有得到利用.提出一种基于混合信号(包括连续和离散)的系统状态估计方法，既利用了连续信号，也用到离散信号的信息。该方法将离散信号的变化视作系统的离散事件，提取其准确的信息并参与系统状态估计，构成具有混合系统特性的新型状态估计器。还讨论了该估计器的稳定性条件和设计方法。仿真实验证明这种所提出的状态估计方法可以有效地改善系统的状态估计性能。本文模拟 MCV 观察器，使用多个候选信号的中位数的状态估计方法来进行观测信号（包括异常值）。
📚2 运行结果
部分代码：
set(0,'DefaultAxesLinewidth',2,'DefaultLineLineWidth',2); set(0,'defaultAxesFontSize',14); set(0,'defaultAxesFontName','arial'); set(0,'defaultTextFontName','arial'); close all clear A = [0.7 0.5 -0.1;0 0.7 0.1;-0.3 0 0.9]; Bd = [0.1;0.1;0.2]; Bu = [-1.2;-0.8;1.4]; C = [1 2 -1;0 -5 -0.2]; D = [0.01;0.02]; E = [1 1 1]; A0 = A; A1 = A^2; A2 = A^3; %A3 = A^4; %A4 = A^5; B0 = sqrt(3)*[zeros(3,1) Bd]; D0 = sqrt(3)*[D zeros(2,1)]; B1 = sqrt(4)*[zeros(3,1) Bd A*Bd]; D1 = sqrt(4)*[D zeros(2,1) zeros(2,1)]; B2 = sqrt(5)*[zeros(3,1) Bd A*Bd A^2*Bd]; D2 = sqrt(5)*[D zeros(2,1) zeros(2,1) zeros(2,1)]; gammaoptimal = 100000; for i = 1:30 for j = 1:30 for k = 1:30 alpha0 = 0.03*i; alpha1 = 0.03*j; alpha2 = 0.03*k; setlmis([]) [gamma,n,sgamma] = lmivar(1,[1 1]); [P,n,sP] = lmivar(1,[3 1]); [Y0,n,sY0] = lmivar(2,[3 2]); [Y1,n,sY1] = lmivar(2,[3 2]); [Y2,n,sY2] = lmivar(2,[3 2]); S0 = newlmi; lmiterm([-S0 1 1 P],(1-alpha0),1) lmiterm([-S0 3 1 P],1,A0) lmiterm([-S0 3 1 Y0],eye(3),C) lmiterm([-S0 3 2 P],1,B0) lmiterm([-S0 2 2 0],alpha0) lmiterm([-S0 3 2 Y0],eye(3),D0) lmiterm([-S0 3 3 P],1,1) S1 = newlmi; lmiterm([-S1 1 1 P],(1-alpha1),1) lmiterm([-S1 3 1 P],1,A1) lmiterm([-S1 3 1 Y1],1,C) lmiterm([-S1 3 2 P],1,B1) lmiterm([-S1 2 2 0],alpha1) lmiterm([-S1 3 2 Y1],1,D1) lmiterm([-S1 3 3 P],1,1) S2 = newlmi; lmiterm([-S2 1 1 P],(1-alpha2),1) lmiterm([-S2 3 1 P],1,A2) lmiterm([-S2 3 1 Y2],1,C) lmiterm([-S2 3 2 P],1,B2) lmiterm([-S2 2 2 0],alpha2) lmiterm([-S2 3 2 Y2],1,D2) lmiterm([-S2 3 3 P],1,1) Sn = newlmi; lmiterm([-Sn 1 1 P],1,1) lmiterm([-Sn 2 1 0],E) lmiterm([-Sn 2 2 gamma],1,1) LMIs = getlmis; c = [1 zeros(1,24)]; [copt,xopt] = mincx(LMIs,c); if isempty(xopt) else Popt = dec2mat(LMIs,xopt,P); Y0opt = dec2mat(LMIs,xopt,Y0); Y1opt = dec2mat(LMIs,xopt,Y1); Y2opt = dec2mat(LMIs,xopt,Y2); gammaopt2 = dec2mat(LMIs,xopt,gamma); L0p = -Popt^(-1)*Y0opt; L1p = -Popt^(-1)*Y1opt; L2p = -Popt^(-1)*Y2opt; al0=abs(max(eig(A0-L0p*C))) al1=abs(max(eig(A1-L1p*C))) al2=abs(max(eig(A2-L2p*C))) if (gammaopt2 < gammaoptimal)&&(alpha0<1-al0^2)&&(alpha1<1-al1^2)&&(alpha2<1-al2^2) gammaoptimal = gammaopt2; L0 = L0p; L1 = L1p; L2 = L2p; iopti = i; jopti = j; kopti = k; end end dellmi(LMIs,S0); dellmi(LMIs,S1); dellmi(LMIs,S2); dellmi(LMIs,Sn); end end end x(1,1) = 0; x(2,1) = 0; x(3,1) = 0; xa(1,1) = 0; xa(2,1) = 0; xa(3,1) = 0; x0(1,1) = 0; x0(2,1) = 0; x0(3,1) = 0; x1(1,1) = 0; x1(2,1) = 0; x1(3,1) = 0; x2(1,1) = 0; x2(2,1) = 0; x2(3,1) = 0; xtotal(1,1) = 0; xtotal(2,1) = 0; xtotal(3,1) = 0; Y(1,1) = x1(1,1); Y(2,1) = x1(2,1); Ts = 0.01; t = 0:1:600; a = 0:0.01:2*pi; u = sin(a); F = []; e_proposed = 0; for k = 1:600; x(:,k+1) = A * x(:,k) + Bu * u(:,k); y(:,k) = C * x(:,k); xa(:,k+1) = A * xa(:,k) + Bu * u(:,k) + Bd*(rand-0.5)*2; ya(:,k) = C * xa(:,k) + D*(rand-0.5)*2; if k < 4 x0(:,k+1) = (A-L0*C)* xtotal(:,k) + Bu * u(:,k) + L0 * ya(:,k); x1(:,k+1) = (A-L0*C)* xtotal(:,k) + Bu * u(:,k) + L0 * ya(:,k); x2(:,k+1) = (A-L0*C)* xtotal(:,k) + Bu * u(:,k) + L0 * ya(:,k); F(1,1) = norm(x0(:,k+1)); F(2,1) = norm(x1(:,k+1)); F(3,1) = norm(x2(:,k+1)); [I1,I2] = sort(F); if I2(2) == 1 xtotal(:,k+1) = x0(:,k+1); elseif I2(2) == 2 xtotal(:,k+1) = x1(:,k+1); elseif I2(2) == 3 xtotal(:,k+1) = x2(:,k+1); end else if rem(k,400) == 0 ya(1,k) = 100; figure(2) plot(k,100,'gs',... 'MarkerSize',10,... 'MarkerEdgeColor','b',... 'MarkerFaceColor',[0.5,0.5,0.5]); hold on; elseif rem(k,200) == 0 ya(1,k) = -100; figure(2) plot(k,-100,'gs',... 'MarkerSize',10,... 'MarkerEdgeColor','b',... 'MarkerFaceColor',[0.5,0.5,0.5]); hold on; elseif rem(k,100) == 0 ya(1,k) = 0; figure(2) plot(k,0,'gs',... 'MarkerSize',10,... 'MarkerEdgeColor','b',... 'MarkerFaceColor',[0.5,0.5,0.5]); hold on; end if rem(k,60) == 5 ya(2,k) = 80; figure(2) plot(k,80,'gs',... 'MarkerSize',10,... 'MarkerEdgeColor','r',... 'MarkerFaceColor',[0.5,0.5,0.5]); hold on; elseif rem(k,30) == 5 ya(2,k) = -80; figure(2) plot(k,-80,'gs',... 'MarkerSize',10,... 'MarkerEdgeColor','r',... 'MarkerFaceColor',[0.5,0.5,0.5]); hold on; end x0(:,k+1) = (A-L0*C)* xtotal(:,k) + Bu * u(:,k) + L0 * ya(:,k); x1(:,k+1) = (A^2-L1*C)*xtotal(:,k-1) + L1*ya(:,k-1) + A*Bu*u(:,k-1) + Bu*u(:,k); x2(:,k+1) = (A^3-L2*C) * xtotal(:,k-2) + L2*ya(:,k-2) + A^2*Bu*u(:,k-2) + A*Bu*u(:,k-1) + Bu*u(:,k); F(1,1) = [1 1] * ya(:,k) - [1 1] * C * xtotal(:,k-1) + [1 1] * C * Bu * u(:,k-1); F(2,1) = [1 1] * ya(:,k-1) - [1 1] * C * xtotal(:,k-2) + [1 1] * C * Bu * u(:,k-2); F(3,1) = [1 1] * ya(:,k-2) - [1 1] * C * xtotal(:,k-3) + [1 1] * C * Bu * u(:,k-3); [I1,I2] = sort(F); if I2(2) == 1 xtotal(:,k+1) = x0(:,k+1); elseif I2(2) == 2 xtotal(:,k+1) = x1(:,k+1); elseif I2(2) == 3 xtotal(:,k+1) = x2(:,k+1); end % xtotal(:,k+1) = (A-L0*C)* xtotal(:,k) + B * u(:,k) + L0 * ya(:,k); end e_proposed = e_proposed + norm(xtotal(:,k)-xa(:,k)); F2(k) = I2(2); end figure(1),stairs(t,xa(1,:),'k','LineStyle','--','LineWidth',2),hold on,stairs(t,xa(2,:),'k','LineStyle','--','LineWidth',2),hold on,stairs(t,xa(3,:),'k','LineStyle','--','LineWidth',2) figure(1),stairs(t,xtotal(1,:),'r'),hold on,stairs(t,xtotal(2,:),'r'),hold on,stairs(t,xtotal(3,:),'r'),axis([0 600 -30 30]); %figure(2),stairs(t,x2(1,:),'b'), hold on %figure(3),stairs(t,x3(1,:),'k'), hold on %figure(4),stairs(t,xtotal(1,:),'k'), hold on ylabel('x_p,x') xlabel('k') ya(1,601) = 0; ya(2,601) = 0; figure(2) stairs(t,ya(1,:),'k') hold on,axis([0 600 -105 105]); stairs(t,ya(2,:),'k'),axis([0 600 -105 105]); hold on ylabel('y') xlabel('k') figure(3) %z,zQ1 stairs(t,E(1)*(xa(1,:)-xtotal(1,:))+E(2)*(xa(2,:)-xtotal(2,:))+E(3)*(xa(3,:)-xtotal(3,:)),'k','LineWidth',0.5); axis([0 600 -2.5 2.5]) %set(gca,'fontsize',14); xlabel('k') ylabel('z_e') figure(4) %z,zQ1 subplot(3,1,1) stairs(t,(xa(1,:)-xtotal(1,:)),'k','LineWidth',0.5); axis([0 600 -.5 .5]) ylabel('e_{1}') xlabel('k') %set(gca,'fontsize',14); subplot(3,1,2) stairs(t,xa(2,:)-xtotal(2,:),'k','LineWidth',0.5); axis([0 600 -.5 .5]) %set(gca,'fontsize',14); ylabel('e_{2}') xlabel('k') subplot(3,1,3) stairs(t,xa(3,:)-xtotal(3,:),'k','LineWidth',0.5); axis([0 600 -.5 .5]) %set(gca,'fontsize',14); ylabel('e_{3}') xlabel('k') gammaopt = sqrt(gammaoptimal) iopti jopti kopti L0 L1 L2
🎉3 参考文献
部分理论来源于网络，如有侵权请联系删除。
[1]Hiroshi Okajima (2023) State estimation method using median of multiple candidates for observation signals including outliers [Source Code].
[2]莫以为,萧德云.基于混合信号的状态估计方法[J].控制理论与应用,2004(03):327-334.
