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£Ûkest,L,P£Ý = kalman(sys,Qn,Rn,Nn)(5.3)
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0.50£¬´Ó¶øµÃµ½¿ØÖÆÆ÷ʽ(5.9)¡£È¡Ö¸ÁîrΪ·½²¨Ðźţ¬r=1.5square(0.25t)£¬·ÂÕæ½á¹ûÈçͼ5.3ºÍͼ5.4Ëùʾ¡£


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·ÂÕæ³ÌÐò£º 
(1) LQG¿ØÖÆÆ÷Éè¼Æ³ÌÐò£º chap5_1LQG.m

%LQG Controller design

clear all;

close all;



%Plant

Ap=£Û0 1;-1 -10£Ý;

Bpu=£Û0;1£Ý;

Bpw0=Bpu;

Cpy=£Û1 0£Ý;

Dpyu=0;

Dpyw0=1;



%LQI design

sys=ss(Ap,Bpu,Cpy,Dpyu);%in state space without noise

Q=£Û5 0 0;0 5 0;0 0 60£Ý;   %Used in lqi for x and xi

R=0.001;

N=0;

K=lqi(sys,Q,R,N);



Kx=£ÛK(1) K(2)£Ý;

Ki=K(3);

%Kalman filter

Plant=ss(Ap,£ÛBpu Bpw0£Ý,Cpy,£ÛDpyu Dpyw0£Ý);%in state space with noise

Qn=100;Rn=100;

£Ûkalmf,L£Ý=kalman(Plant,Qn,Rn);



%LQG controller

Ac=£ÛAp-Bpu*Kx-L*Cpy+L*Dpyu*Kx -Bpu*Ki+L*Dpyu*Ki;

0 0 0£Ý

Bcy=£ÛL;-1£Ý;

Bcw=£Û0 0;0 0;0 1£Ý;



Cc=-K

Dcy=0;

Dcw=£Û0 0£Ý;



save lqg_file Ac Bcy Bcw Cc Dcy Dcw;

(2)  SimulinkÖ÷³ÌÐò£º chap5_1sim.mdl




(3)  ¿ØÖÆÆ÷³ÌÐò£º chap5_1ctrl.m

function £Ûsys,x0,str,ts£Ý=s_function(t,x,u,flag)

switch flag,

case 0,

£Ûsys,x0,str,ts£Ý=mdlInitializeSizes;

case 1,

sys=mdlDerivatives(t,x,u);

case 3,

sys=mdlOutputs(t,x,u);

case {1,2, 4, 9 }

sys = £Û£Ý;

otherwise

error(£Û'Unhandled flag = ',num2str(flag)£Ý);

end

function £Ûsys,x0,str,ts£Ý=mdlInitializeSizes

sizes = simsizes;

sizes.NumContStates= 3;

sizes.NumDiscStates  = 0;

sizes.NumOutputs     = 1;

sizes.NumInputs      = 1;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 1;

sys=simsizes(sizes);

x0=£Û0 0 0£Ý;

str=£Û£Ý;

ts=£Û0 0£Ý;

function sys=mdlDerivatives(t,x,u)

r=1.5*square(0.25*t);

y=u(1);



load lqg_file;  %Ac,Bcy,Bcw

sys(1:3)=Ac*x+Bcy*y+Bcw(:,2)*r;

function sys=mdlOutputs(t,x,u)

r=1.5*square(0.25*t);

y=u(1);

load lqg_file;  %Cc,Dcy,Dcw



xc=£Ûx(1) x(2) x(3)£Ý';

ut=Cc*xc+Dcy*y+Dcw(:,2)*r;



sys(1)=ut;

(4)  ±»¿Ø¶ÔÏó³ÌÐò£º chap5_1plant.m

function £Ûsys,x0,str,ts£Ý=s_function(t,x,u,flag)

switch flag,

case 0,

£Ûsys,x0,str,ts£Ý=mdlInitializeSizes;

case 1,

sys=mdlDerivatives(t,x,u);

case 3,

sys=mdlOutputs(t,x,u);

case {2, 4, 9 }

sys = £Û£Ý;

otherwise

error(£Û'Unhandled flag = ',num2str(flag)£Ý);

end

function £Ûsys,x0,str,ts£Ý=mdlInitializeSizes

sizes = simsizes;

sizes.NumContStates= 2;

sizes.NumDiscStates  = 0;

sizes.NumOutputs     = 1;

sizes.NumInputs      = 1;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 1;

sys=simsizes(sizes);

x0=£Û0 0£Ý;

str=£Û£Ý;

ts=£Û0 0£Ý;

function sys=mdlDerivatives(t,x,u)

ut=u(1);

w0=0.001*sin(10*t);

Ap=£Û0 1;-1 -10£Ý;

Bpu=£Û0;1£Ý;

Bpw0=Bpu;

sys(1:2)=Ap*x+Bpu*ut+Bpw0*w0;

function sys=mdlOutputs(t,x,u)

ut=u(1);%sizes.DirFeedthrough = 1

Cpy=£Û1 0£Ý;

Dpyu=0;

Dpyw0=1;

w0=0.001*sin(10*t);

v0=0.001*rands(1);

sys=Cpy*x+Dpyu*ut+Dpyw0*w0+v0;

(5)  ×÷ͼ³ÌÐò£º chap5_1plot.m

close all;

r=1.5*square(0.25*t);



figure(1);

plot(t,r,'-.r',t,y,'k','linewidth',2);

xlabel('time(s)');ylabel('x1 response');



figure(2);

plot(t,ut(:,1),'r','linewidth',2);

xlabel('time(s)');ylabel('Control input');

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·ÂÕæ³ÌÐò£º 
(1) LMIÉè¼Æ³ÌÐò£º chap5_2LMI.m

% Generic antiwindup LMI program

clear all;

close all;

% Dimension defination

np=2;   % ±»¿Ø¶ÔÏó¶¯Ì¬½×Êý

nc=3;   % ¿ØÖÆÆ÷¶¯Ì¬½×Êý

nu=1;   % ¿ØÖÆÊäÈë¸öÊý

nw=2;   % Íâ¼ÓÊäÈë¸öÊý

nz=1;   % ÐÔÄÜÖ¸±ê¸öÊý

naw=0; % ²¹³¥Æ÷״̬½×Êý

ncl=np+nc;

nv=nu+nc;

npc=np+nc;

n=np+nc+naw;

m=ncl+nu+nw+nz;

% Plant defination

Ap=£Û0 1;-1 -10£Ý;

Bpu=£Û0;1£Ý;

Cpy=£Û1 0£Ý;

Dpyu=0;

 

Bpw0=Bpu;

Bpr=zeros(2,1);

Bpw=£ÛBpw0 Bpr£Ý;



Dpyw0=1;

Dpyr=0;

Dpyw=£ÛDpyw0 Dpyr£Ý;

% error performance 

Cpz=Cpy;

Dpzu=Dpyu;

Dpzw=Dpyw-£Û0 1£Ý;



% other performance

nx=2;   %״̬άÊý

ny=1;   %Êä³öάÊý

sys=ss(Ap,Bpu,Cpy,Dpyu);



%Used in lqi

Q=£Û5 0 0;0 5 0;0 0 60£Ý;   

R1=0.001;

N=0;

K=lqi(sys,Q,R1,N);

Kx=K(1:nx);

Ki=K(nx+1:nx+ny);



Plant=ss(Ap,£ÛBpu Bpw0£Ý,Cpy,£ÛDpyu Dpyw0£Ý);

Qn=10;Rn=10;Nn=0;

£Ûkalmf,L£Ý=kalman(Plant,Qn,Rn,Nn);



Ac=£ÛAp-Bpu*Kx-L*Cpy+L*Dpyu*Kx -Bpu*Ki+L*Dpyu*Ki;

zeros(ny,nx) zeros(ny,ny)£Ý;

Bcy=£ÛL;-eye(ny)£Ý;

Bcwr=£Ûzeros(nx,1);eye(ny)£Ý;

Bcw=£Ûzeros(3,1) Bcwr£Ý;



Cc=-K;

Dcy=0;

Dcw=£Û0 0£Ý;



deltau=inv(eye(1)-Dcy*Dpyu);

deltay=inv(eye(1)-Dpyu*Dcy);



Acl=£ÛAp+Bpu*deltau*Dcy*Cpy Bpu*deltau*Cc;

Bcy*deltay*Cpy Ac+Bcy*deltay*Dpyu*Cc£Ý;

Cclz=£ÛCpz+Dpzu*deltau*Dcy*Cpy Dpzu*deltau*Cc£Ý;

Cclu=£Ûdeltau*Dcy*Cpy deltau*Cc£Ý;

Bclq=£Û-Bpu*deltau;-Bcy*deltay*Dpyu£Ý;

Bclw=£ÛBpw+Bpu*deltau*(Dcw+Dcy*Dpyw);Bcw+Bcy*deltay*(Dpyw+Dpyu*Dcw)£Ý;



Dclzq=-(Dpzu*deltau);

Dclzw=Dpzw+Dpzu*deltau*(Dcw+Dcy*Dpyw);



Dcluw=deltau*(Dcw+Dcy*Dpyw);

Dcluq=eye(1)-deltau;



Bclv=£Ûzeros(np,nc) Bpu*deltau;

eye(nc) Bcy*deltay*Dpyu£Ý;

Dclzv=£Ûzeros(nz,nc) Dpzu*deltau£Ý;

Dcluv=£Ûzeros(nu,nc) deltau£Ý;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

ga=sdpvar(1);

R11=sdpvar(np);

R12=sdpvar(np,nc);

R22=sdpvar(nc);

R=£ÛR11 R12;R12' R22£Ý;



Openloop=£ÛR11*Ap'+Ap*R11 Bpw R11*Cpz';Bpw' -ga*eye(nw) Dpzw';Cpz*R11 Dpzw -ga*eye(nz)£Ý;

Closedloop=£ÛR*Acl'+Acl*R Bclw R*Cclz';Bclw' -ga*eye(nw) Dclzw';Cclz*R Dclzw -ga*eye(nz)£Ý;

 

F=set(Openloop<0)+set(Closedloop<0)+set(R>0);   %First step, LMI to get R and gama

solvesdp(F,ga);

gama=double(ga);



R=double(R)

Q=R;

U=sdpvar(nu);

Au=sdpvar(naw+nv,naw+nu);

ga=sdpvar(1);



fai=£Û(Acl*R)+(Acl*R)' Bclq*U+Q*Cclu' Bclw Q*Cclz';

(Bclq*U+Q*Cclu')' Dcluq*U-U+(Dcluq*U-U)' Dcluw U*Dclzq';

Bclw' Dcluw' -ga*eye(nw) Dclzw';

(Q*Cclz')' (U*Dclzq')' Dclzw -ga*eye(nz)£Ý;



%  H1=£ÛBclv' zeros(naw,nv)£Ý;  %naw=0

H1=Bclv';

H2=Dcluv';

H3=Dclzv';

H=£ÛH1 H2 zeros(naw+nv,nw) H3£Ý;

%  GU=£Ûzeros(naw,nu+nw+nz);zeros(nu,n) eye(nu) zeros(nu,nw+nz)£Ý; %naw=0

GU=£Ûzeros(nu,n) eye(nu) zeros(nu,nw+nz)£Ý;

 

Mu=0; 

Stro=-2*(1-Mu)*U+(Dcluq*U+£Ûzeros(nu,nc) eye(nu)£Ý*Au)+(Dcluq*U+

£Ûzeros(nu,nc) eye(nu)£Ý*Au)';

Anti=fai+GU'*Au'*H+H'*Au*GU;



Fa=set(Stro<0)+set(Anti<0);    %Second step, LMI to get Au and UU

solvesdp(Fa,ga);

gama=double(ga)

Au=double(Au);

UU=double(U);

Daw=Au*inv(UU)



Daw1=Daw(1:2,:);

Daw2=Daw(3,:);



save anti_file Daw;

(2) SimulinkÖ÷³ÌÐò£º chap5_2sim.mdl





(3) ¿ØÖÆÆ÷³ÌÐò£º chap5_2ctrl.m

% LQG controller with AW

function £Ûsys,x0,str,ts£Ý=s_function(t,x,u,flag)

switch flag,

case 0,

£Ûsys,x0,str,ts£Ý=mdlInitializeSizes;

case 1,

sys=mdlDerivatives(t,x,u);

case 3,

sys=mdlOutputs(t,x,u);

case {1,2, 4, 9 }

sys = £Û£Ý;

otherwise

error(£Û'Unhandled flag = ',num2str(flag)£Ý);

end

function £Ûsys,x0,str,ts£Ý=mdlInitializeSizes

sizes = simsizes;

sizes.NumContStates= 3;

sizes.NumDiscStates  = 0;

sizes.NumOutputs     = 2;

sizes.NumInputs      = 5;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 1;

sys=simsizes(sizes);

x0=£Û0 0 0£Ý;

str=£Û£Ý;

ts=£Û0 0£Ý;

function sys=mdlDerivatives(t,x,u)

w=1.5*square(0.25*t);

y=u(5);

v1=£Ûu(1) u(2) u(3)£Ý';



%From chap5_1LQG.m

Ac=£Û-0.0081 1.0000 0;-203.8428  -74.1936  244.9490;

000£Ý;

Bcy=£Û0.0081;0.5;-1£Ý;



Bcw=£Û00;

00;

01£Ý;

sys(1:3)=Ac*x+Bcy*y+Bcw(:,2)*w+v1;

function sys=mdlOutputs(t,x,u)

w=1.5*square(0.25*t);

v2=u(4);

y=u(5);

%From chap5_1LQG.m

Cc=£Û-202.3428  -64.1936  244.9490£Ý;

Dcy=0;

Dcw=£Û0 0£Ý;



xc=£Ûx(1) x(2) x(3)£Ý';

ut=Cc*xc+Dcy*y+Dcw(:,2)*w+v2;  



sys(1)=w;

sys(2)=ut;

(4) ²¹³¥³ÌÐò£º chap5_2cmp.m

function £Ûsys,x0,str,ts£Ý=s_function(t,x,u,flag)

switch flag,

case 0,

£Ûsys,x0,str,ts£Ý=mdlInitializeSizes;

case 3,

sys=mdlOutputs(t,x,u);

case {2, 4, 9 }

sys = £Û£Ý;

otherwise

error(£Û'Unhandled flag = ',num2str(flag)£Ý);

end

function £Ûsys,x0,str,ts£Ý=mdlInitializeSizes

sizes = simsizes;

sizes.NumContStates= 0;

sizes.NumDiscStates  = 0;

sizes.NumOutputs     = 4;

sizes.NumInputs      = 1;

sizes.DirFeedthrough = 1;

sizes.NumSampleTimes = 0;

sys=simsizes(sizes);

x0=£Û£Ý;

str=£Û£Ý;

ts=£Û£Ý;

function sys=mdlOutputs(t,x,u)

persistent Daw

if t==0

load anti_file;  %Daw

end

%Daw =£Û    0.0038   -0.0395   -0.0118    0.9990£Ý;

q=u(1);

v=Daw*q;

sys(1:4)=v;

(5) ±»¿Ø¶ÔÏó³ÌÐò£º chap5_2plant.m

function £Ûsys,x0,str,ts£Ý=s_function(t,x,u,flag)

switch flag,

case 0,

£Ûsys,x0,str,ts£Ý=mdlInitializeSizes;

case 1,

sys=mdlDerivatives(t,x,u);

case 3,

sys=mdlOutputs(t,x,u);

case {2, 4, 9 }

sys = £Û£Ý;

otherwise

error(£Û'Unhandled flag = ',num2str(flag)£Ý);

end

function £Ûsys,x0,str,ts£Ý=mdlInitializeSizes

sizes = simsizes;

sizes.NumContStates= 2;

sizes.NumDiscStates  = 0;

sizes.NumOutputs     = 1;

sizes.NumInputs      = 1;

sizes.DirFeedthrough = 0;

sizes.NumSampleTimes = 1;

sys=simsizes(sizes);

x0=£Û0 0£Ý;

str=£Û£Ý;

ts=£Û0 0£Ý;

function sys=mdlDerivatives(t,x,u)

ut=u(1);

wn=0;%0.001*randn(1,100);

Ap=£Û0 1;-1 -10£Ý;

Bpu=£Û0;1£Ý;

Cpy=£Û1 0£Ý;

Dpyu=0;

Bpw0=Bpu;

sys(1:2)=Ap*x+Bpu*ut+Bpw0*wn;

function sys=mdlOutputs(t,x,u)

Cpy=£Û1 0£Ý;

Dpyw0=1;

vn=0;%0.0001*randn(1,100);

sys=Cpy*x+Dpyw0*vn;   %th

(6) ×÷ͼ³ÌÐò£º chap5_2plot.m

close all;



figure(1);

plot(t,w,'-.r',t,y1,'k','linewidth',2);

xlabel('time(s)');ylabel('x1 response');



figure(2);

plot(t,ut(:,1),'r','linewidth',2);

xlabel('time(s)');ylabel('Control input');

²Î¿¼ÎÄÏ×


£Û1£ÝZaccarian L, Teel A R. Modern antiª²windup synthesis: control augmentation for actuator saturation£ÛM£Ý. Princeton: Princeton University Press, 2011.

£Û2£ÝGrimm G, Hatfield J, Postlethwaite I, et al. Antiwindup for stable linear systems with input saturation: an LMIª²based synthesis£ÛJ£Ý. IEEE Transactions on Automatic Control , 2003,48(9): 1509ª²1525.

£Û3£ÝAilon A. Simple tracking controllers for autonomous VTOL aircraft with bounded inputs£ÛJ£Ý. IEEE Transactions on Automatic Control, 2010,55(3):737ª²743.

£Û4£ÝWen C Y, Zhou J, Liu Z T, et al. Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance£ÛJ£Ý. IEEE Transactions on Automatic Control, 2011, 56(7): 1672ª²1678.

£Û5£ÝHu T S, Teel A, Zaccarian L. Stability and performance for saturated systems via quadratic and nonª²quadratic lyapunov functions£ÛJ£Ý. IEEE Transactions on Automatic Control, 2006,51(3): 1770ª²1786.